Full article: The Jacobi-Sobolev, Laguerre-Sobolev, and Gegenbauer-Sobolev differential equations and their interrelations

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Abstract

Recently, the author determined the higher-order differential operator having the Jacobi-Sobolev polynomials as its eigenfunctions for certain eigenvalues. These polynomials form an orthogonal system with respect to an inner product equipped with the Jacobi measure on the interval $[- 1, 1]$ with parameters $α \in N_{0}, β > - 1$ and two point masses N, S>0 at the right end point of the interval involving functions and their first derivatives. The first purpose of the present paper is to reveal how the Jacobi-Sobolev equation reduces to the differential equation satisfied by the Laguerre-Sobolev polynomials on the positive half line via a confluent limiting process as $β \to \infty$ . Secondly, we explicitly establish the differential equation for the symmetric Gegenbauer-Sobolev polynomials by employing a quadratic transformation of the argument. Each of the three differential operators involved is of order $4 α + 10$ and symmetric with respect to the corresponding Sobolev inner product.

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1. Introduction

Over the past three decades, the theory of orthogonal polynomials with respect to inner products of Sobolev type has witnessed enormous interest and progress, either from a general point of view or with regard to specific classes of polynomials. For various topics and important developments in the field we refer to the profound survey article by F. Marcellán and Y. Xu [Citation1] and, e.g. to [Citation2–5]. In particular, many authors investigated the so-called discrete Sobolev orthogonal polynomials, whose derivatives occur at discrete mass points of the inner product. In this paper, our interest relies on the fact that such polynomial systems often arise as eigenfunctions of a linear differential operator, a feature which proves to be extremely useful for many applications, notably in the harmonic analysis of eigenfunction expansions or in mathematical physics.

In the hierarchy of explicitly determined eigenfunction systems, a prominent role is played by the Jacobi-Sobolev polynomials ${P_{n}^{α, β, N, S} (x)}_{n = 0}^{\infty}$ , with parameters $α, β > - 1, N, S > 0$ . They are orthogonal on $- 1 \leq x \leq 1$ with respect to an inner product defined for suitable functions f, g by (1.1) $(f, g)_{w (α, β, N, S)} = (f, g)_{w (α, β)} + Nf (1) g (1) + S f^{'} (1) g^{'} (1) .$ (1.1) For N = S = 0, it comprises the classical scalar product (1.2) $\begin{aligned} (f, g)_{w (α, β)} = \int_{- 1}^{1} f (x) g (x) w_{α, β} (x) d x, \\ w_{α, β} (x) := \frac{Γ (α + β + 2) (1 - x)^{α} (1 + x)^{β}}{2^{α + β + 1} Γ (α + 1) Γ (β + 1)}, \end{aligned}$ (1.2) associated with the Jacobi polynomials (1.3) $\begin{aligned} P_{n}^{α, β} (x) = \frac{(α + 1)_{n}}{n!} R_{n}^{α, β} (x), \\ R_{n}^{α, β} (x) :=_{2} F_{1} (- n, n + α + β + 1; α + 1; \frac{1 - x}{2}), n \in N_{0} . \end{aligned}$ (1.3) As usual, $(a)_{0} = 1, (a)_{m} = a (a + 1) \dots (a + m - 1), a \in C, m \in N,$ is the Pochhammer symbol. Due to the parameters N, S, it is convenient to split up the Jacobi-Sobolev polynomials into four terms, (1.4) $P_{n}^{α, β, N, S} (x) = P_{n}^{α, β} (x) + N T_{n}^{α, β} (x) + S U_{n}^{α, β} (x) + NS V_{n}^{α, β} (x), - 1 \leq x \leq 1, n \in N_{0},$ (1.4) where $T_{0}^{α, β} (x) = 0, U_{n}^{α, β} (x) = V_{n}^{α, β} (x) = 0$ for $n \leq 1$ , while for higher values of n, (1.5) $\begin{aligned} T_{n}^{α, β} (x) & = - t_{n}^{α, β} (\frac{1 - x}{2}) P_{n - 1}^{α + 2, β} (x), t_{n}^{α, β} = \frac{(α + 2)_{n - 1} (α + β + 2)_{n}}{n! (β + 1)_{n - 1}}, \\ U_{n}^{α, β} (x) & = u_{n, 1}^{α, β} {(\frac{1 - x}{2})}^{2} P_{n - 2}^{α + 4, β} (x) - u_{n, 2}^{α, β} (\frac{1 - x}{2}) P_{n - 1}^{α + 2, β} (x) - u_{n, 3}^{α, β} P_{n}^{α, β} (x) with \\ u_{n, 1}^{α, β} & = \frac{(α + 2)_{n - 1} (n + α + β + 1) (α + β + 2)_{n + 1}}{4 (α + 1)_{3} (n - 1)! (β + 1)_{n - 2}}, \\ u_{n, 2}^{α, β} & = \frac{(α + 2)_{n - 1} (n + α + β + 1) (α + β + 2)_{n + 1}}{4 (α + 1) (α + 3) (n - 2)! (β + 1)_{n - 1}}, \\ u_{n, 3}^{α, β} & = \frac{(α + 2)_{n} (α + β + 2)_{n + 1}}{4 (α + 1)_{3} (n - 2)! (β + 1)_{n - 1}}, \\ V_{n}^{α, β} (x) & = v_{n}^{α, β} {(\frac{1 - x}{2})}^{2} P_{n - 2}^{α + 4, β} (x) with \\ v_{n}^{α, β} & = \frac{(α + 2)_{n - 1} (α + 3)_{n - 1} (α + β + 2)_{n} (α + β + 2)_{n + 1}}{4 (α + 1)_{3} (n - 1)! n! (β + 1)_{n - 2} (β + 1)_{n - 1}} . \end{aligned}$ (1.5) For the representation (Equation1.4(1.4) $P_{n}^{α, β, N, S} (x) = P_{n}^{α, β} (x) + N T_{n}^{α, β} (x) + S U_{n}^{α, β} (x) + NS V_{n}^{α, β} (x), - 1 \leq x \leq 1, n \in N_{0},$ (1.4) )–(Equation1.5(1.5) $\begin{aligned} T_{n}^{α, β} (x) & = - t_{n}^{α, β} (\frac{1 - x}{2}) P_{n - 1}^{α + 2, β} (x), t_{n}^{α, β} = \frac{(α + 2)_{n - 1} (α + β + 2)_{n}}{n! (β + 1)_{n - 1}}, \\ U_{n}^{α, β} (x) & = u_{n, 1}^{α, β} {(\frac{1 - x}{2})}^{2} P_{n - 2}^{α + 4, β} (x) - u_{n, 2}^{α, β} (\frac{1 - x}{2}) P_{n - 1}^{α + 2, β} (x) - u_{n, 3}^{α, β} P_{n}^{α, β} (x) with \\ u_{n, 1}^{α, β} & = \frac{(α + 2)_{n - 1} (n + α + β + 1) (α + β + 2)_{n + 1}}{4 (α + 1)_{3} (n - 1)! (β + 1)_{n - 2}}, \\ u_{n, 2}^{α, β} & = \frac{(α + 2)_{n - 1} (n + α + β + 1) (α + β + 2)_{n + 1}}{4 (α + 1) (α + 3) (n - 2)! (β + 1)_{n - 1}}, \\ u_{n, 3}^{α, β} & = \frac{(α + 2)_{n} (α + β + 2)_{n + 1}}{4 (α + 1)_{3} (n - 2)! (β + 1)_{n - 1}}, \\ V_{n}^{α, β} (x) & = v_{n}^{α, β} {(\frac{1 - x}{2})}^{2} P_{n - 2}^{α + 4, β} (x) with \\ v_{n}^{α, β} & = \frac{(α + 2)_{n - 1} (α + 3)_{n - 1} (α + β + 2)_{n} (α + β + 2)_{n + 1}}{4 (α + 1)_{3} (n - 1)! n! (β + 1)_{n - 2} (β + 1)_{n - 1}} . \end{aligned}$ (1.5) ) of the Jacobi-Sobolev polynomials see [Citation6, Cor.2.1] and the papers cited there. As is well known for a long time, the polynomials satisfy a spectral differential equation of finite order $4 α + 10$ , provided that α is a nonnegative integer, see [Citation7,Citation8]. But only recently, we succeeded to determine this equation explicitly [Citation6, Theorem 2.4].

Theorem 1.1

For $α \in N_{0}, β > - 1, N, S \geq 0$ , the Jacobi-Sobolev polynomials $y_{n} (x) = P_{n}^{α, β, N, S} (x), n \in N_{0},$ are the eigenfunctions of the equation (1.6) $L_{x}^{α, β, N, S} y_{n} (x) = λ_{n}^{α, β, N, S} y_{n} (x), - 1 \leq x \leq 1,$ (1.6) where the differential operator $L_{x}^{α, β, N, S}$ and its eigenvalues $λ_{n}^{α, β, N, S}$ decompose into four parts, as well, namely (1.7) $\begin{aligned} L_{x}^{α, β, N, S} - λ_{n}^{α, β, N, S} = {L_{x}^{α, β} - λ_{n}^{α, β}} + \frac{N {T_{x}^{α, β} - τ_{n}^{α, β}}}{(α + 2)! (β + 1)_{α + 1}} \\ + \frac{S {U_{x}^{α, β} - ϕ_{n}^{α, β}}}{4 (α + 1) (α + 4)! (β + 1)_{α + 1}} + \frac{NS (α + β + 2) {V_{x}^{α, β} - χ_{n}^{α, β}}}{4 (α + 1) (α + 2)! (β + 1)_{α + 1}}, \end{aligned}$ (1.7) where for sufficiently smooth functions $y (x), - 1 \leq x \leq 1$ , (1.8) $\begin{aligned} \begin{aligned} L_{x}^{α, β} y (x) = \frac{1}{(x - 1)^{α} (x + 1)^{β}} D_{x} {(x - 1)^{α + 1} (x + 1)^{β + 1} D_{x} y (x)}, \\ T_{x}^{α, β} y (x) = \frac{x - 1}{(x + 1)^{β}} D_{x}^{α + 2} {(x + 1)^{α + β + 2} D_{x}^{α + 2} [(x - 1)^{α + 1} y (x)]}, \\ U_{x}^{α, β} y (x) = {U_{x, 1}^{α, β} + U_{x, 2}^{α, β} + U_{x, 3}^{α, β}} y (x) with \\ U_{x, 1}^{α, β} y (x) = (α + 2) \frac{(x - 1)^{2}}{(x + 1)^{β}} D_{x}^{α + 4} {(x + 1)^{α + β + 4} D_{x}^{α + 4} [(x - 1)^{α + 2} y (x)]}, \\ U_{x, 2}^{α, β} y (x) = (α + 1) (α + 4) \frac{x - 1}{(x + 1)^{β}} D_{x}^{α + 3} {ψ_{1}^{α, β} (x) D_{x}^{α + 3} [(x - 1)^{α + 1} y (x)]}, \\ U_{x, 3}^{α, β} y (x) = - \frac{2 (α + 1) (α + 3) (α + 4)}{(x + 1)^{β}} D_{x}^{α + 2} {ψ_{2}^{α, β} (x) D_{x}^{α + 2} [(x - 1)^{α} y (x)]}, \\ where ψ_{1}^{α, β} (x) = (x + 1)^{α + β + 3} [2 (α + 2) - β (x - 1)], \\ ψ_{2}^{α, β} (x) = (x + 1)^{α + β + 2} [4 (α + 2) + (α - β + 2) (x - 1)], \end{aligned} \end{aligned}$ (1.8) and (1.9) $\begin{aligned} V_{x}^{α, β} y (x) = \sum_{j = 0}^{α + 2} \frac{(α + 3 - j)_{j} V_{x, j}^{α, β} y (x)}{j! (j + 1)! (j + α + 3) (β + 1)_{j}} with \\ V_{x, j}^{α, β} y (x) = \frac{x - 1}{(x + 1)^{β}} D_{x}^{j + α + 3} {ψ_{3, j}^{α, β} (x) D_{x}^{j + α + 3} [(x - 1)^{α + 1} y (x)]}, 0 \leq j \leq α + 2, \\ where ψ_{3, j}^{α, β} (x) = (x + 1)^{j + α + β + 3} (x - 1)^{j} [x - 1 - \frac{2 (α + 2 - j) (j + α + 3)}{(α + 2) (j + β + 1)}] . \end{aligned}$ (1.9) Throughout, $D_{x}^{i} \equiv (D_{x})^{i}, i \in N$ , denotes an i-fold differentiation with respect to x. Furthermore, the components of the eigenvalues $\begin{aligned} λ_{n}^{α, β, N, S} & = λ_{n}^{α, β} + \frac{N τ_{n}^{α, β}}{(α + 2)! (β + 1)_{α + 1}} \\ + \frac{S ϕ_{n}^{α, β}}{4 (α + 1) (α + 4)! (β + 1)_{α + 1}} + \frac{NS (α + β + 2) χ_{n}^{α, β}}{4 (α + 1) (α + 2)! (β + 1)_{α + 1}} \end{aligned}$ are given, for $n \in N_{0}$ , by (1.10) $\begin{aligned} λ_{n}^{α, β} & = n (n + α + β + 1), τ_{n}^{α, β} = (n)_{α + 2} (n + β)_{α + 2}, \\ ϕ_{n}^{α, β} & = (n - 1)_{α + 3} (n + β)_{α + 3} [(α + 2) (n - 1) (n + α + β + 2) + 2 β], \\ χ_{n}^{α, β} & = \sum_{j = 0}^{min (n - 2, α + 2)} \frac{(α + 3 - j)_{j} (n - 1 - j)_{j + α + 3} (n + β)_{j + α + 3}}{j! (j + 1)! (j + α + 3) (β + 1)_{j}} \\ = \frac{(n - 1)_{α + 3} (n + β)_{α + 3}}{α + 3}_{4} F_{3} (\begin{matrix} 2 - n, - α - 2, α + 3, n + α + β + 3 \\ 2, α + 4, β + 1 \end{matrix}; 1) . \end{aligned}$ (1.10)

Proposition 1.2

[Citation6, Theorem 3.1]

Let $α \in N_{0}, β > - 1, N, S \geq 0$ . The Jacobi-Sobolev differential operator $L_{x}^{α, β, N, S}$ is symmetric with respect to the inner product (Equation1.1(1.1) $(f, g)_{w (α, β, N, S)} = (f, g)_{w (α, β)} + Nf (1) g (1) + S f^{'} (1) g^{'} (1) .$ (1.1) ), i.e. (1.11) $(L_{x}^{α, β, N, S} f, g)_{w (α, β, N, S)} = (f, L_{x}^{α, β, N, S} g)_{w (α, β, N, S)}, f, g \in C^{(4 α + 10)} [- 1, 1] .$ (1.11)

