On a family of q-modified-Laguerre-Appell polynomials

Abstract

This paper aims to introduce a new class of special polynomials called q-modified Laguerre-Appell polynomials. Some definitions and concepts related to this class of polynomials, including generating function and series definition are explored. Also, certain interesting properties related to this class and some of its members are investigated. Furthermore, the class of 2D q-modified Laguerre-Appell polynomials is introduced. Finally, we present certain graphical representations related to some members of these new classes.

Keywords:

• q-Appell polynomials
• determinant definition
• q-modified-Laguerre polynomials

2020 Mathematics Subject Classification:

• 33E20
• 11B68
• 11B83

1. Introduction

The q-calculus, a generalization of ordinary calculus, was developed to study q-analogues of mathematical objects. It has a rich history dating back to the 19^th century and has applications in fields like mechanics, physics, statistical mechanics, quantum groups and transcendental number theory. The q-series, a special case of the q-analogue, is significant in combinatorics, number theory, and particle physics.

The q-analogue of a complex number $\alpha$ is given as (Andrews, Askey, & Roy, 1999): (1.1) $${\left[\alpha \right]}_q=\frac{1-q^{\alpha }}{1-q},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}0<\left|q\right|<1,\alpha \in \mathbb{C}.$$(1.1)

The q-factorial is defined by $${\left[n\right]}_q!=\frac{{(q;q)}_n}{{(1-q)}^n} (n\in \mathbb{N}),{(a;q)}_n=\{\begin{array}{c}1,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}n=0;\hfill \\ \prod _{k=0}^{n-1}(1-a{q}^k),n\in \mathbb{N}\hfill \end{array}$$ and $a\in \mathbb{C},$ see Gasper and Rahman (2004). Also, q-Pochhammer symbol is defined as : (1.2) $${({\left[m\right]}_q)}_n=\{\begin{array}{cc}\frac{(q_n^{m;q})}{{(1-q)}^n},\hfill & n>1\hfill \\ 1,\hfill & n=0.\hfill \end{array}$$(1.2)

The Gauss q-binomial coefficient is defined as (Andrews et al., 1999): (1.3) $${\left[\begin{array}{c}n\\ k\end{array}\right]}_q=\frac{{\left[n\right]}_q!}{{[n-k]}_q!{\left[k\right]}_q!}=\frac{{(q;q)}_n}{{(q;q)}_k{(q;q)}_{n-k}},\mathrm{\hspace{1em}} k=0,1,\dots ,n.$$(1.3)

Jackson (1903) introduced two q-analogs of the exponential function as: (1.4) $$e_q(x)=\frac{1}{{(x(1-q);q)}_{\infty }}=\sum _{n=0}^{\infty }\frac{{(x(1-q))}^n}{{(q;q)}_n},\mathrm{\hspace{1em}}\left|x\right|<\frac{1}{1-q}$$(1.4) and (1.5) $$E_q(x)={(-x(1-q);q)}_{\infty }=\sum _{n=0}^{\infty } \frac{q^{\frac{n(n-1)}{2}}}{{(q;q)}_n}{(x(1-q))}^n,x\in \mathbb{C},$$(1.5) respectively.

The relationship between the two q-exponential functions is given as: (1.6) $$e_q(x)E_q(-x)=1,\mathrm{\hspace{1em}\hspace{1em}}\left|x\right|<\frac{1}{1-q}.$$(1.6)

Jackson (1909) defined the q-difference operator as: (1.7) $$D_{q,x}f(x)=\frac{f(\mathit{\text{qx}})-f(x)}{x(q-1)},\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}} x\ne 0.$$(1.7)

The q-product rule is defined as (1.8) $$D_{q,x}(f(x)g(x))=f(x)D_{q,x}g(x)+g(\mathit{\text{qx}})D_{q,x}f(x).$$(1.8)

It is important to notice that for $q\to 1^{-},$ these q-calculus conclusions reduce to their counterparts in ordinary calculus.

Modified Laguerre polynomials are a generalization of classical Laguerre polynomials, which are widely used in various fields of sciences, including physics, mathematics, and engineering. Louis Weisner’s group-theoretic method is a powerful tool for generating various single and multiple series of generating functions for particular modified Laguerre polynomials. This method has been employed by several authors to study the properties of modified Laguerre polynomials and their applications (see Chongdar & Majumdar, 1993; Singh & Bala, 1986; Sharma, 1990).

The modified Laguerre polynomials has received considerable attention in the literature, and it has been studied extensively by various authors (see Pathan & Khan, 2004; Pittaluga, Sacripante, & Srivastava, 2000; Srivastava & Manocha, 1984).

For, $m,\lambda \in \mathbb{C},$ the q-modified Laguerre polynomials qMLP $f_{n,q}^{(m,\lambda )}(x)$ are defined by the subsequent generating function (Raza, Fadel, Umme Zainab, & Cao): (1.9) $$\frac{1}{{(t;q)}_m}{e}_q(\lambda \mathit{\text{xt}})=\sum _{n=0}^{\infty }{f}_{n,q}^{(m,\lambda )}(x)t^n,\mathrm{\hspace{1em}\hspace{1em}}\left|\lambda \mathit{\text{xt}}\right|<\frac{1}{1-q}$$(1.9) and series definition (1.10) $$f_{n,q}^{(m,\lambda )}(x)=\sum _{k=0}^n\frac{(q_n^{m;q}){(\lambda x)}^{n-k}}{{(q,q)}_k {(q;q)}_{n-k}}.$$(1.10)

The q-Appell polynomials are a family of orthogonal polynomials that generalize the Appell polynomials. These polynomials have been studied extensively by various mathematicians over the years. In 1954, Sharma and Chak (1954) created a q-analogue of the Appell polynomials, which they called the q-harmonic sequence. Later in 1967, Al-Salam (1967) established the q-Appell polynomials family and explored some of its properties. Al-Salam also studied the q-Bernoulli and q-Euler polynomials as specific members of the q-Appell family. In 1982, Srivastava (1982) provided many characterizations of the prominent Appell polynomials and their fundamental analogues. Srivastava (2011) also provided certain q-extensions of the Bernoulli, Euler, and Genocchi polynomials. In 1985, Roman (1985) presented a method resembling the umbral technique under the non-classical category of umbral calculus, which is known as q-umbral calculus. This method has been used to study q-Appell, q-Bernoulli, and q-Euler polynomials extensively. Ernst (2006, 2014) has also provided a comprehensive study of the q-Appell, q-Bernoulli, and q-Euler polynomials in the framework of q-umbral calculus. Overall, the study of q-Appell polynomials and related polynomials has led to many interesting results and applications in various areas of mathematics.

The following generating function is used to define the q-Appell polynomials sequence ${\{{\mathcal{A}}_{n,q}(x)\}}_{n=0}^{\infty }$ (Al-Salam, 1967): (1.11) $${\mathcal{A}}_q(t)e_q(\mathit{\text{xt}})=\sum _{n=0}^{\infty }{\mathcal{A}}_{n,q}(x)\frac{t^n}{{\left[n\right]}_q!},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\mathcal{A}}_q(t)\ne 0,{\mathcal{A}}_{0,q}=1.$$(1.11)

The function ${\mathcal{A}}_q(t)$ may be called the determining function of the set of Appell polynomials, which is defined as (1.12) $${\mathcal{A}}_q(t)=\sum _{n=0}^{\infty }{\mathcal{A}}_{n,q}\frac{t^n}{{\left[n\right]}_q!},{\mathcal{A}}_q(t)\ne 0;\mathrm{\hspace{1em}\hspace{1em}}{\mathcal{A}}_{0,q}=1.$$(1.12)

The q-Appell polynomials ${\mathcal{A}}_{n,q}(x)$ (of degree n) satisfy the following q-recurrence relation: (1.13) $${\widehat{D}}_{q,x}\{{\mathcal{A}}_{n,q}(x)\}={\left[n\right]}_q{\mathcal{A}}_{n-1,q}(x),\hspace{1em}n\in \mathbb{N};x\in \mathbb{C};0<\mid q\mid \mathrm{<}1,$$(1.13) where ${\mathcal{A}}_{n,q}:={\mathcal{A}}_q(0)$ denotes the q-Appell numbers.

Depending on how the function ${\mathcal{A}}_q(t),$ is used, different members of the family of q-Appell polynomials ${\mathcal{A}}_{n,q}(x)$ like q-Bernoulli polynomials ${\mathcal{B}}_{n,q}(x),$ q-Euler polynomials ${\mathcal{E}}_{n,q}(x)$ and q-Genocchi polynomials ${\mathcal{G}}_{n,q}(x),$ are obtained. These representatives are listed in Table 1.

Table 1. A few known members of q-Appell polynomials.