One aim of this paper is to reveal a close relationship between the Jacobi-Sobolev polynomials (Equation1.4(1.4) $P_{n}^{α, β, N, S} (x) = P_{n}^{α, β} (x) + N T_{n}^{α, β} (x) + S U_{n}^{α, β} (x) + NS V_{n}^{α, β} (x), - 1 \leq x \leq 1, n \in N_{0},$ (1.4) )–(Equation1.5(1.5) $\begin{aligned} T_{n}^{α, β} (x) & = - t_{n}^{α, β} (\frac{1 - x}{2}) P_{n - 1}^{α + 2, β} (x), t_{n}^{α, β} = \frac{(α + 2)_{n - 1} (α + β + 2)_{n}}{n! (β + 1)_{n - 1}}, \\ U_{n}^{α, β} (x) & = u_{n, 1}^{α, β} {(\frac{1 - x}{2})}^{2} P_{n - 2}^{α + 4, β} (x) - u_{n, 2}^{α, β} (\frac{1 - x}{2}) P_{n - 1}^{α + 2, β} (x) - u_{n, 3}^{α, β} P_{n}^{α, β} (x) with \\ u_{n, 1}^{α, β} & = \frac{(α + 2)_{n - 1} (n + α + β + 1) (α + β + 2)_{n + 1}}{4 (α + 1)_{3} (n - 1)! (β + 1)_{n - 2}}, \\ u_{n, 2}^{α, β} & = \frac{(α + 2)_{n - 1} (n + α + β + 1) (α + β + 2)_{n + 1}}{4 (α + 1) (α + 3) (n - 2)! (β + 1)_{n - 1}}, \\ u_{n, 3}^{α, β} & = \frac{(α + 2)_{n} (α + β + 2)_{n + 1}}{4 (α + 1)_{3} (n - 2)! (β + 1)_{n - 1}}, \\ V_{n}^{α, β} (x) & = v_{n}^{α, β} {(\frac{1 - x}{2})}^{2} P_{n - 2}^{α + 4, β} (x) with \\ v_{n}^{α, β} & = \frac{(α + 2)_{n - 1} (α + 3)_{n - 1} (α + β + 2)_{n} (α + β + 2)_{n + 1}}{4 (α + 1)_{3} (n - 1)! n! (β + 1)_{n - 2} (β + 1)_{n - 1}} . \end{aligned}$ (1.5) ) and both the Laguerre-Sobolev polynomials ${L_{n}^{α, N, S} (x)}_{n = 0}^{\infty}$ on $0 \leq x < \infty$ , see [Citation9,Citation10], and the Gegenbauer-Sobolev polynomials ${G_{n}^{α, N, S} (x)}_{n = 0}^{\infty}$ on $- 1 \leq x \leq 1$ . These latter polynomials have been introduced in 1989 by H. Bavinck and H.G. Meijer [Citation11] in a form similar to (Equation1.4(1.4) $P_{n}^{α, β, N, S} (x) = P_{n}^{α, β} (x) + N T_{n}^{α, β} (x) + S U_{n}^{α, β} (x) + NS V_{n}^{α, β} (x), - 1 \leq x \leq 1, n \in N_{0},$ (1.4) ), see Proposition 3.1, and may be considered as a prototype within the wider class of symmetric Sobolev-type polynomials [Citation12,Citation13]. Investigations of Fourier expansions into Gegenbauer-Sobolev polynomials and other interesting features of the polynomials can be found, e.g. in [Citation14–17].

For any $α > - 1, N, S \geq 0$ , the two polynomial systems in question are orthogonal with regard to the inner products (1.12) $(f, g)_{w (α, N, S)} = (f, g)_{w (α)} + Nf (0) g (0) + S f^{'} (0) g^{'} (0),$ (1.12) (1.13) $\begin{aligned} (f, g)_{ω (α, N, S)} & = (f, g)_{w (α, α)} \\ + N [f (- 1) g (- 1) + f (1) g (1)] + S [f^{'} (- 1) g^{'} (- 1) + f^{'} (1) g^{'} (1)], \end{aligned}$ (1.13) respectively. On the right-hand side of (Equation1.12(1.12) $(f, g)_{w (α, N, S)} = (f, g)_{w (α)} + Nf (0) g (0) + S f^{'} (0) g^{'} (0),$ (1.12) ), the scalar product $(f, g)_{w (α)} = \int_{0}^{\infty} f (x) g (x) w_{α} (x) d x, w_{α} (x) := \frac{1}{Γ (α + 1)} e^{- x} x^{α},$ is associated with the classical Laguerre polynomials (1.14) $L_{n}^{α} (x) = \frac{(α + 1)_{n}}{n!} R_{n}^{α} (x), R_{n}^{α} (x) :=_{1} F_{1} (- n; α + 1; x), 0 \leq x < \infty, n \in N_{0},$ (1.14) while the scalar product on the right-hand side of (Equation1.13(1.13) $\begin{aligned} (f, g)_{ω (α, N, S)} & = (f, g)_{w (α, α)} \\ + N [f (- 1) g (- 1) + f (1) g (1)] + S [f^{'} (- 1) g^{'} (- 1) + f^{'} (1) g^{'} (1)], \end{aligned}$ (1.13) ) belongs to the particular case $α = β$ of the Jacobi polynomials known as the Gegenbauer (or ultraspherical) polynomials, cf. (Equation1.1(1.1) $(f, g)_{w (α, β, N, S)} = (f, g)_{w (α, β)} + Nf (1) g (1) + S f^{'} (1) g^{'} (1) .$ (1.1) )–(Equation1.3(1.3) $\begin{aligned} P_{n}^{α, β} (x) = \frac{(α + 1)_{n}}{n!} R_{n}^{α, β} (x), \\ R_{n}^{α, β} (x) :=_{2} F_{1} (- n, n + α + β + 1; α + 1; \frac{1 - x}{2}), n \in N_{0} . \end{aligned}$ (1.3) ). Here, the corresponding weight function reduces to $w_{α, α} (x) := \frac{Γ (2 α + 2) (1 - x^{2})^{α}}{2^{2 α + 1} Γ^{2} (α + 1)} = \frac{Γ (α + 3 / 2) (1 - x^{2})^{α}}{Γ (1 / 2) Γ (α + 1)} .$ The three polynomial systems of Jacobi-, Laguerre-, and Gegenbauer-Sobolev type extend the renowned Askey scheme of hypergeometric orthogonal polynomials, see e.g. [Citation18, Section 18]. In this scheme, the Jacobi and Laguerre polynomials are linked to each other via the confluent limit relation [Citation19, 10.12(35)], (1.15) $P_{n}^{α, β} (1 - \frac{2 x}{β}) = \frac{(α + 1)_{n}}{n!}_{2} F_{1} (- n, n + α + β + 1; α + 1; \frac{x}{β}) \to_{β \to \infty}^{} L_{n}^{α} (x) .$ (1.15) Moreover, the (normalized) Jacobi polynomials with second parameter $β = \pm \frac{1}{2}$ are closely related to the (normalized) Gegenbauer polynomials of even and odd degrees, respectively. Indeed, the quadratic transformations of the hypergeometric function [Citation19, 2.11(10)(13)] yield, for any $0 \leq x \leq 1$ , (1.16) $\begin{aligned} R_{n}^{α, - 1 / 2} (2 x^{2} - 1) & =_{2} F_{1} (- n, n + α + \frac{1}{2}; α + 1; 1 - x^{2}) \\ =_{2} F_{1} (- 2 n, 2 n + 2 α + 1; α + 1; \frac{1 - x}{2}) = R_{2 n}^{α, α} (x), \end{aligned}$ (1.16) (1.17) $\begin{aligned} x R_{n}^{α, 1 / 2} (2 x^{2} - 1) & = x_{2} F_{1} (- n, n + α + \frac{3}{2}; α + 1; 1 - x^{2}) \\ =_{2} F_{1} (- 2 n - 1, 2 n + 2 α + 2; α + 1; \frac{1 - x}{2}) = R_{2 n + 1}^{α, α} (x) . \end{aligned}$ (1.17) So it is natural to examine to which extent these relationships carry over to the wider classes of Sobolev type polynomials. But while appropriate generalizations of the formulas (Equation1.15(1.15) $P_{n}^{α, β} (1 - \frac{2 x}{β}) = \frac{(α + 1)_{n}}{n!}_{2} F_{1} (- n, n + α + β + 1; α + 1; \frac{x}{β}) \to_{β \to \infty}^{} L_{n}^{α} (x) .$ (1.15) )–(Equation1.17(1.17) $\begin{aligned} x R_{n}^{α, 1 / 2} (2 x^{2} - 1) & = x_{2} F_{1} (- n, n + α + \frac{3}{2}; α + 1; 1 - x^{2}) \\ =_{2} F_{1} (- 2 n - 1, 2 n + 2 α + 2; α + 1; \frac{1 - x}{2}) = R_{2 n + 1}^{α, α} (x) . \end{aligned}$ (1.17) ) are known already for the subclasses N>0, S = 0, often called Bochner-Krall polynomials [Citation20,Citation21], it requires some more effort to control the impact of all four components of the polynomials (Equation1.4(1.4) $P_{n}^{α, β, N, S} (x) = P_{n}^{α, β} (x) + N T_{n}^{α, β} (x) + S U_{n}^{α, β} (x) + NS V_{n}^{α, β} (x), - 1 \leq x \leq 1, n \in N_{0},$ (1.4) )–(Equation1.5(1.5) $\begin{aligned} T_{n}^{α, β} (x) & = - t_{n}^{α, β} (\frac{1 - x}{2}) P_{n - 1}^{α + 2, β} (x), t_{n}^{α, β} = \frac{(α + 2)_{n - 1} (α + β + 2)_{n}}{n! (β + 1)_{n - 1}}, \\ U_{n}^{α, β} (x) & = u_{n, 1}^{α, β} {(\frac{1 - x}{2})}^{2} P_{n - 2}^{α + 4, β} (x) - u_{n, 2}^{α, β} (\frac{1 - x}{2}) P_{n - 1}^{α + 2, β} (x) - u_{n, 3}^{α, β} P_{n}^{α, β} (x) with \\ u_{n, 1}^{α, β} & = \frac{(α + 2)_{n - 1} (n + α + β + 1) (α + β + 2)_{n + 1}}{4 (α + 1)_{3} (n - 1)! (β + 1)_{n - 2}}, \\ u_{n, 2}^{α, β} & = \frac{(α + 2)_{n - 1} (n + α + β + 1) (α + β + 2)_{n + 1}}{4 (α + 1) (α + 3) (n - 2)! (β + 1)_{n - 1}}, \\ u_{n, 3}^{α, β} & = \frac{(α + 2)_{n} (α + β + 2)_{n + 1}}{4 (α + 1)_{3} (n - 2)! (β + 1)_{n - 1}}, \\ V_{n}^{α, β} (x) & = v_{n}^{α, β} {(\frac{1 - x}{2})}^{2} P_{n - 2}^{α + 4, β} (x) with \\ v_{n}^{α, β} & = \frac{(α + 2)_{n - 1} (α + 3)_{n - 1} (α + β + 2)_{n} (α + β + 2)_{n + 1}}{4 (α + 1)_{3} (n - 1)! n! (β + 1)_{n - 2} (β + 1)_{n - 1}} . \end{aligned}$ (1.5) ) for general $N \geq 0, S > 0$ .

The first goal of this paper is to provide a new confluent limiting process, by means of which the Jacobi-Sobolev Equations (Equation1.6(1.6) $L_{x}^{α, β, N, S} y_{n} (x) = λ_{n}^{α, β, N, S} y_{n} (x), - 1 \leq x \leq 1,$ (1.6) )–(Equation1.10(1.10) $\begin{aligned} λ_{n}^{α, β} & = n (n + α + β + 1), τ_{n}^{α, β} = (n)_{α + 2} (n + β)_{α + 2}, \\ ϕ_{n}^{α, β} & = (n - 1)_{α + 3} (n + β)_{α + 3} [(α + 2) (n - 1) (n + α + β + 2) + 2 β], \\ χ_{n}^{α, β} & = \sum_{j = 0}^{min (n - 2, α + 2)} \frac{(α + 3 - j)_{j} (n - 1 - j)_{j + α + 3} (n + β)_{j + α + 3}}{j! (j + 1)! (j + α + 3) (β + 1)_{j}} \\ = \frac{(n - 1)_{α + 3} (n + β)_{α + 3}}{α + 3}_{4} F_{3} (\begin{matrix} 2 - n, - α - 2, α + 3, n + α + β + 3 \\ 2, α + 4, β + 1 \end{matrix}; 1) . \end{aligned}$ (1.10) ) reduces to the recently established differential Equations (Equation2.4(2.4) $L_{x}^{α, N, S} y_{n} (x) + λ_{n}^{α, N, S} y_{n} (x) = 0, 0 \leq x < \infty,$ (2.4) )–(Equation2.5(2.5) $\begin{aligned} L_{x}^{α, N, S} + λ_{n}^{α, N, S} \\ = {L_{x}^{α} + λ_{n}^{α}} + \frac{N {T_{x}^{α} + τ_{n}^{α}}}{(α + 2)!} + \frac{S {U_{x}^{α} + ϕ_{n}^{α}}}{(α + 1) (α + 4)!} + \frac{NS ({V_{x}^{α} + χ_{n}^{α}}}{(α + 1) (α + 2)!} . \end{aligned}$ (2.5) ) being satisfied by the Laguerre-Sobolev polynomials [Citation10, Theorem 2.1]. This result clearly illustrates how the various parts of the two differential equations fit together nicely.

As our primary result we then establish a new, quite elementary representation of the differential equation for the Gegenbauer-Sobolev polynomials on $- 1 \leq x \leq 1$ . As has been shown by Bavinck [Citation12] and in joint work with J. Koekoek [Citation13], the Gegenbauer-Sobolev polynomials arise as eigenfunctions of a differential operator of finite order $4 α + 10$ , as long as $α \in N_{0}$ . With the freedom of choosing certain values for the lowest eigenvalues, this operator proved to be unique. When written in the form (1.18) $G_{x}^{α, N, S} = L_{x}^{α, α} + \frac{N 2 M_{x}^{α}}{(α + 2) (2 α + 2)!} + \frac{S R_{x}^{α}}{2 (α + 1)_{4} (2 α + 2)!} + \frac{NS 2 (2 α + 3) S_{x}^{α}}{(α + 1)_{2} (2 α + 2)!},$ (1.18) it includes the Gegenbauer differential operator $L_{x}^{α, α} = (x^{2} - 1 D_{x}^{2} + 2 (α + 1) x D_{x}$ , cf. (Equation1.8(1.8) $\begin{aligned} \begin{aligned} L_{x}^{α, β} y (x) = \frac{1}{(x - 1)^{α} (x + 1)^{β}} D_{x} {(x - 1)^{α + 1} (x + 1)^{β + 1} D_{x} y (x)}, \\ T_{x}^{α, β} y (x) = \frac{x - 1}{(x + 1)^{β}} D_{x}^{α + 2} {(x + 1)^{α + β + 2} D_{x}^{α + 2} [(x - 1)^{α + 1} y (x)]}, \\ U_{x}^{α, β} y (x) = {U_{x, 1}^{α, β} + U_{x, 2}^{α, β} + U_{x, 3}^{α, β}} y (x) with \\ U_{x, 1}^{α, β} y (x) = (α + 2) \frac{(x - 1)^{2}}{(x + 1)^{β}} D_{x}^{α + 4} {(x + 1)^{α + β + 4} D_{x}^{α + 4} [(x - 1)^{α + 2} y (x)]}, \\ U_{x, 2}^{α, β} y (x) = (α + 1) (α + 4) \frac{x - 1}{(x + 1)^{β}} D_{x}^{α + 3} {ψ_{1}^{α, β} (x) D_{x}^{α + 3} [(x - 1)^{α + 1} y (x)]}, \\ U_{x, 3}^{α, β} y (x) = - \frac{2 (α + 1) (α + 3) (α + 4)}{(x + 1)^{β}} D_{x}^{α + 2} {ψ_{2}^{α, β} (x) D_{x}^{α + 2} [(x - 1)^{α} y (x)]}, \\ where ψ_{1}^{α, β} (x) = (x + 1)^{α + β + 3} [2 (α + 2) - β (x - 1)], \\ ψ_{2}^{α, β} (x) = (x + 1)^{α + β + 2} [4 (α + 2) + (α - β + 2) (x - 1)], \end{aligned} \end{aligned}$ (1.8) ) for $α = β$ . Moreover, the other three operators are given in [Citation12] as expansions in powers of $D_{x}$ , as well, (1.19) $M_{x}^{α} \equiv \sum_{j = 0}^{2 α + 4} m_{j}^{α} (x) D_{x}^{j}, R_{x}^{α} \equiv \sum_{j = 0}^{2 α + 8} r_{j}^{α} (x) D_{x}^{j}, S_{x}^{α} \equiv \sum_{j = 0}^{4 α + 10} s_{j}^{α} (x) D_{x}^{j},$ (1.19) but, unfortunately, the coefficient functions $m_{j}^{α} (x), r_{j}^{α} (x), s_{j}^{α} (x)$ turned out to be so involved, that only those with highest indices simplified to (1.20) $\begin{aligned} m_{2 α + 4}^{α} (x) & = (x^{2} - 1)^{α + 2}, r_{2 α + 8}^{α} (x) = (α + 2) (x^{2} - 1)^{α + 4}, \\ s_{4 α + 10}^{α} (x) & = \frac{2 (x^{2} - 1)^{2 α + 5}}{(2 α + 6)!} . \end{aligned}$ (1.20) The key of our approach to new representations of the components $M_{x}^{α}, R_{x}^{α}$ and $S_{x}^{α}$ is to observe that the relationship (Equation1.16(1.16) $\begin{aligned} R_{n}^{α, - 1 / 2} (2 x^{2} - 1) & =_{2} F_{1} (- n, n + α + \frac{1}{2}; α + 1; 1 - x^{2}) \\ =_{2} F_{1} (- 2 n, 2 n + 2 α + 1; α + 1; \frac{1 - x}{2}) = R_{2 n}^{α, α} (x), \end{aligned}$ (1.16) ) between the Jacobi polynomials with parameter $β = - \frac{1}{2}$ and the Gegenbauer polynomials of even degree can be generalized to a quadratic transformation resulting in the Gegenbauer-Sobolev polynomials $G_{2 n}^{α, N, S} (x), n \in N_{0}$ . Applying the same transformation to the corresponding Jacobi-Sobolev equation we are then able to determine the Gegenbauer-Sobolev equation first for polynomials of even degree, and then, due to the uniqueness of the differential operator, in general.