S. No	${\mathcal{A}}_q(t)$	Generating functions	Polynomials
I	${\mathcal{A}}_q(t)=\frac{t}{e_q(t)-1}$	$\frac{t}{e_q(t)-1}{e}_q(\mathit{\text{xt}})$ $={\sum }_{n=0}^{\infty }{\mathfrak{B}}_{n,q}(x)\frac{t^n}{{[n]}_q!}$	The q-Bernoulli Polynomials (Al-Salam, 1958; Ernst, 2006)
II	${\mathcal{A}}_q(t)=\frac{{[2]}_q}{e_q(t)+1}$	$\frac{{[2]}_q}{e_q(t)+1}{e}_q(\mathit{\text{xt}})$ $={\sum }_{n=0}^{\infty }{\mathcal{E}}_{n,q}(x)\frac{t^n}{{[n]}_q!}$	The q-Euler Polynomials (Ernst, 2006; Mahmudov & Keleshteri, 2013)
III	${\mathcal{A}}_q(t)=\frac{{[2]}_{q}t}{e_q(t)+1}$	$\frac{{[2]}_{q}t}{e_q(t)+1}{e}_q(\mathit{\text{xt}})$ $={\sum }_{n=0}^{\infty }{\mathcal{G}}_{n,q}(x)\frac{t^n}{{[n]}_q!}$	The q-Genocchi Polynomials (Araci, Acikgoz, Jolany, & He, 2014; Mahmudov & Keleshteri, 2013)

The generating function for the q-analogue of the 2D Appell polynomials ${\mathcal{A}}_{n,q}(x,y)$ is given as (Keleshteri & Mahmudov, 2015): (1.14) $${\mathcal{A}}_q(t)e_q(\mathit{\text{xt}})E_q(\mathit{\text{yt}})=\sum _{n=0}^{\infty }{\mathcal{A}}_{n,q}(x,y)\frac{t^n}{{\left[n\right]}_q!},\mathrm{\hspace{1em}}{\mathcal{A}}_{n,q}={\mathcal{A}}_{n,q}(0,0).$$(1.14)

Table 2 presents selection members of the 2D q-Appell polynomials.

Table 2. Some recognized members of 2D q-Appell polynomials.

S. No	${\mathcal{A}}_q(t)$	Generating functions	Polynomials
I	${\mathcal{A}}_q(t)=\frac{t}{e_q(t)-1}$	$\frac{t}{e_q(t)-1}{e}_q(\mathit{\text{xt}})E_q(\mathit{\text{yt}})={\sum }_{n=0}^{\infty }{\mathfrak{B}}_{n,q}(x,y)\frac{t^n}{{[n]}_q!}$	The 2D q-Bernoulli Polynomials (Keleshteri & Mahmudov, 2015; Mahmudov & Keleshteri, 2013)
II	${\mathcal{A}}_q(t)=\frac{{[2]}_q}{e_q(t)+1}$	$\frac{{[2]}_q}{e_q(t)+1}{e}_q(\mathit{\text{xt}})E_q(\mathit{\text{yt}})={\sum }_{n=0}^{\infty }{\mathcal{E}}_{n,q}(x,y)\frac{t^n}{{[n]}_q!}$	The 2D q-Euler Polynomials (Keleshteri & Mahmudov, 2015; Mahmudov & Keleshteri, 2013)
III	${\mathcal{A}}_q(t)=\frac{{[2]}_{q}t}{e_q(t)+1}$	$\frac{{[2]}_{q}t}{e_q(t)+1}{e}_q(\mathit{\text{xt}})E_q(\mathit{\text{yt}})={\sum }_{n=0}^{\infty }{\mathcal{G}}_{n,q}(x,y)\frac{t^n}{{[n]}_q!}$	The 2D q-Genocchi Polynomials (Keleshteri & Mahmudov, 2015; Mahmudov & Keleshteri, 2013)

Recently, various researcher have considered and investigated the generalized and hybrid type special polynomials, especially those are linked to q-Appell polynomials (see Bildirici, Acikgoz, & Araci, 2014; Cao, Huang, Fadel, & Arjika, 2023; Eweis & Mansour, 2022, 2023; Fadel, Raza, & Du, 2023; Ismail & Mansour, 2023; Khan & Nahid, 2018, 2019, 2022a, 2022b; Muhyi, 2022; Nahid & Choi, 2022; Nahid, Alam & Choi, 2023; Raza, Fadel, Nisar, & Zakarya, 2021; Riyasat & Khan, 2018; Riyasat, Khan, & Mahmudov, 2019; Riyasat, Khan, & Nahid, 2017; Riyasat, Nahid, & Khan, 2021; Srivastava, Khan, & Riyasat, 2019; Yasmin & Muhyi, 2019, 2020; Yasmin, Muhyi, & Araci, 2019).

This article is organized as: In Section 2, the q-modified-Laguerre-Appell polynomials ${}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)$ are presented via a generating function and their features are investigated. In Section 3, some members of the q-modified-Laguerre-Appell polynomials are investigated and their generating functions and series definitions are also addressed. In Section 4, the class of 2D q-modified-Laguerre-Appell polynomials is presented through the generating function and series definition. In Section 5, the graphical representations of certain members of the q-modified-Laguerre-Appell family are plotted for appropriate index values.

2. q-modified-Laguerre-Appell polynomials

In this section, the q-modified-Laguerre-Appell polynomials are introduced by means of the generating function, series definition and determinant definition. Certain related identities are also used to develop some relations for these polynomials.

In order to create the generating function for the q-modified-Laguerre-Appell ${}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x),$ the following result must be proven:

Theorem 2.1.

The respective generating function and series definitions for the q-modified Laguerre-Appell polynomials ${}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)$ are defined as: (2.1) $${\mathcal{A}}_q(t)\frac{1}{{(1-t)}_q^m}{e}_q(\lambda \mathit{\text{xt}})=\sum _{n=0}^{\infty }{}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)\frac{t^n}{{\left[n\right]}_q!},0<\left|q\right|<1,\left|\lambda \mathit{\text{xt}}\right|<\frac{1}{1-q}.$$(2.1) and (2.2) $${}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)=\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q A_{k,q}{f}_{n-k,q}^{(m,\lambda )}(x),A_{0,q}\ne 0,$$(2.2) where ${\mathcal{A}}_q(t)$ is defined in equation (1.12)(1.12) $${\mathcal{A}}_q(t)=\sum _{n=0}^{\infty }{\mathcal{A}}_{n,q}\frac{t^n}{{\left[n\right]}_q!},{\mathcal{A}}_q(t)\ne 0;\mathrm{\hspace{1em}\hspace{1em}}{\mathcal{A}}_{0,q}=1.$$(1.12) and $f_{n,q}^{(m,\lambda )}(x)$ is defined by equation (1.9)(1.9) $$\frac{1}{{(t;q)}_m}{e}_q(\lambda \mathit{\text{xt}})=\sum _{n=0}^{\infty }{f}_{n,q}^{(m,\lambda )}(x)t^n,\mathrm{\hspace{1em}\hspace{1em}}\left|\lambda \mathit{\text{xt}}\right|<\frac{1}{1-q}$$(1.9) .

Proof.

Expanding the exponential function $e_q(\mathit{\text{xt}})$ on the left hand side of the equation (1.11)(1.11) $${\mathcal{A}}_q(t)e_q(\mathit{\text{xt}})=\sum _{n=0}^{\infty }{\mathcal{A}}_{n,q}(x)\frac{t^n}{{\left[n\right]}_q!},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\mathcal{A}}_q(t)\ne 0,{\mathcal{A}}_{0,q}=1.$$(1.11) , then substituting the powers of x i.e. $x^0,x^1,x^2,\dots ,x^n,$ with the appropriate polynomials $f_{0,q}^{(m,\lambda )}(x),f_{1,q}^{(m,\lambda )}(x),f_{2,q}^{(m,\lambda )}(x),$ $\dots ,f_{n,q}^{(m,\lambda )}(x)$ in the right hand side of the resultant equation, we have (2.3) $${\mathcal{A}}_q(t)(1+f_{1,q}^{(m,\lambda )}(x)\frac{t}{{\left[1\right]}_q!}+f_{2,q}^{(m,\lambda )}(x)\frac{t^2}{{\left[2\right]}_q!}+\dots +f_{n,q}^{(m,\lambda )}(x)\frac{t^n}{{\left[n\right]}_q!})=\sum _{n=0}^{\infty }{\mathcal{A}}_{n,q}(f_{1,q}^{(m,\lambda )}(x))\frac{t^n}{{\left[n\right]}_q!}.$$(2.3)

Furthermore, summing up the series in the left hand side and then using equation (1.9)(1.9) $$\frac{1}{{(t;q)}_m}{e}_q(\lambda \mathit{\text{xt}})=\sum _{n=0}^{\infty }{f}_{n,q}^{(m,\lambda )}(x)t^n,\mathrm{\hspace{1em}\hspace{1em}}\left|\lambda \mathit{\text{xt}}\right|<\frac{1}{1-q}$$(1.9) in the resultant equation, gives (2.4) $${\mathcal{A}}_q\left(t\right)\frac{1}{{(1-t)}_q^m}{e}_q\left(\lambda \mathit{\text{xt}}\right)=\sum _{n=0}^{\infty }{\mathcal{A}}_{n,q}\left(f_{1,q}^{(m,\lambda )}\right(x\left)\right)\frac{t^n}{{\left[n\right]}_q!}.$$(2.4)

Finally, denote the result of q-modified-Laguerre-Appell polynomials on the right side of the preceding equation by $${\mathcal{A}}_{n,q}(f_{1,q}^{(m,\lambda )}(x)){=}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x),$$ then simplifying the resultant equation, we reach at assertion (2.1)(2.1) $${\mathcal{A}}_q(t)\frac{1}{{(1-t)}_q^m}{e}_q(\lambda \mathit{\text{xt}})=\sum _{n=0}^{\infty }{}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)\frac{t^n}{{\left[n\right]}_q!},0<\left|q\right|<1,\left|\lambda \mathit{\text{xt}}\right|<\frac{1}{1-q}.$$(2.1) .

In view of equations (1.9)(1.9) $$\frac{1}{{(t;q)}_m}{e}_q(\lambda \mathit{\text{xt}})=\sum _{n=0}^{\infty }{f}_{n,q}^{(m,\lambda )}(x)t^n,\mathrm{\hspace{1em}\hspace{1em}}\left|\lambda \mathit{\text{xt}}\right|<\frac{1}{1-q}$$(1.9) and (1.12)(1.12) $${\mathcal{A}}_q(t)=\sum _{n=0}^{\infty }{\mathcal{A}}_{n,q}\frac{t^n}{{\left[n\right]}_q!},{\mathcal{A}}_q(t)\ne 0;\mathrm{\hspace{1em}\hspace{1em}}{\mathcal{A}}_{0,q}=1.$$(1.12) , and using the Cauchy product rule, equation (2.1)(2.1) $${\mathcal{A}}_q(t)\frac{1}{{(1-t)}_q^m}{e}_q(\lambda \mathit{\text{xt}})=\sum _{n=0}^{\infty }{}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)\frac{t^n}{{\left[n\right]}_q!},0<\left|q\right|<1,\left|\lambda \mathit{\text{xt}}\right|<\frac{1}{1-q}.$$(2.1) can be written as $$\sum _{n=0}^{\infty }\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q{\mathcal{A}}_{k,q} f_{n-k,q}^{(m,\lambda )}(x)t^n=\sum _{n=0}^{\infty }{}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)t^n.$$

Upon comparing the coefficients of equal powers of t on both sides of the above equation, we reach at relation (2.2)(2.2) $${}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)=\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q A_{k,q}{f}_{n-k,q}^{(m,\lambda )}(x),A_{0,q}\ne 0,$$(2.2) . □

Remark 2.1.

Since for $q\to 1^{-},$ the q-modified-Laguerre polynomials $f_{n,q}^{(m,\lambda )}(x)$ reduce to the modified Laguerre polynomials $f_n^{(m,\lambda )}(x)$ (Srivastava & Manocha, 1984). Therefore, for $q\to 1^{-},$ the q-modified-Laguerre-Appell polynomials ${}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)$ reduce to the modified-Laguerre-Appell polynomials ${}_f{\mathcal{A}}_n^{(m,\lambda )}(x)$ such that $${\mathcal{A}}_n(f_1^{(m,\lambda )}(x)){=}_f{\mathcal{A}}_n^{(m,\lambda )}(x).$$

Thus, taking $q\to 1^{-}$ in equations (2.1)(2.1) $${\mathcal{A}}_q(t)\frac{1}{{(1-t)}_q^m}{e}_q(\lambda \mathit{\text{xt}})=\sum _{n=0}^{\infty }{}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)\frac{t^n}{{\left[n\right]}_q!},0<\left|q\right|<1,\left|\lambda \mathit{\text{xt}}\right|<\frac{1}{1-q}.$$(2.1) and (2.2)(2.2) $${}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)=\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q A_{k,q}{f}_{n-k,q}^{(m,\lambda )}(x),A_{0,q}\ne 0,$$(2.2) , we get the following generating function and series definition for the modified-Laguerre-Appell polynomials ${}_f{\mathcal{A}}_n^{(m,\lambda )}\left(x\right).$ (2.5) $$\mathcal{A}\left(t\right)\frac{1}{{(1-t)}^m}\text{exp}\hspace{0.17em}\left(\lambda \mathit{\text{xt}}\right)=\sum _{n=0}^{\infty }{}_f{\mathcal{A}}_n^{(m,\lambda )}\left(x\right)\frac{t^n}{n!}$$(2.5) and $${}_f{\mathcal{A}}_n^{(m,\lambda )}(x)=\sum _{k=0}^n\left(\begin{array}{c}n\\ k\end{array}\right) A_k{f}_{n-k}^{(m,\lambda )}(x),A_0\ne 0.$$ respectively.

The determinant representations of many classical and hybrid special polynomials have been investigated by many researchers (see Eweis et al., 2023; Keleshteri & Mahmudov, 2015; Khan & Nahid, 2018; Riyasat et al., 2019). Here, the determinant definition for qMLAP ${}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)$ is constructed.

Theorem 2.2.

The q-modified-Laguerre-Appell polynomials qMLAP ${}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)$ of degree n are defined by (2.6) $${}_f{\mathcal{A}}_{0,q}^{(m,\lambda )}(x)=\frac{1}{{\mathcal{B}}_{0,q}}$$(2.6) (2.7) $$\begin{array}{l}{}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}\left(x\right)=\frac{{(-1)}^n}{{\left({\mathcal{B}}_{0,q}\right)}^{n+1}}\\ \times \left|\begin{array}{cccccc}1& f_{1,q}^{(m,\lambda )}\left(x\right)& f_{2,q}^{(m,\lambda )}\left(x\right)& \cdots & f_{n-1,q}^{(m,\lambda )}\left(x\right)& f_{n,q}^{(m,\lambda )}\left(x\right)\\ & & & & & \\ {\mathcal{B}}_{0,q}& {\mathcal{B}}_{1,q}& {\mathcal{B}}_{2,q}& \cdots & {\mathcal{B}}_{n-1,q}& {\mathcal{B}}_{n,q}\\ & & & & & \\ 0& {\mathcal{B}}_{0,q}& {\left[\begin{array}{c}2\\ 1\end{array}\right]}_q{\mathcal{B}}_{1,q}& \cdots & {\left[\begin{array}{c}n-1\\ 1\end{array}\right]}_q{\mathcal{B}}_{n-2,q}& {\left[\begin{array}{c}n\\ 1\end{array}\right]}_q{\mathcal{B}}_{n-1,q}\\ & & & & & \\ 0& 0& {\mathcal{B}}_{0,q}& \cdots & {\left[\begin{array}{c}n-1\\ 2\end{array}\right]}_q{\mathcal{B}}_{n-3,q}& {\left[\begin{array}{c}n\\ 2\end{array}\right]}_q{\mathcal{B}}_{n-2,q}\\ & & & & & \\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ & & & & & \\ 0& 0& 0& \cdots & {\mathcal{B}}_{0,q}& {\left[\begin{array}{c}n\\ n-1\end{array}\right]}_q{\mathcal{B}}_{1,q}\end{array}\right|\end{array}$$(2.7) $${\mathcal{B}}_{n,q}=-\frac{1}{{\mathcal{A}}_{n,q}}(\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q {\mathcal{A}}_{k,q} {\mathcal{B}}_{n-k,q}),\hspace{1em}n=1,2,3,\dots ,$$ where $B_{0,q}\ne 0,$ ${\mathcal{B}}_{0,q},{\mathcal{B}}_{1,q},\dots {\mathcal{B}}_{n,q}\in \mathbb{R}.$ ${\mathcal{B}}_{0,q}=\frac{1}{{\mathcal{A}}_{0,q}}$ and $f_{n,q}^{(m,\lambda )}\left(x\right),n=0,1,2,\dots ,$ are the q-modified-Laguerre polynomials of degree n.

Proof.

Consider ${}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)$ to be the sequence of the qMLAP specified by equation (2.1)(2.1) $${\mathcal{A}}_q(t)\frac{1}{{(1-t)}_q^m}{e}_q(\lambda \mathit{\text{xt}})=\sum _{n=0}^{\infty }{}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)\frac{t^n}{{\left[n\right]}_q!},0<\left|q\right|<1,\left|\lambda \mathit{\text{xt}}\right|<\frac{1}{1-q}.$$(2.1) and ${\mathcal{A}}_{n,q},{\mathcal{B}}_{n,q}$ to be the sequences exist two numerical sequences in such a way that (2.8) $${\mathcal{A}}_q(t)=\hspace{0.17em}{\mathcal{A}}_{0,q}+{\mathcal{A}}_{1,q}\frac{t}{{\left[1\right]}_q!}+{\mathcal{A}}_{2,q}\frac{t^2}{{\left[2\right]}_q!}+\dots +{\mathcal{A}}_{n,q}\frac{t^n}{{\left[n\right]}_q!}+\dots ,A_{0,q}\ne 0,$$(2.8) (2.9) $${\widehat{\mathcal{A}}}_q(t)=\hspace{0.17em}{\mathcal{B}}_{0,q}+{\mathcal{B}}_{1,q}\frac{t}{{\left[1\right]}_q!}+{\mathcal{B}}_{2,q}\frac{t^2}{{\left[2\right]}_q!}+\dots +{\mathcal{B}}_{n,q}\frac{t^n}{{\left[n\right]}_q!}+\dots ,B_{0,q}\ne 0$$(2.9) fulfilling (2.10) $${\mathcal{A}}_q(t){\widehat{\mathcal{A}}}_q(t)=1,$$(2.10)