The paper is organized as follows. In Section 2, we determine the limit relation between the Jacobi-Sobolev and Laguerre-Sobolev polynomials as $β \to \infty$ , see Proposition 2.2. Considering each of their four components separately, the originally fixed parameter S has first to be replaced by $S (β) = (2 / β)^{2} S, β > 0$ . Applying now the same substitution to the Jacobi-Sobolev differential operator $L_{x}^{α, β, N, S (β)}$ as well as to its eigenvalues $λ_{n}^{α, β, N, S (β)}$ , they both tend, in the limit, to the respective quantities of the Laguerre-Sobolev equation, see Theorem 2.4. Moreover, the symmetry relation (Equation1.11(1.11) $(L_{x}^{α, β, N, S} f, g)_{w (α, β, N, S)} = (f, L_{x}^{α, β, N, S} g)_{w (α, β, N, S)}, f, g \in C^{(4 α + 10)} [- 1, 1] .$ (1.11) ) of the Jacobi-Sobolev differential operator corresponds to the symmetry of the Laguerre-Sobolev operator stated in Proposition 2.5.

In Section 3 we introduce the Gegenbauer-Sobolev polynomials in Proposition 3.1 and show how the polynomials of even degree are related to the Jacobi-Sobolev polynomials with particularly chosen parameters $α, β = - \frac{1}{2}, 2 N, 2^{5} S$ , see Proposition 3.2. Furthermore, such a relationship is proved to hold between the eigenvalues of the particular Jacobi-Sobolev operator and those of the operator $G_{x}^{α, N, S}$ , suitably denoted by (1.21) $γ_{n}^{α, N, S} = λ_{n}^{α, α} + \frac{N 2 μ_{n}^{α}}{(α + 2) (2 α + 2)!} + \frac{S ρ_{n}^{α}}{2 (α + 1)_{4} (2 α + 2)!} + \frac{NS 2 (2 α + 3) σ_{n}^{α}}{(α + 1)_{2} (2 α + 2)!},$ (1.21) see Proposition 3.4. Finally, the new representation of $G_{x}^{α, N, S}$ is established in Theorem 3.5.

2. Relationship between the Jacobi-Sobolev and Laguerre-Sobolev equations

In [Citation10, (1.2)-(1.3)], we defined the Laguerre-Sobolev polynomials ${L_{n}^{α, N, S} (x)}_{n = 0}^{\infty}$ as follows.

Proposition 2.1

Let $α > - 1, N, S \geq 0$ . In terms of the Laguerre polynomials (Equation1.14(1.14) $L_{n}^{α} (x) = \frac{(α + 1)_{n}}{n!} R_{n}^{α} (x), R_{n}^{α} (x) :=_{1} F_{1} (- n; α + 1; x), 0 \leq x < \infty, n \in N_{0},$ (1.14) ), the Laguerre-Sobolev polynomials are given by (2.1) $L_{n}^{α, N, S} (x) = L_{n}^{α} (x) + N T_{n}^{α} (x) + S U_{n}^{α} (x) + NS V_{n}^{α} (x), 0 \leq x < \infty, n \in N_{0},$ (2.1) where, $T_{0}^{α} (x) = 0, U_{n}^{α} (x) = V_{n}^{α} (x) = 0$ for $n \leq 1$ , and for higher values of n, (2.2) $\begin{aligned} T_{n}^{α} (x) & = - \frac{(α + 2)_{n - 1}}{n!} x L_{n - 1}^{α + 2} (x), \\ U_{n}^{α} (x) & = U_{n, 1}^{α} (x) + U_{n, 2}^{α} (x) + U_{n, 3}^{α} (x) with U_{n, 1}^{α} (x) = \frac{(α + 3)_{n - 2} x^{2} L_{n - 2}^{α + 4} (x)}{(α + 1) (α + 3) (n - 1)!}, \\ U_{n, 2}^{α} (x) & = - \frac{(α + 2)_{n - 1} x L_{n - 1}^{α + 2} (x)}{(α + 1) (α + 3) (n - 2)!}, U_{n, 3}^{α} (x) = - \frac{(α + 3)_{n - 1} L_{n}^{α} (x)}{(α + 1) (α + 3) (n - 2)!}, \\ V_{n}^{α} (x) & = \frac{(α + 3)_{n - 2} (α + 4)_{n - 2}}{(α + 1) (n - 1)! n!} x^{2} L_{n - 2}^{α + 4} (x) . \end{aligned}$ (2.2)

The polynomials (Equation2.1(2.1) $L_{n}^{α, N, S} (x) = L_{n}^{α} (x) + N T_{n}^{α} (x) + S U_{n}^{α} (x) + NS V_{n}^{α} (x), 0 \leq x < \infty, n \in N_{0},$ (2.1) )–(Equation2.2(2.2) $\begin{aligned} T_{n}^{α} (x) & = - \frac{(α + 2)_{n - 1}}{n!} x L_{n - 1}^{α + 2} (x), \\ U_{n}^{α} (x) & = U_{n, 1}^{α} (x) + U_{n, 2}^{α} (x) + U_{n, 3}^{α} (x) with U_{n, 1}^{α} (x) = \frac{(α + 3)_{n - 2} x^{2} L_{n - 2}^{α + 4} (x)}{(α + 1) (α + 3) (n - 1)!}, \\ U_{n, 2}^{α} (x) & = - \frac{(α + 2)_{n - 1} x L_{n - 1}^{α + 2} (x)}{(α + 1) (α + 3) (n - 2)!}, U_{n, 3}^{α} (x) = - \frac{(α + 3)_{n - 1} L_{n}^{α} (x)}{(α + 1) (α + 3) (n - 2)!}, \\ V_{n}^{α} (x) & = \frac{(α + 3)_{n - 2} (α + 4)_{n - 2}}{(α + 1) (n - 1)! n!} x^{2} L_{n - 2}^{α + 4} (x) . \end{aligned}$ (2.2) ) can be linked to the Jacobi-Sobolev polynomials by a generalization of formula (Equation1.15(1.15) $P_{n}^{α, β} (1 - \frac{2 x}{β}) = \frac{(α + 1)_{n}}{n!}_{2} F_{1} (- n, n + α + β + 1; α + 1; \frac{x}{β}) \to_{β \to \infty}^{} L_{n}^{α} (x) .$ (1.15) ), where the parameter S now depends on the Jacobi parameter β.

Proposition 2.2

Let $α > - 1, N, S \geq 0, 0 \leq x < \infty,$ and $S (β) = (2 / β)^{2} S, β > x$ . Then the Jacobi-Sobolev polynomials turn into the Laguerre-Sobolev polynomials for all $n \in N_{0}$ by virtue of the limiting process (2.3) $\begin{aligned} lim_{β \to \infty} P_{n}^{α, β, N, S (β)} (1 - \frac{2 x}{β}) \\ = lim_{β \to \infty} {{P_{n}^{α, β} (ξ) + N T_{n}^{α, β} (ξ) + \frac{4 S}{β^{2}} U_{n}^{α, β} (ξ) + \frac{4 NS}{β^{2}} V_{n}^{α, β} (ξ)} |}_{ξ = 1 - \frac{2 x}{β}} = L_{n}^{α, N, S} (x) . \end{aligned}$ (2.3)

Proof.

While the particular case S = 0 has been stated in [Citation22, (4.8)], we showed in [Citation23, (4.1),(4.5)], that a similar limit relationship holds for the $T_{n} -$ and $U_{n} -$ components, as well. Finally, the $V_{n}$ -components are related to each other by $\begin{aligned} S (β) V_{n}^{α, β} (1 - \frac{2 x}{β}) \\ = \frac{4 S}{β^{2}} \frac{(α + 3)_{n - 2} (α + 4)_{n - 2} (α + β + 2)_{n} (α + β + 2)_{n + 1}}{4 (α + 1) (n - 1)! n! (β + 1)_{n - 2} (β + 1)_{n - 1}} {(\frac{x}{β})}^{2} P_{n - 2}^{α + 4, β} (1 - \frac{2 x}{β}) \\ \to_{β \to \infty}^{} S \frac{(α + 3)_{n - 2} (α + 4)_{n - 2}}{(α + 1) (n - 1)! n!} x^{2} L_{n - 2}^{α + 4} (x) = S V_{n}^{α} (x) \end{aligned}$ for any fixed n.

Next we provide the higher-order Laguerre-Sobolev differential equation in a form most parallel to the Jacobi-Sobolev equation in Theorem 1.1.

Proposition 2.3

[Citation23, Theorem 3.1]

For $α \in N_{0}, N, S \geq 0$ , the Laguerre-Sobolev polynomials $y_{n} (x) = L_{n}^{α, N, S} (x), n \in N_{0},$ are the eigenfunctions of the equation (2.4) $L_{x}^{α, N, S} y_{n} (x) + λ_{n}^{α, N, S} y_{n} (x) = 0, 0 \leq x < \infty,$ (2.4) where the differential operator $L_{x}^{α, N, S}$ and its (negative) eigenvalues $λ_{n}^{α, N, S}$ split up into (2.5) $\begin{aligned} L_{x}^{α, N, S} + λ_{n}^{α, N, S} \\ = {L_{x}^{α} + λ_{n}^{α}} + \frac{N {T_{x}^{α} + τ_{n}^{α}}}{(α + 2)!} + \frac{S {U_{x}^{α} + ϕ_{n}^{α}}}{(α + 1) (α + 4)!} + \frac{NS ({V_{x}^{α} + χ_{n}^{α}}}{(α + 1) (α + 2)!} . \end{aligned}$ (2.5) Here, for sufficiently smooth functions $y (x), 0 \leq x < \infty$ , $\begin{aligned} L_{x}^{α} y (x) = e^{x} x^{- x} D_{x} [e^{- x} x^{α + 1} D_{x} y (x)] = [x D_{x}^{2} + (α + 1 - x) D_{x}] y (x), \\ T_{x}^{α} y (x) = (- 1)^{α + 1} e^{x} x D_{x}^{α + 2} {e^{- x} D_{x}^{α + 2} [x^{α + 1} y (x)]}, \\ U_{x}^{α} y (x) = {U_{x, 1}^{α} + U_{x, 2}^{α} + U_{x, 3}^{α}} y (x) with \\ U_{x, 1}^{α} y (x) = (- 1)^{α + 1} (α + 2) e^{x} x^{2} D_{x}^{α + 4} {e^{- x} D_{x}^{α + 4} [x^{α + 2} y (x)]}, \\ U_{x, 2}^{α} y (x) = (- 1)^{α + 1} (α + 1) (α + 4) e^{x} x D_{x}^{α + 3} {e^{- x} (x + α + 2) D_{x}^{α + 3} [x^{α + 1} y (x)]}, \\ U_{x, 3}^{α} y (x) = (- 1)^{α} (α + 1) (α + 3) (α + 4) e^{x} D_{x}^{α + 2} {e^{- x} (x + 2 α + 4) D_{x}^{α + 2} [x^{α} y (x)]}, \\ V_{x}^{α} y (x) = \sum_{j = 0}^{α + 2} \frac{(α + 3 - j)_{j} (- 1)^{α + j} V_{x, j}^{α} y (x)}{j! (j + 1)! (j + α + 3)} with \\ V_{x, j}^{α} y (x) = e^{x} x D_{x}^{α + 3 + j} {e^{- x} x^{j} [x + \frac{(α + 2 - j) (α + 3 + j)}{α + 2}] D_{x}^{α + 3 + j} [x^{α + 1} y (x)]} . \end{aligned}$ Furthermore, the components of the eigenvalues are given by (2.6) $\begin{aligned} λ_{n}^{α} & = n, τ_{n}^{α} = (n)_{α + 2}, \\ ϕ_{n}^{α} & = (α + 2) (n - 2)_{α + 4} + (α + 4) (n - 1)_{α + 3} \\ = (n - 1)_{α + 3} [(α + 2) (n - 1) + 2], \\ χ_{n}^{α} & = \sum_{j = 0}^{min (n - 2, α + 2)} \frac{(α + 3 - j)_{j} (n - 1 - j)_{j + α + 3}}{j! (j + 1)! (j + α + 3)} \\ = \frac{(n - 1)_{α + 3}}{α + 3}_{3} F_{2} (\begin{matrix} 2 - n, - α - 2, α + 3 \\ 2, α + 4 \end{matrix}; 1) . \end{aligned}$ (2.6)

As suggested by the limit relation (Equation2.3(2.3) $\begin{aligned} lim_{β \to \infty} P_{n}^{α, β, N, S (β)} (1 - \frac{2 x}{β}) \\ = lim_{β \to \infty} {{P_{n}^{α, β} (ξ) + N T_{n}^{α, β} (ξ) + \frac{4 S}{β^{2}} U_{n}^{α, β} (ξ) + \frac{4 NS}{β^{2}} V_{n}^{α, β} (ξ)} |}_{ξ = 1 - \frac{2 x}{β}} = L_{n}^{α, N, S} (x) . \end{aligned}$ (2.3) ), the Jacobi-Sobolev and Laguerre-Sobolev equations are closely related to each other, as well. In fact, after dividing Equation (Equation1.6(1.6) $L_{x}^{α, β, N, S} y_{n} (x) = λ_{n}^{α, β, N, S} y_{n} (x), - 1 \leq x \leq 1,$ (1.6) ) by the parameter β, we obtain our first main result.

Theorem 2.4

Let $α \in N_{0}, N, S \geq 0, 0 \leq x < \infty,$ and $S (β) = (2 / β)^{2} S, β > x$ . For all $n \in N_{0}$ , there holds (2.7) $\begin{aligned} - lim_{β \to \infty} \frac{1}{β} {L_{ξ}^{α, β, N, S (β)} - λ_{n}^{α, β, N, S (β)}} P_{n}^{α, β, N, S (β)} (ξ) |_{ξ = 1 - 2 x / β} \\ = {L_{x}^{α, N, S} + λ_{n}^{α, N, S}} L_{n}^{α, N, S} (x) . \end{aligned}$ (2.7)

Proof.

For the particular case N = 0 see [Citation23, Theorem 4.3,(4.14)]. The relationship between the eigenvalues, i.e. $lim_{β \to \infty} λ_{n}^{α, N, S (β)} / β = λ_{n}^{α, N, S}$ , follows by observing that the components in (Equation1.10(1.10) $\begin{aligned} λ_{n}^{α, β} & = n (n + α + β + 1), τ_{n}^{α, β} = (n)_{α + 2} (n + β)_{α + 2}, \\ ϕ_{n}^{α, β} & = (n - 1)_{α + 3} (n + β)_{α + 3} [(α + 2) (n - 1) (n + α + β + 2) + 2 β], \\ χ_{n}^{α, β} & = \sum_{j = 0}^{min (n - 2, α + 2)} \frac{(α + 3 - j)_{j} (n - 1 - j)_{j + α + 3} (n + β)_{j + α + 3}}{j! (j + 1)! (j + α + 3) (β + 1)_{j}} \\ = \frac{(n - 1)_{α + 3} (n + β)_{α + 3}}{α + 3}_{4} F_{3} (\begin{matrix} 2 - n, - α - 2, α + 3, n + α + β + 3 \\ 2, α + 4, β + 1 \end{matrix}; 1) . \end{aligned}$ (1.10) ) tend to those in (Equation2.6(2.6) $\begin{aligned} λ_{n}^{α} & = n, τ_{n}^{α} = (n)_{α + 2}, \\ ϕ_{n}^{α} & = (α + 2) (n - 2)_{α + 4} + (α + 4) (n - 1)_{α + 3} \\ = (n - 1)_{α + 3} [(α + 2) (n - 1) + 2], \\ χ_{n}^{α} & = \sum_{j = 0}^{min (n - 2, α + 2)} \frac{(α + 3 - j)_{j} (n - 1 - j)_{j + α + 3}}{j! (j + 1)! (j + α + 3)} \\ = \frac{(n - 1)_{α + 3}}{α + 3}_{3} F_{2} (\begin{matrix} 2 - n, - α - 2, α + 3 \\ 2, α + 4 \end{matrix}; 1) . \end{aligned}$ (2.6) ) by means of $\begin{aligned} \frac{1}{β} λ_{n}^{α, β} = \frac{n (n + α + β + 1)}{β} \to_{β \to \infty}^{} n = λ_{n}^{α}, \\ \frac{N}{β} \frac{τ_{n}^{α, β}}{(α + 2)! (β + 1)_{α + 1}} = \frac{N}{β} \frac{(n)_{α + 2} (n + β)_{α + 2}}{(α + 2)! (β + 1)_{α + 1}} \to_{β \to \infty}^{} \frac{N (n)_{α + 2}}{(α + 2)!} = \frac{N τ_{n}^{α}}{(α + 2)!}, \\ \frac{S (β)}{β} \frac{ϕ_{n}^{α, β}}{4 (α + 1) (α + 4)! (β + 1)_{α + 1}} \\ = \frac{S (n - 1)_{α + 3} (n + β)_{α + 3} [(α + 2) (n - 1) (n + α + β + 2) + 2 β]}{β^{3} (α + 1) (α + 4)! (β + 1)_{α + 1}} \\ \to_{β \to \infty}^{} \frac{S (n - 1)_{α + 3} [(α + 2) (n - 1) + 2]}{(α + 1) (α + 4)!} = \frac{S ϕ_{n}^{α}}{(α + 1) (α + 4)!}, \\ \frac{NS (β)}{β} \frac{(α + β + 2) χ_{n}^{α, β}}{4 (α + 1) (α + 2)! (β + 1)_{α + 1}} \\ = \frac{NS (α + β + 2) (n - 1)_{α + 3} (n + β)_{α + 3}}{β^{3} (α + 1) (α + 3)! (β + 1)_{α + 1}}_{4} F_{3} (\begin{matrix} 2 - n, - α - 2, α + 3, n + α + β + 3 \\ 2, α + 4, β + 1 \end{matrix}; 1) \\ \to_{β \to \infty}^{} \frac{NS (n - 1)_{α + 3}}{(α + 1) (α + 3)!}_{3} F_{2} (\begin{matrix} 2 - n, - α - 2, α + 3 \\ 2, α + 4 \end{matrix}; 1) = \frac{NS χ_{n}^{α}}{(α + 1) (α + 2)!} . \end{aligned}$ Similarly, we consider the four components of the expression $L_{ξ}^{α, β, N, S (β)} P_{n}^{α, β, N, S (β)} (ξ)$ , separately. In view of the substitution $ξ = 1 - 2 x / β$ , each differentiation $D_{ξ}$ can formally be replaced by $- (β / 2) D_{x}$ . So, by invoking the limit relation (Equation2.3(2.3) $\begin{aligned} lim_{β \to \infty} P_{n}^{α, β, N, S (β)} (1 - \frac{2 x}{β}) \\ = lim_{β \to \infty} {{P_{n}^{α, β} (ξ) + N T_{n}^{α, β} (ξ) + \frac{4 S}{β^{2}} U_{n}^{α, β} (ξ) + \frac{4 NS}{β^{2}} V_{n}^{α, β} (ξ)} |}_{ξ = 1 - \frac{2 x}{β}} = L_{n}^{α, N, S} (x) . \end{aligned}$ (2.3) ), we achieve $\begin{aligned} - {\frac{1}{β} L_{ξ}^{α, β} P_{n}^{α, β, N, S (β)} (ξ) |}_{ξ = 1 - 2 x / β} \\ = {[\frac{4 x}{β} - {\frac{2 x}{β}}^{2}] \frac{β}{4} D_{x}^{2} + [2 α + 2 - (α + β + 2) \frac{2 x}{β}] \frac{1}{2} D_{x}} P_{n}^{α, β, N, S (β)} (1 - \frac{2 x}{β}) \\ \to_{β \to \infty}^{} {x D_{x}^{2} + (α + 1 - x) D_{x}} L_{n}^{α, N, S} (x) = L_{ξ}^{α} L_{n}^{α, N, S} (x), \\ - {\frac{N}{β} \frac{T_{ξ}^{α, β} P_{n}^{α, β, N, S (β)} (ξ)}{(β + 1)_{α + 1}} |}_{ξ = 1 - 2 x / β} = \frac{N 2 x}{β^{2} (β + 1)_{α + 1} (2 - 2 x / β)^{β}} {(- \frac{β}{2} D_{x})}^{α + 2} \\ \times {{(2 - \frac{2 x}{β})}^{α + β + 2} {(- \frac{β}{2} D_{x})}^{α + 2} [{(- \frac{2 x}{β})}^{α + 1} P_{n}^{α, β, N, S (β)} (1 - \frac{2 x}{β})]} \\ \to_{β \to \infty}^{} N (- 1)^{α + 1} e^{x} x D_{x}^{α + 2} {e^{- x} D_{x}^{α + 2} [x^{α + 1} L_{n}^{α, N, S} (x)]} = N T_{x}^{α} L_{n}^{α, N, S} (x) . \end{aligned}$ Furthermore, the three components of the operator $U_{ξ}^{α, β}$ given in (1.9), yield $\begin{aligned} - {\frac{S (β)}{β} \frac{U_{ξ, 1}^{α, β} P_{n}^{α, β, N, S (β)} (ξ)}{4 (β + 1)_{α + 1}} |}_{ξ = 1 - 2 x / β} = - \frac{S (α + 2)}{β^{3} (β + 1)_{α + 1}} {(- \frac{2 x}{β})}^{2} {(2 - \frac{2 x}{β})}^{- β} \\ \cdot {(\frac{β}{2})}^{2 α + 8} D_{x}^{α + 4} {{(2 - \frac{2 x}{β})}^{α + β + 4} D_{x}^{α + 4} [{(- \frac{2 x}{β})}^{α + 2} P_{n}^{α, β, N, S (β)} (1 - \frac{2 x}{β})]} \\ \to_{β \to \infty}^{} S (- 1)^{α + 1} (α + 2) e^{x} x^{2} D_{x}^{α + 4} {e^{- x} D_{x}^{α + 4} [x^{α + 2} L_{n}^{α, N, S} (x)]} = S U_{x, 1}^{α} L_{n}^{α, N, S} (x), \end{aligned}$ and, analogously, $\begin{aligned} - {\frac{S (β)}{β} \frac{U_{ξ, 2}^{α, β} P_{n}^{α, β, N, S (β)} (ξ)}{4 (β + 1)_{α + 1}} |}_{ξ = 1 - 2 x / β} \to_{β \to \infty}^{} S (- 1)^{α + 1} (α + 1) (α + 4) e^{x} x \\ \times D_{x}^{α + 3} {e^{- x} (x + α + 2) D_{x}^{α + 3} [x^{α + 1} L_{n}^{α, N, S} (x)]} = S U_{x, 2}^{α} L_{n}^{α, N, S} (x), \\ - {\frac{S (β)}{β} \frac{U_{ξ, 3}^{α, β} P_{n}^{α, β, N, S (β)} (ξ)}{4 (β + 1)_{α + 1}} |}_{ξ = 1 - 2 x / β} \to_{β \to \infty}^{} S (- 1)^{α} (α + 1) (α + 3) (α + 4) e^{x} \\ \times D_{x}^{α + 2} {e^{- x} (x + 2 α + 4) D_{x}^{α + 2} [x^{α} L_{n}^{α, N, S} (x)]} = S U_{x, 3}^{α} L_{n}^{α, N, S} (x) . \end{aligned}$ Finally, by definition (Equation1.9(1.9) $\begin{aligned} V_{x}^{α, β} y (x) = \sum_{j = 0}^{α + 2} \frac{(α + 3 - j)_{j} V_{x, j}^{α, β} y (x)}{j! (j + 1)! (j + α + 3) (β + 1)_{j}} with \\ V_{x, j}^{α, β} y (x) = \frac{x - 1}{(x + 1)^{β}} D_{x}^{j + α + 3} {ψ_{3, j}^{α, β} (x) D_{x}^{j + α + 3} [(x - 1)^{α + 1} y (x)]}, 0 \leq j \leq α + 2, \\ where ψ_{3, j}^{α, β} (x) = (x + 1)^{j + α + β + 3} (x - 1)^{j} [x - 1 - \frac{2 (α + 2 - j) (j + α + 3)}{(α + 2) (j + β + 1)}] . \end{aligned}$ (1.9) ), we obtain $\begin{aligned} - {\frac{N S (β)}{β} \frac{(α + β + 2) V_{ξ}^{α, β} P_{n}^{α, β, N, S (β)} (ξ)}{4 (β + 1)_{α + 1}} |}_{ξ = 1 - 2 x / β} \\ = \frac{NS (α + β + 2) β^{α + 4} x}{β^{4} (β + 1)_{α + 1} (1 - x / β)^{β}} \sum_{j = 0}^{α + 2} \frac{(α + 3 - j)_{j} β^{2 j + 1} 2^{- 2 j - α - β - 4}}{j! (j + 1)! (j + α + 3) (β + 1)_{j}} \\ \times D_{x}^{α + 3 + j} {ψ_{3, j}^{α, β} (1 - \frac{2 x}{β}) D_{x}^{α + 3 + j} [(- x)^{α + 1} P_{n}^{α, β, N, S (β)} (1 - \frac{2 x}{β})]} \to_{β \to \infty}^{} \\ NS e^{x} x \sum_{j = 0}^{α + 2} \frac{(α + 3 - j)_{j} (- 1)^{α + j}}{j! (j + 1)! (j + α + 3)} D_{x}^{α + 3 + j} {e^{- x} x^{j} \cdot \\ \cdot [x + \frac{(α + 2 - j) (α + 3 + j)}{α + 2}] D_{x}^{α + 3 + j} [x^{α + 1} L_{n}^{α, N, S} (x)]} = NS V_{x}^{α} L_{n}^{α, N, S} (x), \end{aligned}$ since $\begin{aligned} \frac{β^{2 j + 1}}{(β + 1)_{j} 2^{2 j + α + β + 4}} ψ_{3, j}^{α, β} (1 - \frac{2 x}{β}) \\ = \frac{β^{2 j + 1}}{(β + 1)_{j} 2^{2 j + α + β + 4}} {(2 - \frac{2 x}{β})}^{j + α + β + 3} {(- \frac{2 x}{β})}^{j} [- \frac{2 x}{β} - \frac{2 (α + 2 - j) (α + 3 + j)}{(α + 2) (j + β + 1)}] \\ \to_{β \to \infty}^{} (- 1)^{j + 1} e^{- x} x^{j} [x + \frac{(α + 2 - j) (α + 3 + j)}{α + 2}] . \end{aligned}$ Altogether, this settles the proof of identity (Equation2.7(2.7) $\begin{aligned} - lim_{β \to \infty} \frac{1}{β} {L_{ξ}^{α, β, N, S (β)} - λ_{n}^{α, β, N, S (β)}} P_{n}^{α, β, N, S (β)} (ξ) |_{ξ = 1 - 2 x / β} \\ = {L_{x}^{α, N, S} + λ_{n}^{α, N, S}} L_{n}^{α, N, S} (x) . \end{aligned}$ (2.7) ) in Theorem 2.4.

By applying Proposition 1.2 to the particular Jacobi-Sobolev operator $L_{ξ}^{α, β, N, S (β)}$ , we are able to confirm a similar feature of the Laguerre-Sobolev operator, as well.

Proposition 2.5

[Citation10, Theorem 3.1]

For $α \in N_{0}, N, S \geq 0$ , the Laguerre-Sobolev differential operator $L_{x}^{α, N, S}$ is symmetric with respect to the inner product (Equation1.12(1.12) $(f, g)_{w (α, N, S)} = (f, g)_{w (α)} + Nf (0) g (0) + S f^{'} (0) g^{'} (0),$ (1.12) ), i.e. $(L_{x}^{α, N, S} F, G)_{w (α, N, S)} = (F, L_{x}^{α, N, S} G)_{w (α, N, S)}, F, G \in C^{(4 α + 10)} [0, \infty) .$

3. A new representation of the differential equation for the Gegenbauer-Sobolev polynomials

According to [Citation12, Section 2], the Gegenbauer-Sobolev polynomials may be represented in terms of the classical Gegenbauer polynomials ${P_{n}^{α, α)} (x)}_{n = 0}^{\infty}$ as follows.