Therefore, applying the Cauchy product rule for the preceding equation, yields (2.11) $$\begin{array}{l}{\mathcal{A}}_q\left(t\right){\widehat{\mathcal{A}}}_q\left(t\right)=\sum _{n=0}^{\infty }{\mathcal{A}}_{n,q}\frac{t^n}{{\left[n\right]}_q!}\sum _{k=0}^{\infty }{\mathcal{B}}_{k,q}\frac{t^k}{{\left[k\right]}_q!}\\ =\sum _{k=0}^{\infty }\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q{\mathcal{A}}_{k,q} {\mathcal{B}}_{n-k,q}\frac{t^n}{{\left[n\right]}_q!},\end{array}$$(2.11) accordingly (2.12) $$\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q{A}_{k,q} {\mathcal{B}}_{n-k,q}=\{\begin{array}{cc}1,\mathrm{\hspace{1em}\hspace{1em}}\mathit{\text{if}} n=0,\hfill & \hfill \\ 0,\mathrm{\hspace{1em}\hspace{1em}}\mathit{\text{if}} n>0.\hfill & \hfill \end{array}$$(2.12)

Particularly (2.13) $$\{\begin{array}{cc}{\mathcal{B}}_{0,q}=\frac{1}{{\mathcal{A}}_{0,q}},\hfill & ,\hfill \\ {\mathcal{B}}_{n,q}=-\frac{1}{{\mathcal{A}}_{0,q}}(\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q{\mathcal{A}}_{k,q} {\mathcal{B}}_{n-k,q},),\mathrm{\hspace{1em}}n=1,2,\dots \hfill & \hfill \end{array}$$(2.13)

Next, multiplying both sides of equation (2.1)(2.1) $${\mathcal{A}}_q(t)\frac{1}{{(1-t)}_q^m}{e}_q(\lambda \mathit{\text{xt}})=\sum _{n=0}^{\infty }{}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)\frac{t^n}{{\left[n\right]}_q!},0<\left|q\right|<1,\left|\lambda \mathit{\text{xt}}\right|<\frac{1}{1-q}.$$(2.1) by ${\widehat{\mathcal{A}}}_q(t),$ we get (2.14) $${\mathcal{A}}_q(t){\widehat{\mathcal{A}}}_q(t)\frac{1}{{(1-t)}_q^m}{e}_q(\lambda \mathit{\text{xt}})={\widehat{\mathcal{A}}}_q(t)\sum _{n=0}^{\infty }{}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)\frac{t^n}{{\left[n\right]}_q!}.$$(2.14)

Using equations (2.1)(2.1) $${\mathcal{A}}_q(t)\frac{1}{{(1-t)}_q^m}{e}_q(\lambda \mathit{\text{xt}})=\sum _{n=0}^{\infty }{}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)\frac{t^n}{{\left[n\right]}_q!},0<\left|q\right|<1,\left|\lambda \mathit{\text{xt}}\right|<\frac{1}{1-q}.$$(2.1) , (2.9)(2.9) $${\widehat{\mathcal{A}}}_q(t)=\hspace{0.17em}{\mathcal{B}}_{0,q}+{\mathcal{B}}_{1,q}\frac{t}{{\left[1\right]}_q!}+{\mathcal{B}}_{2,q}\frac{t^2}{{\left[2\right]}_q!}+\dots +{\mathcal{B}}_{n,q}\frac{t^n}{{\left[n\right]}_q!}+\dots ,B_{0,q}\ne 0$$(2.9) and (2.12)(2.12) $$\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q{A}_{k,q} {\mathcal{B}}_{n-k,q}=\{\begin{array}{cc}1,\mathrm{\hspace{1em}\hspace{1em}}\mathit{\text{if}} n=0,\hfill & \hfill \\ 0,\mathrm{\hspace{1em}\hspace{1em}}\mathit{\text{if}} n>0.\hfill & \hfill \end{array}$$(2.12) , the preceding equation yields (2.15) $$\sum _{n=0}^{\infty }{f}_{n,q}^{(m,\lambda )}(x)t^n=\sum _{n=0}^{\infty }{\mathcal{B}}_{n,q}\frac{t^n}{{\left[n\right]}_q!}\sum _{n=0}^{\infty }{}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)\frac{t^n}{{\left[n\right]}_q!}.$$(2.15)

When we apply the Cauchy product rule to the two series on the right-hand side of (2.15)(2.15) $$\sum _{n=0}^{\infty }{f}_{n,q}^{(m,\lambda )}(x)t^n=\sum _{n=0}^{\infty }{\mathcal{B}}_{n,q}\frac{t^n}{{\left[n\right]}_q!}\sum _{n=0}^{\infty }{}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)\frac{t^n}{{\left[n\right]}_q!}.$$(2.15) , we get the following infinite system for the unknowns ${}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x):$ (2.16) $$\{\begin{array}{ll} {\mathcal{B}}_{0,q}{ }_f{\mathcal{A}}_{0,q}^{(m,\lambda )}(x)=1,\hfill & \hfill \\ \hfill & \hfill \\ {\mathcal{B}}_{1,q}{ }_f{\mathcal{A}}_{0,q}^{(m,\lambda )}(x)+{\mathcal{B}}_{0,q}{ }_f{\mathcal{A}}_{1,q}^{(m,\lambda )}(x)=f_{1,q}^{(m,\lambda )}(x)\hfill & ,\hfill \\ \hfill & \hfill \\ {\mathcal{B}}_{2,q}{ }_f{\mathcal{A}}_{0,q}^{(m,\lambda )}(x)+{\left[\begin{array}{c}2\\ 1\end{array}\right]}_q{\mathcal{B}}_{1,q}{ }_f{\mathcal{A}}_{1,q}^{(m,\lambda )}(x)+{\mathcal{B}}_{0,q}{ }_f{\mathcal{A}}_{2,q}^{(m,\lambda )}(x)=f_{2,q}^{(m,\lambda )}(x),\hfill & \hfill \\ \hfill & \hfill \\ ⋮\hfill & \hfill \\ \hfill & \hfill \\ {\mathcal{B}}_{n-1,q}{ }_f{\mathcal{A}}_{0,q}^{(m,\lambda )}(x)+{\left[\begin{array}{c}n-1\\ 1\end{array}\right]}_q{\mathcal{B}}_{n-2,q}{ }_f{\mathcal{A}}_{1,q}^{(m,\lambda )}(x)+\dots +{\mathcal{B}}_{0,q}{ }_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)=f_{n-1,q}^{(m,\lambda )}(x)\hfill & \hfill \\ \hfill & \hfill \\ {\mathcal{B}}_{n,q}{ }_f{\mathcal{A}}_{0,q}^{(m,\lambda )}(x)+{\left[\begin{array}{c}n\\ 1\end{array}\right]}_q{\mathcal{B}}_{n-1,q}{ }_f{\mathcal{A}}_{1,q}^{(m,\lambda )}(x)+\dots +{\mathcal{B}}_{0,q}{ }_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)=f_{n,q}^{(m,\lambda )}(x),\hfill & \hfill \\ ⋮\hfill & \hfill \end{array}$$(2.16)

Clearly, the first equation of system (2.16)(2.16) $$\{\begin{array}{ll} {\mathcal{B}}_{0,q}{ }_f{\mathcal{A}}_{0,q}^{(m,\lambda )}(x)=1,\hfill & \hfill \\ \hfill & \hfill \\ {\mathcal{B}}_{1,q}{ }_f{\mathcal{A}}_{0,q}^{(m,\lambda )}(x)+{\mathcal{B}}_{0,q}{ }_f{\mathcal{A}}_{1,q}^{(m,\lambda )}(x)=f_{1,q}^{(m,\lambda )}(x)\hfill & ,\hfill \\ \hfill & \hfill \\ {\mathcal{B}}_{2,q}{ }_f{\mathcal{A}}_{0,q}^{(m,\lambda )}(x)+{\left[\begin{array}{c}2\\ 1\end{array}\right]}_q{\mathcal{B}}_{1,q}{ }_f{\mathcal{A}}_{1,q}^{(m,\lambda )}(x)+{\mathcal{B}}_{0,q}{ }_f{\mathcal{A}}_{2,q}^{(m,\lambda )}(x)=f_{2,q}^{(m,\lambda )}(x),\hfill & \hfill \\ \hfill & \hfill \\ ⋮\hfill & \hfill \\ \hfill & \hfill \\ {\mathcal{B}}_{n-1,q}{ }_f{\mathcal{A}}_{0,q}^{(m,\lambda )}(x)+{\left[\begin{array}{c}n-1\\ 1\end{array}\right]}_q{\mathcal{B}}_{n-2,q}{ }_f{\mathcal{A}}_{1,q}^{(m,\lambda )}(x)+\dots +{\mathcal{B}}_{0,q}{ }_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)=f_{n-1,q}^{(m,\lambda )}(x)\hfill & \hfill \\ \hfill & \hfill \\ {\mathcal{B}}_{n,q}{ }_f{\mathcal{A}}_{0,q}^{(m,\lambda )}(x)+{\left[\begin{array}{c}n\\ 1\end{array}\right]}_q{\mathcal{B}}_{n-1,q}{ }_f{\mathcal{A}}_{1,q}^{(m,\lambda )}(x)+\dots +{\mathcal{B}}_{0,q}{ }_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)=f_{n,q}^{(m,\lambda )}(x),\hfill & \hfill \\ ⋮\hfill & \hfill \end{array}$$(2.16) , gives the first assertion (2.6)(2.6) $${}_f{\mathcal{A}}_{0,q}^{(m,\lambda )}(x)=\frac{1}{{\mathcal{B}}_{0,q}}$$(2.6) .