Proposition 3.1

Let $α > - 1, N, S \geq 0$ . For $n \in N_{0}, - 1 \leq x \leq 1$ , the orthogonal polynomials with respect to the inner product (Equation1.13(1.13) $\begin{aligned} (f, g)_{ω (α, N, S)} & = (f, g)_{w (α, α)} \\ + N [f (- 1) g (- 1) + f (1) g (1)] + S [f^{'} (- 1) g^{'} (- 1) + f^{'} (1) g^{'} (1)], \end{aligned}$ (1.13) ) are given by (with the usual convention that $P_{n}^{α, α} (x) \equiv 0$ for n<0) $\begin{aligned} G_{n}^{α, N, S} (x) & = P_{n}^{α, α} (x) + N A_{n}^{α} (x) + S B_{n}^{α} (x) + NS C_{n}^{α} (x), where \\ A_{n}^{α} (x) & = - a_{n}^{α} (\frac{1 - x^{2}}{4}) P_{n - 2}^{α + 2, α + 2} (x), a_{n}^{α} = \frac{2 (2 α + 3)_{n}}{n!}, \\ B_{n}^{α} (x) & = b_{n, 1}^{α} {(\frac{1 - x^{2}}{4})}^{2} P_{n - 4}^{α + 4, α + 4} (x) \\ - b_{n, 2}^{α} (\frac{1 - x^{2}}{4}) P_{n - 2}^{α + 2, α + 2} (x) - b_{n, 3}^{α} P_{n}^{α, α} (x) with \\ b_{n, 1}^{α} & = \frac{(n + 2 α + 1) (2 α + 3)_{n + 2}}{2 (α + 1)_{3} (n - 1)!}, b_{n, 2}^{α} = \frac{(α + 2) (n + 2 α + 1) (2 α + 3)_{n + 1}}{2 (α + 1)_{3} (n - 1) (n - 3)!}, \\ b_{n, 3}^{α} & = \frac{(2 α + 3)_{n + 1}}{2 (α + 1)_{3} (n - 3)!}, \\ C_{n}^{α} (x) & = c_{n}^{α} {(\frac{1 - x^{2}}{4})}^{2} P_{n - 4}^{α + 4, α + 4} (x), c_{n}^{α} = \frac{(2 α + 3)_{n + 2} (2 α + 3)_{n}}{(α + 1)_{3} (α + 2) n! (n - 2)!} . \end{aligned}$

The first task of this section is to find a formula linking the Jacobi-Sobolev polynomials (Equation1.4(1.4) $P_{n}^{α, β, N, S} (x) = P_{n}^{α, β} (x) + N T_{n}^{α, β} (x) + S U_{n}^{α, β} (x) + NS V_{n}^{α, β} (x), - 1 \leq x \leq 1, n \in N_{0},$ (1.4) )–(Equation1.5(1.5) $\begin{aligned} T_{n}^{α, β} (x) & = - t_{n}^{α, β} (\frac{1 - x}{2}) P_{n - 1}^{α + 2, β} (x), t_{n}^{α, β} = \frac{(α + 2)_{n - 1} (α + β + 2)_{n}}{n! (β + 1)_{n - 1}}, \\ U_{n}^{α, β} (x) & = u_{n, 1}^{α, β} {(\frac{1 - x}{2})}^{2} P_{n - 2}^{α + 4, β} (x) - u_{n, 2}^{α, β} (\frac{1 - x}{2}) P_{n - 1}^{α + 2, β} (x) - u_{n, 3}^{α, β} P_{n}^{α, β} (x) with \\ u_{n, 1}^{α, β} & = \frac{(α + 2)_{n - 1} (n + α + β + 1) (α + β + 2)_{n + 1}}{4 (α + 1)_{3} (n - 1)! (β + 1)_{n - 2}}, \\ u_{n, 2}^{α, β} & = \frac{(α + 2)_{n - 1} (n + α + β + 1) (α + β + 2)_{n + 1}}{4 (α + 1) (α + 3) (n - 2)! (β + 1)_{n - 1}}, \\ u_{n, 3}^{α, β} & = \frac{(α + 2)_{n} (α + β + 2)_{n + 1}}{4 (α + 1)_{3} (n - 2)! (β + 1)_{n - 1}}, \\ V_{n}^{α, β} (x) & = v_{n}^{α, β} {(\frac{1 - x}{2})}^{2} P_{n - 2}^{α + 4, β} (x) with \\ v_{n}^{α, β} & = \frac{(α + 2)_{n - 1} (α + 3)_{n - 1} (α + β + 2)_{n} (α + β + 2)_{n + 1}}{4 (α + 1)_{3} (n - 1)! n! (β + 1)_{n - 2} (β + 1)_{n - 1}} . \end{aligned}$ (1.5) ) with parameter $β = - \frac{1}{2}$ to the Gegenbauer-Sobolev polynomials of even degree by utilizing the quadratic transformation (Equation1.16(1.16) $\begin{aligned} R_{n}^{α, - 1 / 2} (2 x^{2} - 1) & =_{2} F_{1} (- n, n + α + \frac{1}{2}; α + 1; 1 - x^{2}) \\ =_{2} F_{1} (- 2 n, 2 n + 2 α + 1; α + 1; \frac{1 - x}{2}) = R_{2 n}^{α, α} (x), \end{aligned}$ (1.16) ) between their classical counterparts. To this end, we first normalize both sides as in (Equation1.3(1.3) $\begin{aligned} P_{n}^{α, β} (x) = \frac{(α + 1)_{n}}{n!} R_{n}^{α, β} (x), \\ R_{n}^{α, β} (x) :=_{2} F_{1} (- n, n + α + β + 1; α + 1; \frac{1 - x}{2}), n \in N_{0} . \end{aligned}$ (1.3) ). Observing that $2^{2 n - j} (a)_{n} (a + \frac{1}{2})_{n - j} = (2 a)_{2 n - j}$ for j = 0, 1 and $(1)_{n} = n!$ , we get $\begin{aligned} {\hat{P}}_{n}^{α, - \frac{1}{2}, N, S} (x) & = \frac{n! P_{n}^{α, - \frac{1}{2}, N, S} (x)}{(α + 1)_{n}} \\ = R_{n}^{α, - \frac{1}{2}} (x) + N {\hat{T}}_{n}^{α, - \frac{1}{2}} (x) + S {\hat{U}}_{n}^{α, - \frac{1}{2}} (x) + NS {\hat{V}}_{n}^{α, - \frac{1}{2}} (x), \end{aligned}$ where $\begin{aligned} {\hat{T}}_{n}^{α, - \frac{1}{2}} (x) & = - {\hat{t}}_{n}^{α, - \frac{1}{2}} (\frac{1 - x}{2}) R_{n - 1}^{α + 2, - \frac{1}{2}} (x), \\ {\hat{t}}_{n}^{α, - \frac{1}{2}} & = \frac{n! (α + 3)_{n - 1}}{(α + 1)_{n} (n - 1)!} t_{n}^{α, - \frac{1}{2}} = \frac{(2 α + 3)_{2 n}}{4 (α + 1)_{2} (2 n - 2)!}, \\ {\hat{U}}_{n}^{α, - \frac{1}{2}} (x) & = {\hat{u}}_{n, 1}^{α, - \frac{1}{2}} {(\frac{1 - x}{2})}^{2} R_{n - 2}^{α + 4, - \frac{1}{2}} (x) \\ - {\hat{u}}_{n, 2}^{α, - \frac{1}{2}} (\frac{1 - x}{2}) R_{n - 1}^{α + 2, - \frac{1}{2}} (x) - {\hat{u}}_{n, 3}^{α, - \frac{1}{2}} R_{n}^{α, - \frac{1}{2}} (x), \\ {\hat{u}}_{n, 1}^{α, - \frac{1}{2}} & = \frac{n! (α + 5)_{n - 2}}{(α + 1)_{n} (n - 2)!} u_{n, 1}^{α, - \frac{1}{2}} = \frac{2 n (2 n + α + 1) (2 α + 3)_{2 n + 2}}{2^{10} (α + 1)_{3} (α + 1)_{4} (2 n - 4)!}, \\ {\hat{u}}_{n, 2}^{α, - \frac{1}{2}} & = \frac{n! (α + 3)_{n - 1}}{(α + 1)_{n} (n - 1)!} u_{n, 2}^{α, - \frac{1}{2}} = \frac{2 n (2 n + α + 1) (2 α + 3)_{2 n + 1}}{2^{8} (α + 1) (α + 1)_{3} (2 n - 3)!}, \\ {\hat{u}}_{n, 3}^{α, - \frac{1}{2}} & = u_{n, 3}^{α, - \frac{1}{2}} = \frac{(2 α + 3)_{2 n + 1}}{2^{6} (α + 1)_{3} (2 n - 3)!}, \end{aligned}$ and $\begin{aligned} {\hat{V}}_{n}^{α, - \frac{1}{2}} (x) & = {\hat{v}}_{n}^{α, - \frac{1}{2}} {(\frac{1 - x}{2})}^{2} R_{n - 2}^{α + 4, - \frac{1}{2}} (x), \\ {\hat{v}}_{n}^{α, - \frac{1}{2}} & = \frac{n! (α + 5)_{n - 2}}{(α + 1)_{n} (n - 2)!} v_{n}^{α, - \frac{1}{2}} = \frac{(2 α + 3)_{2 n + 2} (2 α + 3)_{2 n}}{2^{10} (α + 2) (α + 1)_{3} (α + 1)_{4} (2 n - 2)! (2 n - 4)!} . \end{aligned}$ On the other hand, $\begin{aligned} {\hat{G}}_{n}^{α, N, S} (x) = \frac{n! G_{n}^{α, N, S} (x)}{(α + 1)_{n}} = R_{n}^{α, α} (x) + N {\hat{A}}_{n}^{α} (x) + S {\hat{B}}_{n}^{α} (x) + NS {\hat{C}}_{n}^{α} (x), where \\ {\hat{A}}_{n}^{α} (x) = - {\hat{a}}_{n}^{α} (\frac{1 - x^{2}}{4}) R_{n - 2}^{α + 2, α + 2} (x), {\hat{a}}_{n}^{α} = \frac{2 (2 α + 3)_{n}}{(α + 1)_{2} (n - 2)!}, \\ {\hat{B}}_{n}^{α} (x) = {\hat{b}}_{n, 1}^{α} {(\frac{1 - x^{2}}{4})}^{2} R_{n - 4}^{α + 4, α + 4} (x) - {\hat{b}}_{n, 2}^{α} (\frac{1 - x^{2}}{4}) R_{n - 2}^{α + 2, α + 2} (x) - {\hat{b}}_{n, 3}^{α} R_{n}^{α, α} (x) with \\ {\hat{b}}_{n, 1}^{α} = \frac{n (n + 2 α + 1) (2 α + 3)_{n + 2}}{2 (α + 1)_{3} (α + 1)_{4} (n - 4)!}, {\hat{b}}_{n, 2}^{α} = \frac{n (n + 2 α + 1) (2 α + 3)_{n + 1}}{2 (α + 1) (α + 1)_{3} (n - 3)!}, \\ {\hat{b}}_{n, 3}^{α} = \frac{(2 α + 3)_{n + 1}}{2 (α + 1)_{3} (n - 3)!}, \\ {\hat{C}}_{n}^{α} (x) = {\hat{c}}_{n}^{α} {(\frac{1 - x^{2}}{4})}^{2} R_{n - 4}^{α + 4, α + 4} (x), {\hat{c}}_{n}^{α} = \frac{(2 α + 3)_{n + 2} (2 α + 3)_{n}}{(α + 2) (α + 1)_{3} (α + 1)_{4} (n - 2)! (n - 4)!} . \end{aligned}$

Proposition 3.2

Let $α > - 1, N, S \geq 0$ . For $n \in N_{0}, 0 \leq x \leq 1$ , there holds (3.1) $\begin{aligned} {\hat{P}}_{n}^{α, - \frac{1}{2}, 2 N, 2^{5} S} (2 x^{2} - 1) \\ = {{R_{n}^{α, - \frac{1}{2}} (ξ) + 2 N {\hat{T}}_{n}^{α, - \frac{1}{2}} (ξ) + 2^{5} S {\hat{U}}_{n}^{α, - \frac{1}{2}} (ξ) + 2^{6} NS {\hat{V}}_{n}^{α, - \frac{1}{2}} (ξ)} |}_{ξ = 2 x^{2} - 1} \\ = R_{2 n}^{α, α} (x) + N {\hat{A}}_{2 n}^{α} (x) + S {\hat{B}}_{2 n}^{α} (x) + NS {\hat{C}}_{2 n}^{α} (ξ) = {\hat{G}}_{2 n}^{α, N, S} (x) . \end{aligned}$ (3.1)

Proof.

While $R_{n}^{α, - \frac{1}{2}} (2 x^{2} - 1) = R_{2 n}^{α, α} (x)$ in view of (Equation1.16(1.16) $\begin{aligned} R_{n}^{α, - 1 / 2} (2 x^{2} - 1) & =_{2} F_{1} (- n, n + α + \frac{1}{2}; α + 1; 1 - x^{2}) \\ =_{2} F_{1} (- 2 n, 2 n + 2 α + 1; α + 1; \frac{1 - x}{2}) = R_{2 n}^{α, α} (x), \end{aligned}$ (1.16) ), it follows that $\begin{aligned} {\hat{T}}_{n}^{α, - \frac{1}{2}} (2 x^{2} - 1) = - {\hat{t}}_{n}^{α, - \frac{1}{2}} (1 - x^{2}) R_{n - 1}^{α + 2, - \frac{1}{2}} (2 x^{2} - 1) \\ = - \frac{1}{2} {\hat{a}}_{2 n}^{α} (\frac{1 - x^{2}}{4}) R_{2 n - 2}^{α + 2, α + 2} (x) = \frac{1}{2} {\hat{A}}_{2 n}^{α} (x), \\ {\hat{U}}_{n}^{α, - \frac{1}{2}} (2 x^{2} - 1) \\ = 16 {\hat{u}}_{n, 1}^{α, - \frac{1}{2}} {(\frac{1 - x^{2}}{4})}^{2} R_{2 n - 4}^{α + 4, α + 4} (x) - 4 {\hat{u}}_{n, 2}^{α, - \frac{1}{2}} (\frac{1 - x^{2}}{4}) R_{2 n - 2}^{α + 2, α + 2} (x) - {\hat{u}}_{n, 3}^{α, - \frac{1}{2}} R_{2 n}^{α, α} (x) \\ = \frac{1}{2^{5}} {{\hat{b}}_{2 n, 1}^{α} {(\frac{1 - x^{2}}{4})}^{2} R_{2 n - 4}^{α + 4, α + 4} (x) - {\hat{b}}_{2 n, 2}^{α} (\frac{1 - x^{2}}{4}) R_{2 n - 2}^{α + 2, α + 2} (x) - {\hat{b}}_{2 n, 3}^{α} R_{2 n}^{α, α} (x)} \\ = \frac{1}{2^{5}} {\hat{B}}_{2 n}^{α} (x), \\ {\hat{V}}_{n}^{α, - \frac{1}{2}} (2 x^{2} - 1) = \frac{1}{2^{6}} {\hat{c}}_{2 n}^{α} {(\frac{1 - x^{2}}{4})}^{2} R_{2 n - 4}^{α + 4, α + 4} (x) = \frac{1}{2^{6}} {\hat{C}}_{2 n}^{α} (x) . \end{aligned}$ Putting things together then yields identity (Equation3.1(3.1) $\begin{aligned} {\hat{P}}_{n}^{α, - \frac{1}{2}, 2 N, 2^{5} S} (2 x^{2} - 1) \\ = {{R_{n}^{α, - \frac{1}{2}} (ξ) + 2 N {\hat{T}}_{n}^{α, - \frac{1}{2}} (ξ) + 2^{5} S {\hat{U}}_{n}^{α, - \frac{1}{2}} (ξ) + 2^{6} NS {\hat{V}}_{n}^{α, - \frac{1}{2}} (ξ)} |}_{ξ = 2 x^{2} - 1} \\ = R_{2 n}^{α, α} (x) + N {\hat{A}}_{2 n}^{α} (x) + S {\hat{B}}_{2 n}^{α} (x) + NS {\hat{C}}_{2 n}^{α} (ξ) = {\hat{G}}_{2 n}^{α, N, S} (x) . \end{aligned}$ (3.1) ).

As stated in Section 1, the (normalized) Gegenbauer-Sobolev polynomials $y_{n} (x) = {\hat{G}}_{n}^{α, N, S} (x), n \in N_{0}$ , satisfy a unique differential equation (3.2) $G_{x}^{α, N, S} y_{n} (x) = γ_{n}^{α, N, S} y_{n} (x),$ (3.2) where the differential operator $G_{x}^{α, N, S}$ and its eigenvalues $γ_{n}^{α, N, S}$ are given as a linear combination of four terms stated in (Equation1.18(1.18) $G_{x}^{α, N, S} = L_{x}^{α, α} + \frac{N 2 M_{x}^{α}}{(α + 2) (2 α + 2)!} + \frac{S R_{x}^{α}}{2 (α + 1)_{4} (2 α + 2)!} + \frac{NS 2 (2 α + 3) S_{x}^{α}}{(α + 1)_{2} (2 α + 2)!},$ (1.18) ) and (Equation1.21(1.21) $γ_{n}^{α, N, S} = λ_{n}^{α, α} + \frac{N 2 μ_{n}^{α}}{(α + 2) (2 α + 2)!} + \frac{S ρ_{n}^{α}}{2 (α + 1)_{4} (2 α + 2)!} + \frac{NS 2 (2 α + 3) σ_{n}^{α}}{(α + 1)_{2} (2 α + 2)!},$ (1.21) ), respectively.

Proposition 3.3

Let $α \in N_{0}$ and set $μ_{1}^{α} = ρ_{1}^{α} = ρ_{2}^{α} = σ_{1}^{α} = σ_{2}^{α} = σ_{3}^{α} = 0$ . The eigenvalues $γ_{n}^{α, N, S}$ possess the representation (Equation1.21(1.21) $γ_{n}^{α, N, S} = λ_{n}^{α, α} + \frac{N 2 μ_{n}^{α}}{(α + 2) (2 α + 2)!} + \frac{S ρ_{n}^{α}}{2 (α + 1)_{4} (2 α + 2)!} + \frac{NS 2 (2 α + 3) σ_{n}^{α}}{(α + 1)_{2} (2 α + 2)!},$ (1.21) ) with $\begin{aligned} λ_{n}^{α, α} & = n (n + 2 α + 1), n \geq 1, μ_{n}^{α} = (n - 1)_{2 α + 4}, n \geq 2, \\ ρ_{n}^{α} & = (α + 2) (n - 3)_{2 α + 8} + 2 (α + 1) (α + 4) (n - 2)_{2 α + 6} \\ = (n - 2)_{2 α + 6} [(α + 2) (n + 2 α + 3) (n - 2) - 4], n \geq 3, \\ σ_{n}^{α} & = \frac{(n - 3)_{2 α + 8}}{2 (α + 3)_{2}}_{4} F_{3} (\begin{matrix} 2 - \frac{n}{2}, - α - 1, \frac{n}{2} + α + \frac{5}{2}, α + 4 \\ 2, \frac{3}{2}, α + 5 \end{matrix}; 1) \\ = \frac{(n - 2)_{2 α + 6}}{α + 3}_{4} F_{3} (\begin{matrix} 2 - \frac{n}{2}, - α - 2, \frac{n}{2} + α + \frac{5}{2}, α + 3 \\ 2, \frac{1}{2}, α + 4 \end{matrix}; 1), n \geq 4. \end{aligned}$

Proof.

While $λ_{n}^{α, α}, n \in N_{0}$ , are the eigenvalues of the classical Gegenbauer operator $L_{x}^{α, α}$ , the $μ_{n}^{α}, ρ_{n}^{α}$ , and the first line of $σ_{n}^{α}$ coincide with the values stated in [Citation12, Section 5]. In order to verify the second identity of $σ_{n}^{α}$ , we make use of Whipple's transformation for terminating and balanced $_{4} F_{3}$ hypergeometric functions with unit argument [Citation18, 16.4.14], i.e. $_{4} F_{3} (\begin{matrix} - m, a, b, c \\ d, e, f \end{matrix}; 1) = \frac{(e - a)_{m} (f - a)_{m}}{(e)_{m} (f)_{m}}_{4} F_{3} (\begin{matrix} - m, a, d - b, d - c \\ d, a - e - m + 1, a - f - m + 1 \end{matrix}; 1)$ where a + b + c−m + 1 = d + e + f. In fact, just choose $\begin{aligned} m & = \frac{n}{2} - 2, a = \frac{n}{2} + α + \frac{5}{2}, b = - α - 1, c = α + 4, d = 2, \\ e & = \frac{3}{2}, f = α + 5. \end{aligned}$

Fortunately, there exists a relationship between the eigenvalues of the Jacobi-Sobolev and Gegenbauer-Sobolev equations, which corresponds to the identity (Equation3.1(3.1) $\begin{aligned} {\hat{P}}_{n}^{α, - \frac{1}{2}, 2 N, 2^{5} S} (2 x^{2} - 1) \\ = {{R_{n}^{α, - \frac{1}{2}} (ξ) + 2 N {\hat{T}}_{n}^{α, - \frac{1}{2}} (ξ) + 2^{5} S {\hat{U}}_{n}^{α, - \frac{1}{2}} (ξ) + 2^{6} NS {\hat{V}}_{n}^{α, - \frac{1}{2}} (ξ)} |}_{ξ = 2 x^{2} - 1} \\ = R_{2 n}^{α, α} (x) + N {\hat{A}}_{2 n}^{α} (x) + S {\hat{B}}_{2 n}^{α} (x) + NS {\hat{C}}_{2 n}^{α} (ξ) = {\hat{G}}_{2 n}^{α, N, S} (x) . \end{aligned}$ (3.1) ) of their eigenfunctions.

Proposition 3.4

For $α \in N_{0}, N, S \geq 0$ let the eigenvalues $λ_{n}^{α, - \frac{1}{2}, N, S}, n \in N_{0}$ , be given as in (Equation1.7(1.7) $\begin{aligned} L_{x}^{α, β, N, S} - λ_{n}^{α, β, N, S} = {L_{x}^{α, β} - λ_{n}^{α, β}} + \frac{N {T_{x}^{α, β} - τ_{n}^{α, β}}}{(α + 2)! (β + 1)_{α + 1}} \\ + \frac{S {U_{x}^{α, β} - ϕ_{n}^{α, β}}}{4 (α + 1) (α + 4)! (β + 1)_{α + 1}} + \frac{NS (α + β + 2) {V_{x}^{α, β} - χ_{n}^{α, β}}}{4 (α + 1) (α + 2)! (β + 1)_{α + 1}}, \end{aligned}$ (1.7) ) with components (Equation1.10(1.10) $\begin{aligned} λ_{n}^{α, β} & = n (n + α + β + 1), τ_{n}^{α, β} = (n)_{α + 2} (n + β)_{α + 2}, \\ ϕ_{n}^{α, β} & = (n - 1)_{α + 3} (n + β)_{α + 3} [(α + 2) (n - 1) (n + α + β + 2) + 2 β], \\ χ_{n}^{α, β} & = \sum_{j = 0}^{min (n - 2, α + 2)} \frac{(α + 3 - j)_{j} (n - 1 - j)_{j + α + 3} (n + β)_{j + α + 3}}{j! (j + 1)! (j + α + 3) (β + 1)_{j}} \\ = \frac{(n - 1)_{α + 3} (n + β)_{α + 3}}{α + 3}_{4} F_{3} (\begin{matrix} 2 - n, - α - 2, α + 3, n + α + β + 3 \\ 2, α + 4, β + 1 \end{matrix}; 1) . \end{aligned}$ (1.10) ) for $β = - \frac{1}{2}$ . Then $\begin{aligned} 4 λ_{n}^{α, - \frac{1}{2}, 2 N, 2^{5} S} \\ = 4 λ_{n}^{α, - \frac{1}{2}} + \frac{N 8 τ_{n}^{α, - \frac{1}{2}}}{(α + 2)! {(\frac{1}{2})}_{α + 1}} + \frac{S 2^{5} ϕ_{n}^{α, - \frac{1}{2}}}{(α + 1) (α + 4)! {(\frac{1}{2})}_{α + 1}} + \frac{NS 2^{6} (α + \frac{3}{2}) χ_{n}^{α, - \frac{1}{2}}}{(α + 1) (α + 2)! {(\frac{1}{2})}_{α + 1}} \\ = λ_{2 n}^{α, α} + \frac{N 2 μ_{2 n}^{α}}{(α + 2) (2 α + 2)!} + \frac{S ρ_{2 n}^{α}}{2 (α + 1)_{4} (2 α + 2)!} + \frac{NS 2 (2 α + 3) σ_{2 n}^{α}}{(α + 1)_{2} (2 α + 2)!} = γ_{2 n}^{α, N, S} . \end{aligned}$

Proof.

It remains to observe that $\begin{aligned} 4 λ_{n}^{α, - \frac{1}{2}} = 4 n (n + α + \frac{1}{2}) = 2 n (2 n + 2 α + 1) = λ_{2 n}^{α, α}, \\ \frac{8 τ_{n}^{α, - \frac{1}{2}}}{(α + 2)! {(\frac{1}{2})}_{α + 1}} = \frac{8 (n)_{α + 2} (n - \frac{1}{2})_{α + 2}}{(α + 2) (1)_{α + 1} {(\frac{1}{2})}_{α + 1}} = \frac{2 (2 n - 1)_{2 α + 4}}{(α + 2) (2 α + 2)!} = \frac{2 μ_{2 n}^{α}}{(α + 2) (2 α + 2)!}, \\ \frac{2^{5} ϕ_{n}^{α, - \frac{1}{2}}}{(α + 1) (α + 4)! {(\frac{1}{2})}_{α + 1}} = \frac{2^{5} (n - 1)_{α + 3} (n - \frac{1}{2})_{α + 3} [(α + 2) (n - 1) (n + α + \frac{3}{2}) - 1]}{(α + 1)_{4} (1)_{α + 1} {(\frac{1}{2})}_{α + 1}} \\ = \frac{(2 n - 2)_{2 α + 6} [(α + 2) (2 n - 2) (2 n + 2 α + 3) - 4]}{2 (α + 1)_{4} (2 α + 2)!} = \frac{ρ_{2 n}^{α}}{2 (α + 1)_{4} (2 α + 2)!}, \\ \frac{2^{6} (α + \frac{3}{2}) χ_{n}^{α, - \frac{1}{2}}}{(α + 1) (α + 2)! {(\frac{1}{2})}_{α + 1}} \\ = \frac{2^{5} (2 α + 3) (n - 1)_{α + 3} (n - \frac{1}{2})_{α + 3}}{(α + 1)_{3} (1)_{α + 1} {(\frac{1}{2})}_{α + 1}}_{4} F_{3} (\begin{matrix} 2 - n, - α - 2, n + α + \frac{5}{2}, α + 3 \\ 2, \frac{1}{2}, α + 4 \end{matrix}; 1) \\ = \frac{2 (2 α + 3) (2 n - 2)_{2 α + 6}}{(α + 1)_{3} (2 α + 2)!}_{4} F_{3} (\begin{matrix} \dots \\ \dots \end{matrix}; 1) = \frac{2 (2 α + 3) σ_{2 n}^{α}}{(α + 1)_{2} (2 α + 2)!} . \end{aligned}$ The assertion then follows by definition (Equation1.21(1.21) $γ_{n}^{α, N, S} = λ_{n}^{α, α} + \frac{N 2 μ_{n}^{α}}{(α + 2) (2 α + 2)!} + \frac{S ρ_{n}^{α}}{2 (α + 1)_{4} (2 α + 2)!} + \frac{NS 2 (2 α + 3) σ_{n}^{α}}{(α + 1)_{2} (2 α + 2)!},$ (1.21) ).

As suggested by Propositions 3.2 and 3.4, we apply both the transformation of the argument and the modification of the parameters, which occur on the left-hand side of (Equation3.1(3.1) $\begin{aligned} {\hat{P}}_{n}^{α, - \frac{1}{2}, 2 N, 2^{5} S} (2 x^{2} - 1) \\ = {{R_{n}^{α, - \frac{1}{2}} (ξ) + 2 N {\hat{T}}_{n}^{α, - \frac{1}{2}} (ξ) + 2^{5} S {\hat{U}}_{n}^{α, - \frac{1}{2}} (ξ) + 2^{6} NS {\hat{V}}_{n}^{α, - \frac{1}{2}} (ξ)} |}_{ξ = 2 x^{2} - 1} \\ = R_{2 n}^{α, α} (x) + N {\hat{A}}_{2 n}^{α} (x) + S {\hat{B}}_{2 n}^{α} (x) + NS {\hat{C}}_{2 n}^{α} (ξ) = {\hat{G}}_{2 n}^{α, N, S} (x) . \end{aligned}$ (3.1) ), to the Jacobi-Sobolev differential operator defined in Proposition 1.1.