The lower triangular coefficient matrix of system (2.6)(2.6) $${}_f{\mathcal{A}}_{0,q}^{(m,\lambda )}(x)=\frac{1}{{\mathcal{B}}_{0,q}}$$(2.6) enables us to extract the unknowns ${}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)$ by applying the Cramer rule to the first $n+1$ equation (2.16)(2.16) $$\{\begin{array}{ll} {\mathcal{B}}_{0,q}{ }_f{\mathcal{A}}_{0,q}^{(m,\lambda )}(x)=1,\hfill & \hfill \\ \hfill & \hfill \\ {\mathcal{B}}_{1,q}{ }_f{\mathcal{A}}_{0,q}^{(m,\lambda )}(x)+{\mathcal{B}}_{0,q}{ }_f{\mathcal{A}}_{1,q}^{(m,\lambda )}(x)=f_{1,q}^{(m,\lambda )}(x)\hfill & ,\hfill \\ \hfill & \hfill \\ {\mathcal{B}}_{2,q}{ }_f{\mathcal{A}}_{0,q}^{(m,\lambda )}(x)+{\left[\begin{array}{c}2\\ 1\end{array}\right]}_q{\mathcal{B}}_{1,q}{ }_f{\mathcal{A}}_{1,q}^{(m,\lambda )}(x)+{\mathcal{B}}_{0,q}{ }_f{\mathcal{A}}_{2,q}^{(m,\lambda )}(x)=f_{2,q}^{(m,\lambda )}(x),\hfill & \hfill \\ \hfill & \hfill \\ ⋮\hfill & \hfill \\ \hfill & \hfill \\ {\mathcal{B}}_{n-1,q}{ }_f{\mathcal{A}}_{0,q}^{(m,\lambda )}(x)+{\left[\begin{array}{c}n-1\\ 1\end{array}\right]}_q{\mathcal{B}}_{n-2,q}{ }_f{\mathcal{A}}_{1,q}^{(m,\lambda )}(x)+\dots +{\mathcal{B}}_{0,q}{ }_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)=f_{n-1,q}^{(m,\lambda )}(x)\hfill & \hfill \\ \hfill & \hfill \\ {\mathcal{B}}_{n,q}{ }_f{\mathcal{A}}_{0,q}^{(m,\lambda )}(x)+{\left[\begin{array}{c}n\\ 1\end{array}\right]}_q{\mathcal{B}}_{n-1,q}{ }_f{\mathcal{A}}_{1,q}^{(m,\lambda )}(x)+\dots +{\mathcal{B}}_{0,q}{ }_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)=f_{n,q}^{(m,\lambda )}(x),\hfill & \hfill \\ ⋮\hfill & \hfill \end{array}$$(2.16) . Thus, we may conclude $${}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)=\frac{\left|\begin{array}{cccccc}{\mathcal{B}}_{0,q}& 0& 0& \cdots & 0& 1\\ & & & & & \\ \mathrm{\hspace{1em}\hspace{1em}}{\mathcal{B}}_{1,q}& {\mathcal{B}}_{0,q}& 0& \cdots & 0& f_{1,q}^{(m,\lambda )}(x)\\ & & & & & \\ {\mathcal{B}}_{2,q}& {\left[\begin{array}{c}2\\ 1\end{array}\right]}_q{\mathcal{B}}_{1,q}& {\mathcal{B}}_{0,q}& \cdots & 0& f_{2,q}^{(m,\lambda )}(x)\\ & & & & & \\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ & & & & & \\ {\mathcal{B}}_{n-1,q}& {\left[\begin{array}{c}n-1\\ 1\end{array}\right]}_q{\mathcal{B}}_{n-2,q}& {\left[\begin{array}{c}n-1\\ 2\end{array}\right]}_q{\mathcal{B}}_{n-3,q}& \cdots & {\mathcal{B}}_{0,q}& f_{n-1,q}^{(m,\lambda )}(x)\\ & & & & & \\ {\mathcal{B}}_{n,q}& {\left[\begin{array}{c}n\\ 1\end{array}\right]}_q{\mathcal{B}}_{n-1,q}& {\left[\begin{array}{c}n\\ 2\end{array}\right]}_q{\mathcal{B}}_{n-2,q}& \cdots & {\left[\begin{array}{c}n\\ n-1\end{array}\right]}_q{\mathcal{B}}_{1,q}& f_{n,q}^{(m,\lambda )}(x)\end{array}\right|}{\left|\begin{array}{cccccc}{\mathcal{B}}_{0,q}& 0& 0& \cdots & 0& 1\\ & & & & & \\ {\mathcal{B}}_{1,q}& {\mathcal{B}}_{0,q}& 0& \cdots & 0& 0\\ & & & & & \\ {\mathcal{B}}_{2,q}& {\left[\begin{array}{c}2\\ 1\end{array}\right]}_q{\mathcal{B}}_{1,q}& {\mathcal{B}}_{0,q}& \cdots & 0& 0\\ & & & & & \\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ & & & & & \\ {\mathcal{B}}_{n-1,q}& {\left[\begin{array}{c}n-1\\ 1\end{array}\right]}_q{\mathcal{B}}_{n-2,q}& {\left[\begin{array}{c}n-1\\ 2\end{array}\right]}_q{\mathcal{B}}_{n-3,q}& \cdots & {\mathcal{B}}_{0,q}& 0\\ & & & & & \\ {\mathcal{B}}_{n,q}& {\left[\begin{array}{c}n\\ 1\end{array}\right]}_q{\mathcal{B}}_{n-1,q}& {\left[\begin{array}{c}n\\ 2\end{array}\right]}_q{\mathcal{B}}_{n-2,q}& \cdots & {\left[\begin{array}{c}n\\ n-1\end{array}\right]}_q{\mathcal{B}}_{1,q}& {\mathcal{B}}_{0,q}\end{array}\right|},$$ where $n=1,2,3\dots ,$ which on expanding the determinant and taking the transpose of the determinant in the numerator, yields to (2.17) $${}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)=\frac{1}{{({\mathcal{B}}_{0,q})}^{n+1}}\times \left|\begin{array}{cccccc}{\mathcal{B}}_{0,q}& {\mathcal{B}}_{1,q}& {\mathcal{B}}_{2,q}& \cdots & {\mathcal{B}}_{n-1,q}& {\mathcal{B}}_{n,q}\\ & & & & & \\ 0& {\mathcal{B}}_{0,q}& {\left[\begin{array}{c}2\\ 1\end{array}\right]}_q{\mathcal{B}}_{1,q}& \cdots & {\left[\begin{array}{c}n-1\\ 1\end{array}\right]}_q{\mathcal{B}}_{n-2,q}& {\left[\begin{array}{c}n\\ 1\end{array}\right]}_q{\mathcal{B}}_{n-1,q}\\ & & & & & \\ 0& 0& {\mathcal{B}}_{0,q}& \cdots & {\left[\begin{array}{c}n-1\\ 2\end{array}\right]}_q{\mathcal{B}}_{n-3,q}& {\left[\begin{array}{c}n\\ 2\end{array}\right]}_q{\mathcal{B}}_{n-2,q}\\ & & & & & \\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ & & & & & \\ 0& 0& 0& \cdots & {\mathcal{B}}_{0,q}& {\left[\begin{array}{c}n\\ n-1\end{array}\right]}_q{\mathcal{B}}_{1,q}\\ & & & & & \\ 1& f_{1,q}^{(m,\lambda )}(x)& f_{2,q}^{(m,\lambda )}(x)& \cdots & f_{n-1,q}^{(m,\lambda )}(x)& f_{n,q}^{(m,\lambda )}(x)\end{array}\right|$$(2.17)

After n circular row exchanges, i.e. after changing the $i^{\mathit{\text{th}}}$ row to the ${(i+1)}^{\mathit{\text{th}}}$ location for $i=1,2,3,\dots ,n-1,$ the assertion (2.7)(2.7) $$\begin{array}{l}{}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}\left(x\right)=\frac{{(-1)}^n}{{\left({\mathcal{B}}_{0,q}\right)}^{n+1}}\\ \times \left|\begin{array}{cccccc}1& f_{1,q}^{(m,\lambda )}\left(x\right)& f_{2,q}^{(m,\lambda )}\left(x\right)& \cdots & f_{n-1,q}^{(m,\lambda )}\left(x\right)& f_{n,q}^{(m,\lambda )}\left(x\right)\\ & & & & & \\ {\mathcal{B}}_{0,q}& {\mathcal{B}}_{1,q}& {\mathcal{B}}_{2,q}& \cdots & {\mathcal{B}}_{n-1,q}& {\mathcal{B}}_{n,q}\\ & & & & & \\ 0& {\mathcal{B}}_{0,q}& {\left[\begin{array}{c}2\\ 1\end{array}\right]}_q{\mathcal{B}}_{1,q}& \cdots & {\left[\begin{array}{c}n-1\\ 1\end{array}\right]}_q{\mathcal{B}}_{n-2,q}& {\left[\begin{array}{c}n\\ 1\end{array}\right]}_q{\mathcal{B}}_{n-1,q}\\ & & & & & \\ 0& 0& {\mathcal{B}}_{0,q}& \cdots & {\left[\begin{array}{c}n-1\\ 2\end{array}\right]}_q{\mathcal{B}}_{n-3,q}& {\left[\begin{array}{c}n\\ 2\end{array}\right]}_q{\mathcal{B}}_{n-2,q}\\ & & & & & \\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ & & & & & \\ 0& 0& 0& \cdots & {\mathcal{B}}_{0,q}& {\left[\begin{array}{c}n\\ n-1\end{array}\right]}_q{\mathcal{B}}_{1,q}\end{array}\right|\end{array}$$(2.7) is reached.