Theorem 3.5

For $α \in N_{0}, N, S \geq 0$ , let a differential operator $G_{x}^{α, N, S}$ be given as in (Equation1.18(1.18) $G_{x}^{α, N, S} = L_{x}^{α, α} + \frac{N 2 M_{x}^{α}}{(α + 2) (2 α + 2)!} + \frac{S R_{x}^{α}}{2 (α + 1)_{4} (2 α + 2)!} + \frac{NS 2 (2 α + 3) S_{x}^{α}}{(α + 1)_{2} (2 α + 2)!},$ (1.18) ), where its components are defined, for sufficiently smooth functions $y (x), - 1 \leq x \leq 1$ , by (3.3) $\begin{aligned} \begin{aligned} L_{x}^{α, α} y (x) = (x^{2} - 1)^{- α} D_{x} [(x^{2} - 1)^{α + 1} D_{x} y (x)] \\ M_{x}^{α} (x) = (x^{2} - 1) D_{x}^{2 α + 4} [(x^{2} - 1)^{α + 1} y (x)], \\ R_{x}^{α} y (x) = (R_{x, 1}^{α} + R_{x, 2}^{α} + R_{x, 3}^{α}) y (x) with \\ R_{x, 1}^{α} y (x) = (α + 2) (x^{2} - 1)^{2} D_{x}^{2 α + 8} [(x^{2} - 1)^{α + 2} y (x)], \\ R_{x, 2}^{α} y (x) = 2 (α + 1) (α + 4) (x^{2} - 1) \\ \times {(2 α + 3) D_{x}^{2 α + 6} [(x^{2} - 1)^{α + 1} y (x)] + x D_{x}^{2 α + 6} [x (x^{2} - 1)^{α + 1} y (x)]}, \\ R_{x, 3}^{α} y (x) = - 8 (α + 1) (α + 3) (α + 4) \\ \times {(2 α + 3) D_{x}^{2 α + 4} [(x^{2} - 1)^{α} y (x)] + (2 α + 5) x D_{x}^{2 α + 4} [x (x^{2} - 1)^{α} y (x)]}, \end{aligned} \end{aligned}$ (3.3) and, setting $δ_{x} := \frac{1}{x} D_{x}$ , (3.4) $\begin{aligned} S_{x}^{α} y (x) = \sum_{j = 0}^{α + 2} \frac{(α + 3 - j)_{j} S_{x, j}^{α} y (x)}{(2 j)! (j + 1)! (j + α + 3)} with \\ S_{x, j}^{α} y (x) = (x^{2} - 1) x δ_{x}^{j + α + 3} {ψ_{j}^{α} (x) δ_{x}^{j + α + 3} [(x^{2} - 1)^{α + 1} y (x)]}, \\ ψ_{j}^{α} (x) = x^{2 j + 2 α + 5} (x^{2} - 1)^{j} [x^{2} - 1 - \frac{2 (α + 2 - j) (α + 3 + j)}{(α + 2) (2 j + 1)}] . \end{aligned}$ (3.4) (a) Choosing for $y (ξ), - 1 \leq ξ \leq 1$ , the Jacobi-Sobolev polynomial ${\hat{P}}_{m}^{α, - \frac{1}{2}, 2 N, 2^{5} S} (ξ)$ , $m \in N_{0}$ , so that $\tilde{y} (x) := y (2 x^{2} - 1), 0 \leq x \leq 1$ , becomes the Gegenbauer-Sobolev polynomial ${\hat{G}}_{2 m}^{α, N, S} (x)$ by Proposition 3.2, there holds $\begin{aligned} 4 L_{ξ}^{α, - \frac{1}{2}, 2 N, 2^{5} S} y (ξ) |_{ξ = 2 x^{2} - 1} = 4 {L_{ξ}^{α, - \frac{1}{2}} + \frac{2 N T_{ξ}^{α, - \frac{1}{2}}}{(α + 2)! {(\frac{1}{2})}_{α + 1}} + \\ + \frac{2^{5} S U_{ξ}^{α, - \frac{1}{2}}}{4 (α + 1) (α + 4)! {(\frac{1}{2})}_{α + 1}} + \frac{2^{6} NS (α + \frac{3}{2}) V_{ξ}^{α, - \frac{1}{2}}}{4 (α + 1) (α + 2! {(\frac{1}{2})}_{α + 1}}} y (ξ) |_{ξ = 2 x^{2} - 1} \\ = {G_{x}^{α} + \frac{N 2 M_{x}^{α}}{(α + 2) (2 α + 2)!} + \frac{S R_{x}^{α}}{2 (α + 1)_{4} (2 α + 2)!} + \frac{NS 2 (2 α + 3) S_{x}^{α}}{(α + 1)_{2} (2 α + 2)!}} \tilde{y} (x) \\ = G_{x}^{α, N, S} \tilde{y} (x) . \end{aligned}$ (b) For all $n \in N_{0}$ , the Gegenbauer-Sobolev polynomials $G_{n}^{α, N, S} (x)$ are, regardless of their normalization, the eigenfunctions of the differential operator $G_{x}^{α, N, S}$ . In particular, the components $M_{x}^{α}, R_{x}^{α}, S_{x}^{α}$ in Bavinck's representation (Equation1.19(1.19) $M_{x}^{α} \equiv \sum_{j = 0}^{2 α + 4} m_{j}^{α} (x) D_{x}^{j}, R_{x}^{α} \equiv \sum_{j = 0}^{2 α + 8} r_{j}^{α} (x) D_{x}^{j}, S_{x}^{α} \equiv \sum_{j = 0}^{4 α + 10} s_{j}^{α} (x) D_{x}^{j},$ (1.19) ) of the operator coincide with those defined in (Equation3.3(3.3) $\begin{aligned} \begin{aligned} L_{x}^{α, α} y (x) = (x^{2} - 1)^{- α} D_{x} [(x^{2} - 1)^{α + 1} D_{x} y (x)] \\ M_{x}^{α} (x) = (x^{2} - 1) D_{x}^{2 α + 4} [(x^{2} - 1)^{α + 1} y (x)], \\ R_{x}^{α} y (x) = (R_{x, 1}^{α} + R_{x, 2}^{α} + R_{x, 3}^{α}) y (x) with \\ R_{x, 1}^{α} y (x) = (α + 2) (x^{2} - 1)^{2} D_{x}^{2 α + 8} [(x^{2} - 1)^{α + 2} y (x)], \\ R_{x, 2}^{α} y (x) = 2 (α + 1) (α + 4) (x^{2} - 1) \\ \times {(2 α + 3) D_{x}^{2 α + 6} [(x^{2} - 1)^{α + 1} y (x)] + x D_{x}^{2 α + 6} [x (x^{2} - 1)^{α + 1} y (x)]}, \\ R_{x, 3}^{α} y (x) = - 8 (α + 1) (α + 3) (α + 4) \\ \times {(2 α + 3) D_{x}^{2 α + 4} [(x^{2} - 1)^{α} y (x)] + (2 α + 5) x D_{x}^{2 α + 4} [x (x^{2} - 1)^{α} y (x)]}, \end{aligned} \end{aligned}$ (3.3) )–(Equation3.4(3.4) $\begin{aligned} S_{x}^{α} y (x) = \sum_{j = 0}^{α + 2} \frac{(α + 3 - j)_{j} S_{x, j}^{α} y (x)}{(2 j)! (j + 1)! (j + α + 3)} with \\ S_{x, j}^{α} y (x) = (x^{2} - 1) x δ_{x}^{j + α + 3} {ψ_{j}^{α} (x) δ_{x}^{j + α + 3} [(x^{2} - 1)^{α + 1} y (x)]}, \\ ψ_{j}^{α} (x) = x^{2 j + 2 α + 5} (x^{2} - 1)^{j} [x^{2} - 1 - \frac{2 (α + 2 - j) (α + 3 + j)}{(α + 2) (2 j + 1)}] . \end{aligned}$ (3.4) ). When expanding the new expressions into finite power series over $D_{x}^{i}$ , the highest coefficient functions agree with those in (Equation1.20(1.20) $\begin{aligned} m_{2 α + 4}^{α} (x) & = (x^{2} - 1)^{α + 2}, r_{2 α + 8}^{α} (x) = (α + 2) (x^{2} - 1)^{α + 4}, \\ s_{4 α + 10}^{α} (x) & = \frac{2 (x^{2} - 1)^{2 α + 5}}{(2 α + 6)!} . \end{aligned}$ (1.20) ).

Proof.

(a) We separately consider the four terms of the Jacobi-Sobolev operator $L_{ξ}^{α, - \frac{1}{2}, 2 N, 2^{5} S}$ . Substituting $ξ = x^{2} - 1$ , we can formally replace each differentiation in the occurring differential expressions by $D_{ξ} = \frac{1}{4 x} D_{x} =: \frac{1}{4} δ_{x}$ . First of all, this yields the well known identity $\begin{aligned} 4 L_{ξ}^{α, - \frac{1}{2}} y (ξ) & = \frac{4 D_{ξ} [(ξ - 1)^{α + 1} (ξ + 1)^{\frac{1}{2}} D_{ξ} y (ξ)]}{(ξ - 1)^{α} (ξ + 1)^{- \frac{1}{2}}} \\ = \frac{2^{α - \frac{1}{2}} D_{x} [(x^{2} - 1)^{α + 1} D_{x} \tilde{y} (x)]}{x 2^{α - \frac{1}{2}} (x^{2} - 1)^{α} (x^{2})^{- \frac{1}{2}}} = \frac{D_{x} [(x^{2} - 1)^{α + 1} D_{x} \tilde{y} (x)]}{(x^{2} - 1)^{α}} = L_{x}^{α, α} \tilde{y} (x) . \end{aligned}$ Concerning the other three components, we frequently make use of

Lemma 3.6

[Citation24, (6.6)]

For all $m \in N_{0}$ and admissible functions $ϕ (x)$ , $\begin{aligned} δ_{x}^{m} [x^{2 m + 1} δ_{x}^{m + 1} ϕ (x)] = D_{x}^{2 m + 1} ϕ (x), x δ_{x}^{m + 1} [x^{2 m + 1} δ_{x}^{m + 1} ϕ (x)] = D_{x}^{2 m + 2} ϕ (x), \\ x δ_{x}^{m + 1} [x^{2 m + 3} δ_{x}^{m + 1} ϕ (x)] = x^{2} D_{x}^{2 m + 2} ϕ (x) + 2 (m + 1) x D_{x}^{2 m + 1} ϕ (x) = x D_{x}^{2 m + 2} [xϕ (x)] . \end{aligned}$

Hence, $\begin{aligned} \frac{8 T_{ξ}^{α, - \frac{1}{2}} y (ξ)}{(α + 2)! {(\frac{1}{2})}_{α + 1}} = \frac{8 (ξ - 1) (ξ + 1)^{\frac{1}{2}}}{(α + 2)! {(\frac{1}{2})}_{α + 1}} D_{ξ}^{α + 2} {(ξ + 1)^{α + \frac{3}{2}} D_{ξ}^{α + 2} [(ξ - 1)^{α + 1} y (ξ)]} \\ = \frac{2 (x^{2} - 1)}{2^{2 α + 2} (α + 2)! {(\frac{1}{2})}_{α + 1}} x δ_{x}^{α + 2} {x^{2 α + 3} δ_{x}^{α + 2} [(x^{2} - 1)^{α + 1} \tilde{y} (x)]} \\ = \frac{2 (x^{2} - 1) D_{x}^{2 α + 4} [(x^{2} - 1)^{α + 1} \tilde{y} (x)]}{(α + 2) (2 α + 2)!} = \frac{2 M_{x}^{α} \tilde{y} (x)}{(α + 2) (2 α + 2)!} . \end{aligned}$ Next, the three parts of $U_{ξ}^{α, - \frac{1}{2}} y (ξ)$ lead to $\begin{aligned} \frac{2^{7} U_{ξ, 1}^{α, - \frac{1}{2}} y (ξ)}{4 (α + 1) (α + 4)! {(\frac{1}{2})}_{α + 1}} = \\ = \frac{2^{5} (α + 2) (ξ - 1)^{2} (ξ + 1)^{\frac{1}{2}}}{(α + 1) (α + 4)! {(\frac{1}{2})}_{α + 1}} D_{ξ}^{α + 4} {(ξ + 1)^{α + \frac{7}{2}} D_{ξ}^{α + 4} [(ξ - 1)^{α + 2} y (ξ)]} \\ = \frac{(α + 2) (x^{2} - 1)^{2}}{2^{2 α + 3} (α + 1)_{4} (1)_{α + 1} {(\frac{1}{2})}_{α + 1}} x δ_{x}^{α + 4} {x^{2 α + 7} δ_{x}^{α + 4} [(x^{2} - 1)^{α + 2} \tilde{y} (x)]} \\ = \frac{(α + 2) (x^{2} - 1)^{2}}{2 (α + 1)_{4} (2 α + 2)!} D_{x}^{2 α + 8} [(x^{2} - 1)^{α + 2} \tilde{y} (x)] = \frac{R_{x, 1}^{α} \tilde{y} (x)}{2 (α + 1)_{4} (2 α + 2)!}, \\ \frac{2^{7} U_{ξ, 2}^{α, - \frac{1}{2}} y (ξ)}{4 (α + 1) (α + 4)! {(\frac{1}{2})}_{α + 1}} \\ = \frac{2^{5} (ξ - 1) (ξ + 1)^{\frac{1}{2}}}{(α + 3)! {(\frac{1}{2})}_{α + 1}} D_{ξ}^{α + 3} {ψ_{1}^{α, - \frac{1}{2}} (ξ) D_{ξ}^{α + 3} [(ξ - 1)^{α + 1} y (ξ)]} \\ = \frac{(x^{2} - 1) x δ_{x}^{α + 3} {x^{2 α + 5} [2 α + 3 + x^{2}] δ_{x}^{α + 3} [(x^{2} - 1)^{α + 1} \tilde{y} (x)]}}{2^{2 α + 2} (α + 2)_{2} (1)_{α + 1} {(\frac{1}{2})}_{α + 1}} \\ = \frac{(x^{2} - 1) {(2 α + 3) D_{x}^{2 α + 6} [(x^{2} - 1)^{α + 1} \tilde{y} (x)] + x D_{x}^{2 α + 6} [x (x^{2} - 1)^{α + 1} \tilde{y} (x)]}}{(α + 2)_{2} (2 α + 2)!} \\ = \frac{R_{x, 2}^{α} \tilde{y} (x)}{2 (α + 1)_{4} (2 α + 2)!}, \end{aligned}$ and $\begin{aligned} \frac{2^{7} U_{ξ, 3}^{α, - \frac{1}{2}} y (ξ)}{4 (α + 1) (α + 4)! {(\frac{1}{2})}_{α + 1}} = - \frac{2^{6} (ξ + 1)^{\frac{1}{2}}}{(α + 2)! {(\frac{1}{2})}_{α + 1}} D_{ξ}^{α + 2} {ψ_{2}^{α, - \frac{1}{2}} (ξ) D_{ξ}^{α + 2} [(ξ - 1)^{α} y (ξ)]} \\ = - \frac{x δ_{x}^{α + 2} {x^{2 α + 3} [2 α + 3 + (2 α + 5) x^{2}] δ_{x}^{α + 2} [(x^{2} - 1)^{α} \tilde{y} (x)]}}{2^{2 α} (α + 2) (1)_{α + 1} {(\frac{1}{2})}_{α + 1}} \\ = - \frac{4 {(2 α + 3) D_{x}^{2 α + 4} [(x^{2} - 1)^{α} \tilde{y} (x)] + (2 α + 5) x D_{x}^{2 α + 4} [x (x^{2} - 1)^{α} \tilde{y} (x)]}}{(α + 2) (2 α + 2)!} \\ = \frac{R_{x, 3}^{α} \tilde{y} (x)}{2 (α + 1)_{4} (2 α + 2)!} . \end{aligned}$ Finally, we get $\begin{aligned} \frac{2^{8} (α + \frac{3}{2}) V_{ξ}^{α, - \frac{1}{2}} y (ξ)}{4 (α + 1) (α + 2)! {(\frac{1}{2})}_{α + 1}} & = \frac{2 (2 α + 3)}{(α + 1)_{2} (2 α + 2)!} \sum_{j = 0}^{α + 2} \frac{(α + 3 - j)_{j} 2^{2 j + 2 α + 6} V_{x, j}^{α, - \frac{1}{2}} y (ξ)}{(2 j)! (j + 1)! (j + α + 3)} \\ = \frac{2 (2 α + 3) S_{x}^{α} \tilde{y} (x)}{(α + 1)_{2} (2 α + 2)!}, \end{aligned}$ since, by the definitions of $ψ_{3, j}^{α, - \frac{1}{2}} (x)$ in (Equation1.9(1.9) $\begin{aligned} V_{x}^{α, β} y (x) = \sum_{j = 0}^{α + 2} \frac{(α + 3 - j)_{j} V_{x, j}^{α, β} y (x)}{j! (j + 1)! (j + α + 3) (β + 1)_{j}} with \\ V_{x, j}^{α, β} y (x) = \frac{x - 1}{(x + 1)^{β}} D_{x}^{j + α + 3} {ψ_{3, j}^{α, β} (x) D_{x}^{j + α + 3} [(x - 1)^{α + 1} y (x)]}, 0 \leq j \leq α + 2, \\ where ψ_{3, j}^{α, β} (x) = (x + 1)^{j + α + β + 3} (x - 1)^{j} [x - 1 - \frac{2 (α + 2 - j) (j + α + 3)}{(α + 2) (j + β + 1)}] . \end{aligned}$ (1.9) ) and of $ψ_{j}^{α} (x)$ in (Equation3.4(3.4) $\begin{aligned} S_{x}^{α} y (x) = \sum_{j = 0}^{α + 2} \frac{(α + 3 - j)_{j} S_{x, j}^{α} y (x)}{(2 j)! (j + 1)! (j + α + 3)} with \\ S_{x, j}^{α} y (x) = (x^{2} - 1) x δ_{x}^{j + α + 3} {ψ_{j}^{α} (x) δ_{x}^{j + α + 3} [(x^{2} - 1)^{α + 1} y (x)]}, \\ ψ_{j}^{α} (x) = x^{2 j + 2 α + 5} (x^{2} - 1)^{j} [x^{2} - 1 - \frac{2 (α + 2 - j) (α + 3 + j)}{(α + 2) (2 j + 1)}] . \end{aligned}$ (3.4) ), $\begin{aligned} 2^{2 j + 2 α + 6} V_{ξ, j}^{α, - \frac{1}{2}} y (ξ) & = \frac{2^{2 j + 2 α + 6} (ξ - 1)}{(ξ + 1)^{- \frac{1}{2}}} D_{ξ}^{j + α + 3} {ψ_{3, j}^{α, - \frac{1}{2}} (ξ) D_{ξ}^{j + α + 3} [(ξ - 1)^{α + 1} y (ξ)]} \\ = (x^{2} - 1) x δ_{x}^{j + α + 3} {ψ_{j}^{α} (x) δ_{x}^{j + α + 3} [(x^{2} - 1)^{α + 1} \tilde{y} (x)]} = S_{x, j}^{α} \tilde{y} (x) . \end{aligned}$ (b) Given the Jacobi-Sobolev Equation (Equation1.6(1.6) $L_{x}^{α, β, N, S} y_{n} (x) = λ_{n}^{α, β, N, S} y_{n} (x), - 1 \leq x \leq 1,$ (1.6) ), it follows by part (a) and Proposition 3.4 that the Gegenbauer-Sobolev polynomials of even degree satisfy the differential Equation (Equation3.2(3.2) $G_{x}^{α, N, S} y_{n} (x) = γ_{n}^{α, N, S} y_{n} (x),$ (3.2) ) for $0 \leq x \leq 1$ , i.e. $\begin{aligned} 0 & = 4 {L_{ξ}^{α, - \frac{1}{2}, 2 N, 2^{5} S} - λ_{m}^{α, - \frac{1}{2}, 2 N, 2^{5} S}} {\hat{P}}_{m}^{α, - \frac{1}{2}, 2 N, 2^{5} S} (ξ) |_{ξ = 2 x^{2} - 1} \\ = {G_{x}^{α, N, S} - γ_{2 m}^{α, N, S}} {\hat{G}}_{2 m}^{α, N, S} (x) . \end{aligned}$ Clearly, the latter equation can be extended to the whole interval $- 1 \leq x \leq 1$ , and since $G_{x}^{α, N, S}$ is independent of the index $m \in N_{0}$ , the operator must coincide with Bavinck's unique representation based on the components (Equation1.19(1.19) $M_{x}^{α} \equiv \sum_{j = 0}^{2 α + 4} m_{j}^{α} (x) D_{x}^{j}, R_{x}^{α} \equiv \sum_{j = 0}^{2 α + 8} r_{j}^{α} (x) D_{x}^{j}, S_{x}^{α} \equiv \sum_{j = 0}^{4 α + 10} s_{j}^{α} (x) D_{x}^{j},$ (1.19) ). Concerning their highest coefficients, it readily follows by definition (Equation3.3(3.3) $\begin{aligned} \begin{aligned} L_{x}^{α, α} y (x) = (x^{2} - 1)^{- α} D_{x} [(x^{2} - 1)^{α + 1} D_{x} y (x)] \\ M_{x}^{α} (x) = (x^{2} - 1) D_{x}^{2 α + 4} [(x^{2} - 1)^{α + 1} y (x)], \\ R_{x}^{α} y (x) = (R_{x, 1}^{α} + R_{x, 2}^{α} + R_{x, 3}^{α}) y (x) with \\ R_{x, 1}^{α} y (x) = (α + 2) (x^{2} - 1)^{2} D_{x}^{2 α + 8} [(x^{2} - 1)^{α + 2} y (x)], \\ R_{x, 2}^{α} y (x) = 2 (α + 1) (α + 4) (x^{2} - 1) \\ \times {(2 α + 3) D_{x}^{2 α + 6} [(x^{2} - 1)^{α + 1} y (x)] + x D_{x}^{2 α + 6} [x (x^{2} - 1)^{α + 1} y (x)]}, \\ R_{x, 3}^{α} y (x) = - 8 (α + 1) (α + 3) (α + 4) \\ \times {(2 α + 3) D_{x}^{2 α + 4} [(x^{2} - 1)^{α} y (x)] + (2 α + 5) x D_{x}^{2 α + 4} [x (x^{2} - 1)^{α} y (x)]}, \end{aligned} \end{aligned}$ (3.3) )–(Equation3.4(3.4) $\begin{aligned} S_{x}^{α} y (x) = \sum_{j = 0}^{α + 2} \frac{(α + 3 - j)_{j} S_{x, j}^{α} y (x)}{(2 j)! (j + 1)! (j + α + 3)} with \\ S_{x, j}^{α} y (x) = (x^{2} - 1) x δ_{x}^{j + α + 3} {ψ_{j}^{α} (x) δ_{x}^{j + α + 3} [(x^{2} - 1)^{α + 1} y (x)]}, \\ ψ_{j}^{α} (x) = x^{2 j + 2 α + 5} (x^{2} - 1)^{j} [x^{2} - 1 - \frac{2 (α + 2 - j) (α + 3 + j)}{(α + 2) (2 j + 1)}] . \end{aligned}$ (3.4) ) that $\begin{aligned} M_{x}^{α} y (x) & = (x^{2} - 1) D_{x}^{2 α + 4} [(x^{2} - 1)^{α + 1} y (x)] \\ = (x^{2} - 1)^{α + 2} D_{x}^{2 α + 4} y (x) + \sum_{j = 1}^{2 α + 3} m_{j}^{α} (x) D_{x}^{j} y (x), \\ R_{x}^{α} y (x) & = (α + 2) (x^{2} - 1)^{2} D_{x}^{2 α + 8} [(x^{2} - 1)^{α + 2} y (x)] + (R_{x, 2}^{α} + R_{x, 3}^{α}) y (x) \\ = (α + 2) (x^{2} - 1)^{α + 4} D_{x}^{2 α + 8} y (x) + \sum_{j = 1}^{2 α + 7} r_{j}^{α} (x) D_{x}^{j} y (x), \\ S_{x}^{α} y (x) & = \frac{2 S_{x, α + 2}^{α} y (x)}{(2 α + 6)!} + \sum_{j = 0}^{α + 1} \frac{(α + 3 - j)_{j} S_{x, j}^{α} y (x)}{(2 j)! (j + 1)! (j + α + 3)} \\ = \frac{2 (x^{2} - 1) x}{(2 α + 6)!} {(\frac{1}{x} D_{x})}^{2 α + 5} \\ \times {x^{4 α + 9} (x^{2} -)^{α + 3} {(\frac{1}{x} D_{x})}^{2 α + 5} [(x^{2} - 1)^{α + 1} y (x)]} + lower terms \\ = \frac{2 (x^{2} - 1)^{2 α + 5}}{(2 α + 6)!} D_{x}^{4 α + 10} y (x) + \sum_{j = 1}^{4 α + 9} s_{j}^{α} (x) D_{x}^{j} y (x) . \end{aligned}$ This confirms the values in (Equation1.20(1.20) $\begin{aligned} m_{2 α + 4}^{α} (x) & = (x^{2} - 1)^{α + 2}, r_{2 α + 8}^{α} (x) = (α + 2) (x^{2} - 1)^{α + 4}, \\ s_{4 α + 10}^{α} (x) & = \frac{2 (x^{2} - 1)^{2 α + 5}}{(2 α + 6)!} . \end{aligned}$ (1.20) ) and concludes the proof of Theorem 3.5.

Remark 3.7

(a) The particular case S = 0 of Theorem 3.5 has been treated already in [Citation25]. There we proved that the ultraspherical-type polynomials $G_{n}^{α, N, 0} (x) = P_{n}^{α, α} (x) + N A_{n}^{α} (x), n \in N_{0}$ , satisfy the differential equation $\begin{aligned} {G_{x}^{α, N, 0} - γ_{n}^{α, N, 0}} G_{n}^{α, N, 0} (x) \\ = {[L_{x}^{α, α} - n (n + 2 α + 1)] + N \frac{2 [M_{x}^{α} - (n - 1)_{2 α + 4}]}{(α + 2) (2 α + 2)!}} G_{n}^{α, N, 0} (x) = 0 \end{aligned}$ on $- 1 \leq x \leq 1$ with $L_{x}^{α, α}, M_{x}^{α}$ being given in (Equation3.3(3.3) $\begin{aligned} \begin{aligned} L_{x}^{α, α} y (x) = (x^{2} - 1)^{- α} D_{x} [(x^{2} - 1)^{α + 1} D_{x} y (x)] \\ M_{x}^{α} (x) = (x^{2} - 1) D_{x}^{2 α + 4} [(x^{2} - 1)^{α + 1} y (x)], \\ R_{x}^{α} y (x) = (R_{x, 1}^{α} + R_{x, 2}^{α} + R_{x, 3}^{α}) y (x) with \\ R_{x, 1}^{α} y (x) = (α + 2) (x^{2} - 1)^{2} D_{x}^{2 α + 8} [(x^{2} - 1)^{α + 2} y (x)], \\ R_{x, 2}^{α} y (x) = 2 (α + 1) (α + 4) (x^{2} - 1) \\ \times {(2 α + 3) D_{x}^{2 α + 6} [(x^{2} - 1)^{α + 1} y (x)] + x D_{x}^{2 α + 6} [x (x^{2} - 1)^{α + 1} y (x)]}, \\ R_{x, 3}^{α} y (x) = - 8 (α + 1) (α + 3) (α + 4) \\ \times {(2 α + 3) D_{x}^{2 α + 4} [(x^{2} - 1)^{α} y (x)] + (2 α + 5) x D_{x}^{2 α + 4} [x (x^{2} - 1)^{α} y (x)]}, \end{aligned} \end{aligned}$ (3.3) ). Of course, the new representations (3.3)-(Equation3.4(3.4) $\begin{aligned} S_{x}^{α} y (x) = \sum_{j = 0}^{α + 2} \frac{(α + 3 - j)_{j} S_{x, j}^{α} y (x)}{(2 j)! (j + 1)! (j + α + 3)} with \\ S_{x, j}^{α} y (x) = (x^{2} - 1) x δ_{x}^{j + α + 3} {ψ_{j}^{α} (x) δ_{x}^{j + α + 3} [(x^{2} - 1)^{α + 1} y (x)]}, \\ ψ_{j}^{α} (x) = x^{2 j + 2 α + 5} (x^{2} - 1)^{j} [x^{2} - 1 - \frac{2 (α + 2 - j) (α + 3 + j)}{(α + 2) (2 j + 1)}] . \end{aligned}$ (3.4) ) of the operators $R_{x}^{α}, S_{x}^{α}$ related to the additional 'Sobolev parameter' S>0, are quite sophisticated. Nevertheless, they can easily be implemented in a symbolic computer programme and so are accessible for further applications. Indeed, we checked the Gegenbauer-Sobolev equation in Theorem 3.5 via MAPLE for both even and odd Gegenbauer-Sobolev polynomials by considering, e.g. the cases $(n, α) = (4, 0), (5, 1)$ and $(7, 1)$ with arbitrary $N, S \geq 0$ .

(b) Though not required in the proof of Theorem 3.5, it would be worthwhile to find an extension of the quadratic transfomation (Equation1.17(1.17) $\begin{aligned} x R_{n}^{α, 1 / 2} (2 x^{2} - 1) & = x_{2} F_{1} (- n, n + α + \frac{3}{2}; α + 1; 1 - x^{2}) \\ =_{2} F_{1} (- 2 n - 1, 2 n + 2 α + 2; α + 1; \frac{1 - x}{2}) = R_{2 n + 1}^{α, α} (x) . \end{aligned}$ (1.17) ) for the Gegenbauer-Sobolev polynomials of odd degree. However, there seems to be no relationship as simple as identity (Equation3.1(3.1) $\begin{aligned} {\hat{P}}_{n}^{α, - \frac{1}{2}, 2 N, 2^{5} S} (2 x^{2} - 1) \\ = {{R_{n}^{α, - \frac{1}{2}} (ξ) + 2 N {\hat{T}}_{n}^{α, - \frac{1}{2}} (ξ) + 2^{5} S {\hat{U}}_{n}^{α, - \frac{1}{2}} (ξ) + 2^{6} NS {\hat{V}}_{n}^{α, - \frac{1}{2}} (ξ)} |}_{ξ = 2 x^{2} - 1} \\ = R_{2 n}^{α, α} (x) + N {\hat{A}}_{2 n}^{α} (x) + S {\hat{B}}_{2 n}^{α} (x) + NS {\hat{C}}_{2 n}^{α} (ξ) = {\hat{G}}_{2 n}^{α, N, S} (x) . \end{aligned}$ (3.1) ) in case of even polynomials.

Corollary 3.8

For $α \in N_{0}, N, S \geq 0$ , the Gegenbauer-Sobolev operator $G_{x}^{α, N, S}$ is symmetric with respect to the inner product (Equation1.13(1.13) $\begin{aligned} (f, g)_{ω (α, N, S)} & = (f, g)_{w (α, α)} \\ + N [f (- 1) g (- 1) + f (1) g (1)] + S [f^{'} (- 1) g^{'} (- 1) + f^{'} (1) g^{'} (1)], \end{aligned}$ (1.13) ), i.e. (3.5) $(G_{x}^{α, N, S} F, G)_{ω (α, N, S)} = (F, G_{x}^{α, N, S} G)_{ω (α, N, S)}, F, G \in C^{(4 α + 10)} [- 1, 1] .$ (3.5)

Proof.

Since the Gegenbauer-Sobolev polynomials are the eigenfunctions of $G_{x}^{α, N, S}$ with different eigenvalues, the symmetry relation (Equation3.5(3.5) $(G_{x}^{α, N, S} F, G)_{ω (α, N, S)} = (F, G_{x}^{α, N, S} G)_{ω (α, N, S)}, F, G \in C^{(4 α + 10)} [- 1, 1] .$ (3.5) ) is an immediate consequence of the fact that they form an orthogonal polynomial system which is dense in the corresponding inner product space, cf. [Citation14]. Especially for even functions F, G, this result can also be deduced from the symmetry property (Equation1.11(1.11) $(L_{x}^{α, β, N, S} f, g)_{w (α, β, N, S)} = (f, L_{x}^{α, β, N, S} g)_{w (α, β, N, S)}, f, g \in C^{(4 α + 10)} [- 1, 1] .$ (1.11) ) of the Jacobi-Sobolev differential operator with parameters $α, β = - \frac{1}{2}, 2 N, 2^{5} S$ .

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The Jacobi-Sobolev, Laguerre-Sobolev, and Gegenbauer-Sobolev differential equations and their interrelations

Abstract

1. Introduction

[Citation6, Theorem 3.1]

2. Relationship between the Jacobi-Sobolev and Laguerre-Sobolev equations

[Citation23, Theorem 3.1]

[Citation10, Theorem 3.1]

3. A new representation of the differential equation for the Gegenbauer-Sobolev polynomials

[Citation24, (6.6)]

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References

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The Jacobi-Sobolev, Laguerre-Sobolev, and Gegenbauer-Sobolev differential equations and their interrelations

Abstract

1. Introduction

[Citation6, Theorem 3.1]

2. Relationship between the Jacobi-Sobolev and Laguerre-Sobolev equations

[Citation23, Theorem 3.1]

[Citation10, Theorem 3.1]

3. A new representation of the differential equation for the Gegenbauer-Sobolev polynomials

[Citation24, (6.6)]

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