The determinant definition for the modified Laguerre-Appell polynomials ${}_f{\mathcal{A}}_n^{(m,\lambda )}(x)$ can be found from Theorem 2.2 by taking $q\to 1^{-}.$

Corollary 2.1.

The modified Laguerre-Appell polynomials of degree n ${}_f{\mathcal{A}}_n^{(m,\lambda )}(x)$ are defined by (2.18) $${}_f{\mathcal{A}}_0^{(m,\lambda )}(x)=\frac{1}{{\mathcal{B}}_0}$$(2.18) (2.19) $$\begin{array}{l}{}_f{\mathcal{A}}_n^{(m,\lambda )}(x)=\frac{{(-1)}^n}{{({\mathcal{B}}_0)}^{n+1}}\\ \times \left|\begin{array}{cccccc}1& f_1^{(m,\lambda )}(x)& f_2^{(m,\lambda )}(x)& \cdots & f_{n-1}^{(m,\lambda )}(x)& f_n^{(m,\lambda )}(x)\\ & & & & & \\ {\mathcal{B}}_0& {\mathcal{B}}_1& {\mathcal{B}}_2& \cdots & {\mathcal{B}}_{n-1}& {\mathcal{B}}_n\\ & & & & & \\ 0& {\mathcal{B}}_0& \left[\begin{array}{c}2\\ 1\end{array}\right]{\mathcal{B}}_1& \cdots & \left[\begin{array}{c}n-1\\ 1\end{array}\right]{\mathcal{B}}_{n-2}& \left[\begin{array}{c}n\\ 1\end{array}\right]{\mathcal{B}}_{n-1}\\ & & & & & \\ 0& 0& {\mathcal{B}}_0& \cdots & \left[\begin{array}{c}n-1\\ 2\end{array}\right]{\mathcal{B}}_{n-3}& \left[\begin{array}{c}n\\ 2\end{array}\right]{\mathcal{B}}_{n-2}\\ & & & & & \\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ & & & & & \\ 0& 0& 0& \cdots & {\mathcal{B}}_0& \left[\begin{array}{c}n\\ n-1\end{array}\right]{\mathcal{B}}_1\\ & & & & & \end{array}\right|\end{array}$$(2.19) $${\mathcal{B}}_n=-\frac{1}{{\mathcal{A}}_0}(\sum _{k=0}^n\left[\begin{array}{c}n\\ k\end{array}\right] {\mathcal{A}}_k{\mathcal{B}}_{n-k}),n=1,2,3,\dots ,$$

Theorem 2.3.

The following identity for the qMLAP ${}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)$ holds true: (2.20) $${}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)=\frac{1}{{\mathcal{B}}_{0,p,q}}(f_{n,q}^{(m,\lambda )}(x)-\sum _{k=0}^{n-1}{\left[\begin{array}{c}n\\ k\end{array}\right]}_q {\mathcal{B}}_{n-k,q}{ }_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)),n=1,2,3,\dots .$$(2.20)

Proof.

We obtain the needed findings by expanding the determinant in equation (2.7)(2.7) $$\begin{array}{l}{}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}\left(x\right)=\frac{{(-1)}^n}{{\left({\mathcal{B}}_{0,q}\right)}^{n+1}}\\ \times \left|\begin{array}{cccccc}1& f_{1,q}^{(m,\lambda )}\left(x\right)& f_{2,q}^{(m,\lambda )}\left(x\right)& \cdots & f_{n-1,q}^{(m,\lambda )}\left(x\right)& f_{n,q}^{(m,\lambda )}\left(x\right)\\ & & & & & \\ {\mathcal{B}}_{0,q}& {\mathcal{B}}_{1,q}& {\mathcal{B}}_{2,q}& \cdots & {\mathcal{B}}_{n-1,q}& {\mathcal{B}}_{n,q}\\ & & & & & \\ 0& {\mathcal{B}}_{0,q}& {\left[\begin{array}{c}2\\ 1\end{array}\right]}_q{\mathcal{B}}_{1,q}& \cdots & {\left[\begin{array}{c}n-1\\ 1\end{array}\right]}_q{\mathcal{B}}_{n-2,q}& {\left[\begin{array}{c}n\\ 1\end{array}\right]}_q{\mathcal{B}}_{n-1,q}\\ & & & & & \\ 0& 0& {\mathcal{B}}_{0,q}& \cdots & {\left[\begin{array}{c}n-1\\ 2\end{array}\right]}_q{\mathcal{B}}_{n-3,q}& {\left[\begin{array}{c}n\\ 2\end{array}\right]}_q{\mathcal{B}}_{n-2,q}\\ & & & & & \\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ & & & & & \\ 0& 0& 0& \cdots & {\mathcal{B}}_{0,q}& {\left[\begin{array}{c}n\\ n-1\end{array}\right]}_q{\mathcal{B}}_{1,q}\end{array}\right|\end{array}$$(2.7) with respect to the $(n+1)\mathit{\text{th}}$ row and using the same method as in Keleshteri and Mahmudov (2015). □

Taking $q\to 1^{-}$ from Theorem 2.3, we get the following result for modified Laguerre-Appell polynomials ${}_{f^{(m,\lambda )}}{\mathcal{A}}_n(x):$

Corollary 2.2.

The following identity for MLAP ${}_f{\mathcal{A}}_n^{(m,\lambda )}(x)$ holds true: (2.21) $${}_f{\mathcal{A}}_n^{(m,\lambda )}(x)=\frac{1}{{\mathcal{B}}_0}(f_n^{(m,\lambda )}(x)-\sum _{k=0}^{n-1}\left[\begin{array}{c}n\\ k\end{array}\right] {\mathcal{B}}_{n-k}{ }_f{\mathcal{A}}_n^{(m,\lambda )}(x)),n=1,2,3,\dots ,$$(2.21)

3. Certain members of the q-modified-Laguerre-Appell family

In this section, we present some members of the q-modified-Laguerre-Appell family by choosing appropriate values for the function ${\mathcal{A}}_{n,q}(t).$

3.1. The q-modified-Laguerre-Bernoulli polynomials

Since, for ${\mathcal{A}}_q(t)$=$\frac{t}{e_q(t)-1},$ the q-Appell polynomials ${\mathcal{A}}_{n,q}(x)$ reduce to the q-Bernoulli polynomials ${\mathfrak{B}}_{n,q}(x)$ (Table 1 (I)). Therefore, for the choice of ${\mathcal{A}}_q(t),$ the qMLAP ${}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)$ reduce to qMLBP ${}_f{\mathfrak{B}}_{n,q}^{(m,\lambda )}(x),$ which are defined by means of the following generating function: (3.1) $$\frac{t}{e_q\left(t\right)-1}\frac{1}{{(1-t)}_q^m}{e}_q\left(\lambda \mathit{\text{xt}}\right)=\sum _{n=0}^{\infty }{}_f{\mathfrak{B}}_{n,q}^{(m,\lambda )}\left(x\right)\frac{t^n}{{\left[n\right]}_q!},\left|t\right|<2\pi ,\left|\lambda \mathit{\text{xt}}\right|<\frac{1}{1-q}.$$(3.1)

The qMLBP ${}_f{\mathfrak{B}}_{n,q}^{(m,\lambda )}(x)$ of degree n are defined by the series (3.2) $${}_f{\mathfrak{B}}_{n,q}^{(m,\lambda )}(x)=\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q {\mathfrak{B}}_{k,q} f_{n-k,q}^{(m,\lambda )}(x),\mathrm{\hspace{1em}\hspace{1em}}{\mathfrak{B}}_{0,q}\ne 0.$$(3.2)

The following identity for the qMLBP ${}_f{\mathfrak{B}}_{n,q}^{(m,\lambda )}(x)$ holds true: (3.3) $${}_f{\mathfrak{B}}_{n,q}^{(m,\lambda )}(x)=\frac{1}{{\mathcal{B}}_{0,q}}(f_{n,q}^{(m,\lambda )}(x)-\sum _{k=0}^{n-1}{\left[\begin{array}{c}n\\ k\end{array}\right]}_q {\mathcal{B}}_{n-k,q}{ }_{f^{(m,\lambda )}}{\mathfrak{B}}_{n,q}(x)),n=1,2,3,\dots .$$(3.3)

Furthermore, by taking ${\mathcal{B}}_{0,q}=1$ and ${\mathcal{B}}_{i,q}=\frac{1}{{[i+1]}_q},$ i = 1,2,3…, in equations (2.6)(2.6) $${}_f{\mathcal{A}}_{0,q}^{(m,\lambda )}(x)=\frac{1}{{\mathcal{B}}_{0,q}}$$(2.6) and (2.7)(2.7) $$\begin{array}{l}{}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}\left(x\right)=\frac{{(-1)}^n}{{\left({\mathcal{B}}_{0,q}\right)}^{n+1}}\\ \times \left|\begin{array}{cccccc}1& f_{1,q}^{(m,\lambda )}\left(x\right)& f_{2,q}^{(m,\lambda )}\left(x\right)& \cdots & f_{n-1,q}^{(m,\lambda )}\left(x\right)& f_{n,q}^{(m,\lambda )}\left(x\right)\\ & & & & & \\ {\mathcal{B}}_{0,q}& {\mathcal{B}}_{1,q}& {\mathcal{B}}_{2,q}& \cdots & {\mathcal{B}}_{n-1,q}& {\mathcal{B}}_{n,q}\\ & & & & & \\ 0& {\mathcal{B}}_{0,q}& {\left[\begin{array}{c}2\\ 1\end{array}\right]}_q{\mathcal{B}}_{1,q}& \cdots & {\left[\begin{array}{c}n-1\\ 1\end{array}\right]}_q{\mathcal{B}}_{n-2,q}& {\left[\begin{array}{c}n\\ 1\end{array}\right]}_q{\mathcal{B}}_{n-1,q}\\ & & & & & \\ 0& 0& {\mathcal{B}}_{0,q}& \cdots & {\left[\begin{array}{c}n-1\\ 2\end{array}\right]}_q{\mathcal{B}}_{n-3,q}& {\left[\begin{array}{c}n\\ 2\end{array}\right]}_q{\mathcal{B}}_{n-2,q}\\ & & & & & \\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ & & & & & \\ 0& 0& 0& \cdots & {\mathcal{B}}_{0,q}& {\left[\begin{array}{c}n\\ n-1\end{array}\right]}_q{\mathcal{B}}_{1,q}\end{array}\right|\end{array}$$(2.7) , we obtain the determinant definition of the qMLBP ${}_f{\mathfrak{B}}_{n,q}^{(m,\lambda )}(x).$

Definition 3.1.

The q-modified-Laguerre-Bernoulli polynomials qMLBP ${}_f{\mathfrak{B}}_{n,q}^{(m,\lambda )}(x)$ of degree n are defined by (3.4) $${}_f{\mathfrak{B}}_{0,q}^{(m,\lambda )}(x)=1$$(3.4) (3.5) $${}_f{\mathfrak{B}}_{n,q}^{(m,\lambda )}(x)={(-1)}^n\left|\begin{array}{cccccc}1& f_{1,q}^{(m,\lambda )}(x)& f_{2,q}^{(m,\lambda )}(x)& \cdots & f_{n-1,q}^{(m,\lambda )}(x)& f_{n,q}^{(m,\lambda )}(x)\\ & & & & & \\ 1& \frac{1}{{\left[2\right]}_q}& \frac{1}{{\left[3\right]}_q}& \cdots & \frac{1}{{\left[n\right]}_q}& \frac{1}{{[n+1]}_q}\\ & & & & & \\ 0& 1& {\left[\begin{array}{c}2\\ 1\end{array}\right]}_q\frac{1}{{\left[2\right]}_q}& \cdots & {\left[\begin{array}{c}n-1\\ k\end{array}\right]}_q\frac{1}{{[n-1]}_q}& {\left[\begin{array}{c}n\\ 1\end{array}\right]}_q\frac{1}{{\left[n\right]}_q}\\ & & & & & \\ 0& 0& 1& \cdots & {\left[\begin{array}{c}n-1\\ 2\end{array}\right]}_q\frac{1}{{[n-2]}_q}& {\left[\begin{array}{c}n\\ 2\end{array}\right]}_q\frac{1}{{[n-1]}_q}\\ & & & & & \\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ & & & & & \\ 0& 0& 0& \dots & 1& {\left[\begin{array}{c}n\\ n-1\end{array}\right]}_q\frac{1}{{\left[2\right]}_q}\end{array}\right|$$(3.5) where $f_{n,q}^{(m,\lambda )}(x),$ $n=0,1,2,\dots$ are the q-modified-Laguerre polynomials of degree n.

3.2. The q-modified-Laguerre-Euler polynomials

Since, for ${\mathcal{A}}_q(t)$=$\frac{{[2]}_q}{e_q(t)+1},$ the q-Appell polynomials ${\mathcal{A}}_{n,q}(x)$ reduce to the q-Euler polynomials ${\mathcal{E}}_{n,q}(x)$ (Table 1 (II)). Therefore, for the choice of ${\mathcal{A}}_q(t),$ the qMLAP ${}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)$ reduce to qMLEP ${}_f{\mathcal{E}}_{n,q}^{(m,\lambda )}(x),$ which are defined by means of the following generating function: (3.6) $$\frac{{\left[2\right]}_q}{e_q(t)+1}\frac{1}{{(1-t)}_q^m}{e}_q(\lambda \mathit{\text{xt}})=\sum _{n=0}^{\infty }{}_f{\mathcal{E}}_{n,q}^{(m,\lambda )}(x)\frac{t^n}{{\left[n\right]}_q!},\left|t\right|<\pi ,\left|\lambda \mathit{\text{xt}}\right|<\frac{1}{1-q}.$$(3.6)

The qMLEP ${}_f{\mathcal{E}}_{n,q}^{(m,\lambda )}(x)$ of degree n are defined by the series (3.7) $${}_f{\mathcal{E}}_{n,q}^{(m,\lambda )}(x)=\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q {\mathcal{E}}_{n,q} f_{n-k,q}^{(m,\lambda )}(x),\mathrm{\hspace{1em}\hspace{1em}}{\mathcal{E}}_{0,q}\ne 0.$$(3.7)

The following identity for the qMLEP ${}_f{\mathcal{E}}_{n,q}^{(m,\lambda )}(x)$ holds true: (3.8) $${}_f{\mathcal{E}}_{n,q}^{(m,\lambda )}(x)=\frac{1}{{\mathcal{E}}_{0,q}}(f_{n,q}^{(m,\lambda )}-\sum _{k=0}^{n-1}{\left[\begin{array}{c}n\\ k\end{array}\right]}_q {\mathcal{B}}_{n-k,q}{ }_f{\mathcal{E}}_{n,q}^{(m,\lambda )}(x)),n=1,2,3,\dots .$$(3.8)

Furthermore, by taking ${\mathcal{B}}_{0,q}=1$ and ${\mathcal{B}}_{i,q}=\frac{1}{2},$ $i=1,2,3$…, in equations (2.6)(2.6) $${}_f{\mathcal{A}}_{0,q}^{(m,\lambda )}(x)=\frac{1}{{\mathcal{B}}_{0,q}}$$(2.6) and (2.7)(2.7) $$\begin{array}{l}{}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}\left(x\right)=\frac{{(-1)}^n}{{\left({\mathcal{B}}_{0,q}\right)}^{n+1}}\\ \times \left|\begin{array}{cccccc}1& f_{1,q}^{(m,\lambda )}\left(x\right)& f_{2,q}^{(m,\lambda )}\left(x\right)& \cdots & f_{n-1,q}^{(m,\lambda )}\left(x\right)& f_{n,q}^{(m,\lambda )}\left(x\right)\\ & & & & & \\ {\mathcal{B}}_{0,q}& {\mathcal{B}}_{1,q}& {\mathcal{B}}_{2,q}& \cdots & {\mathcal{B}}_{n-1,q}& {\mathcal{B}}_{n,q}\\ & & & & & \\ 0& {\mathcal{B}}_{0,q}& {\left[\begin{array}{c}2\\ 1\end{array}\right]}_q{\mathcal{B}}_{1,q}& \cdots & {\left[\begin{array}{c}n-1\\ 1\end{array}\right]}_q{\mathcal{B}}_{n-2,q}& {\left[\begin{array}{c}n\\ 1\end{array}\right]}_q{\mathcal{B}}_{n-1,q}\\ & & & & & \\ 0& 0& {\mathcal{B}}_{0,q}& \cdots & {\left[\begin{array}{c}n-1\\ 2\end{array}\right]}_q{\mathcal{B}}_{n-3,q}& {\left[\begin{array}{c}n\\ 2\end{array}\right]}_q{\mathcal{B}}_{n-2,q}\\ & & & & & \\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ & & & & & \\ 0& 0& 0& \cdots & {\mathcal{B}}_{0,q}& {\left[\begin{array}{c}n\\ n-1\end{array}\right]}_q{\mathcal{B}}_{1,q}\end{array}\right|\end{array}$$(2.7) , we obtain the determinant definition of the qMLEP ${}_f{\mathcal{E}}_{n,q}^{(m,\lambda )}(x).$

Definition 3.2.

The q-modified-Laguerre-Euler polynomials qMLBP ${}_f{\mathcal{E}}_{n,q}^{(m,\lambda )}(x)$ of degree n are defined by (3.9) $${}_f{\mathcal{E}}_{0,q}^{(m,\lambda )}(x)=1$$(3.9) (3.10) $${}_f{\mathcal{E}}_{n,q}^{(m,\lambda )}(x)={(-1)}^n\left|\begin{array}{cccccc}1& f_{1,q}^{(m,\lambda )}(x)& f_{2,q}^{(m,\lambda )}(x)& \cdots & f_{n-1,q}^{(m,\lambda )}(x)& f_{n,q}^{(m,\lambda )}(x)\\ & & & & & \\ 1& \frac{1}{2}& \frac{1}{2}& \cdots & \frac{1}{2}& \frac{1}{2}\\ & & & & & \\ 0& 1& {\left[\begin{array}{c}2\\ 1\end{array}\right]}_q\frac{1}{2}& \cdots & {\left[\begin{array}{c}n-1\\ 1\end{array}\right]}_q\frac{1}{2}& {\left[\begin{array}{c}n\\ 1\end{array}\right]}_q\frac{1}{2}\\ & & & & & \\ 0& 0& 1& \cdots & {\left[\begin{array}{c}n-1\\ 2\end{array}\right]}_q\frac{1}{2}& {\left[\begin{array}{c}n\\ 2\end{array}\right]}_q\frac{1}{2}\\ & & & & & \\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ & & & & & \\ 0& 0& 0& \dots & 1& {\left[\begin{array}{c}n\\ n-1\end{array}\right]}_q\frac{1}{2}\\ & & & & & \end{array}\right|,$$(3.10) where $f_{n,q}^{(m,\lambda )}(x),$ $n=0,1,2,\dots$ are the q-modified Laguerre polynomials of degree n.

3.3. The q-modified-Laguerre-Genocchi polynomials

Since, for ${\mathcal{A}}_q(t)$=$\frac{{[2]}_{q}t}{e_q(t)+1},$ the q-Appell polynomials ${\mathcal{A}}_{n,q}(x)$ reduce to the q-Genocchi polynomials ${\mathcal{G}}_{n,q}(x)$ (Table 1 (III)). Therefore, for the choice of ${\mathcal{A}}_q(t),$ the qMLAP ${}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)$ reduce to qMLGP ${}_f{\mathcal{G}}_{n,q}^{(m,\lambda )}(x),$ which are defined by means of the following generating function: (3.11) $$\frac{{\left[2\right]}_{q}t}{e_q(t)+1}\frac{1}{{(1-t)}_q^m}{e}_q(\lambda \mathit{\text{xt}})=\sum _{n=0}^{\infty }{}_f{\mathcal{G}}_{n,q}^{(m,\lambda )}(x)\frac{t^n}{{\left[n\right]}_q!}\left|t\right|<\pi ,\left|\lambda \mathit{\text{xt}}\right|<\frac{1}{1-q}.$$(3.11)

The qMLGP ${}_f{\mathcal{G}}_{n,q}^{(m,\lambda )}(x)$ of degree n are defined by the series (3.12) $${}_f{\mathcal{G}}_{n,q}^{(m,\lambda )}(x)=\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q {\mathcal{G}}_{k,q} f_{n-k,q}^{(m,\lambda )}(x),\mathrm{\hspace{1em}\hspace{1em}}{\mathcal{G}}_{0,q}\ne 0.$$(3.12)

The following identity for the qMLGP ${}_f{\mathcal{G}}_{n,q}^{(m,\lambda )}(x)$ holds true: (3.13) $${}_f{\mathcal{G}}_{n,q}^{(m,\lambda )}(x)=\frac{1}{{\mathcal{B}}_{0,q}}(f_{n,q}^{(m,\lambda )}(x)-\sum _{k=0}^{n-1}{\left[\begin{array}{c}n\\ k\end{array}\right]}_q{\mathcal{B}}_{n-k,q}{ }_f{\mathcal{G}}_{n,q}^{(m,\lambda )}(x)),n=1,2,3,\dots .$$(3.13)

In the next section, we introduce the 2D q-modified-Laguerre-Appell polynomials and study certain significant related properties.

4. 2D q-modified-Laguerre-Appell polynomials

The approach used in the previous section is further exploited to introduce the 2D q-modified-Laguerre-Appell polynomials (2DqMLAP). We focus on deriving its generating function and series definition.

In order to establish the generating function and series definition for the 2DqMLAP ${}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x,y),$ the following result are proved.

Theorem 4.1.

The following respective generating function and series definition for the 2D q-modified-Laguerre-Appell polynomials ${}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x,y)$ hold true (4.1) $${\mathcal{A}}_q\left(t\right)\frac{1}{{(1-t)}_q^m}{e}_q\left(\lambda \mathit{\text{xt}}\right)E_q\left(\mathit{\text{yt}}\right)=\sum _{n=0}^{\infty }{}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x,y)\frac{t^n}{{\left[n\right]}_q!},0<\left|q\right|<1,\left|\lambda \mathit{\text{xt}}\right|<\frac{1}{1-q},$$(4.1) (4.2) $${}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x,y)=\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q q^{\left({}_2^k\right)} y^k{}_f{\mathcal{A}}_{n-k,q}^{(m,\lambda )}(x).$$(4.2)

Proof.

Expanding the first exponential function $e_q(\lambda \mathit{\text{xt}})$ in the left hand side of equation (1.14)(1.14) $${\mathcal{A}}_q(t)e_q(\mathit{\text{xt}})E_q(\mathit{\text{yt}})=\sum _{n=0}^{\infty }{\mathcal{A}}_{n,q}(x,y)\frac{t^n}{{\left[n\right]}_q!},\mathrm{\hspace{1em}}{\mathcal{A}}_{n,q}={\mathcal{A}}_{n,q}(0,0).$$(1.14) , then replacing the powers of x i.e. $x^0,x^1,x^2,\dots ,x^n$ by the corresponding polynomials $f_{0,q}^{(m,\lambda )}(x),f_{1,q}^{(m,\lambda )}(x),$ $f_{2,q}^{(m,\lambda )}(x),\dots ,f_{n,q}^{(m,\lambda )}(x)$ in the left hand side and x by $f_{1,q}^{(m,\lambda )}(x)$ in the right hand side of the resultant equation, we have (4.3) $${\mathcal{A}}_q(t)(1+f_{1,q}^{(m,\lambda )}(x)\frac{t}{{\left[1\right]}_q!}+f_{2,q}^{(m,\lambda )}(x)\frac{t^2}{{\left[2\right]}_q!}+\dots +f_{n,q}^{(m,\lambda )}(x)\frac{t^n}{{\left[n\right]}_q!})E_q(\mathit{\text{yt}})=\sum _{n=0}^{\infty }{\mathcal{A}}_{n,q}(f_{1,q}^{(m,\lambda )}(x),y)\frac{t^n}{{\left[n\right]}_q!}.$$(4.3)

Now, summation up the series in the left hand side and then using equation (2.1)(2.1) $${\mathcal{A}}_q(t)\frac{1}{{(1-t)}_q^m}{e}_q(\lambda \mathit{\text{xt}})=\sum _{n=0}^{\infty }{}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)\frac{t^n}{{\left[n\right]}_q!},0<\left|q\right|<1,\left|\lambda \mathit{\text{xt}}\right|<\frac{1}{1-q}.$$(2.1) in the resultant equation, gives: (4.4) $${\mathcal{A}}_q\left(t\right)\frac{1}{{(1-t)}_q^m}{e}_q\left(\lambda \mathit{\text{xt}}\right)E_q\left(\mathit{\text{yt}}\right)=\sum _{n=0}^{\infty }{\mathcal{A}}_{n,q}\left(f_{1,q}^{(m,\lambda )}\right(x),y)\frac{t^n}{{\left[n\right]}_q!}.$$(4.4)

If denoting the resultant of 2DqMLAP in the right hand side of the above equation by ${}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x,y),$ gives $${\mathcal{A}}_{n,q}(f_{1,q}^{(m,\lambda )}(x),y){=}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x,y).$$

Thus, the assertion (4.1)(4.1) $${\mathcal{A}}_q\left(t\right)\frac{1}{{(1-t)}_q^m}{e}_q\left(\lambda \mathit{\text{xt}}\right)E_q\left(\mathit{\text{yt}}\right)=\sum _{n=0}^{\infty }{}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x,y)\frac{t^n}{{\left[n\right]}_q!},0<\left|q\right|<1,\left|\lambda \mathit{\text{xt}}\right|<\frac{1}{1-q},$$(4.1) is proved.

In view equations (1.5)(1.5) $$E_q(x)={(-x(1-q);q)}_{\infty }=\sum _{n=0}^{\infty } \frac{q^{\frac{n(n-1)}{2}}}{{(q;q)}_n}{(x(1-q))}^n,x\in \mathbb{C},$$(1.5) and (2.1)(2.1) $${\mathcal{A}}_q(t)\frac{1}{{(1-t)}_q^m}{e}_q(\lambda \mathit{\text{xt}})=\sum _{n=0}^{\infty }{}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)\frac{t^n}{{\left[n\right]}_q!},0<\left|q\right|<1,\left|\lambda \mathit{\text{xt}}\right|<\frac{1}{1-q}.$$(2.1) then equation (4.1)(4.1) $${\mathcal{A}}_q\left(t\right)\frac{1}{{(1-t)}_q^m}{e}_q\left(\lambda \mathit{\text{xt}}\right)E_q\left(\mathit{\text{yt}}\right)=\sum _{n=0}^{\infty }{}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x,y)\frac{t^n}{{\left[n\right]}_q!},0<\left|q\right|<1,\left|\lambda \mathit{\text{xt}}\right|<\frac{1}{1-q},$$(4.1) can be written as $$\sum _{n=0}^{\infty }{}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x)t^n\sum _{k=0}^{\infty }{q}^{\left({}_2^k\right)} y^k\frac{t^k}{{\left[k\right]}_q!}=\sum _{n=0}^{\infty }{}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x,y)t^n,$$ which on using the Cauchy product rule gives $$\sum _{n=0}^{\infty }\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q{q}^{\left({}_2^k\right)} y^k{}_f{\mathcal{A}}_{n-k,q}^{(m,\lambda )}(x)t^n=\sum _{n=0}^{\infty }{}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x,y)t^n.$$

Equating the coefficients of equal powers of t in the both sides of the above equation, gives assertion (4.2)(4.2) $${}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x,y)=\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q q^{\left({}_2^k\right)} y^k{}_f{\mathcal{A}}_{n-k,q}^{(m,\lambda )}(x).$$(4.2) .□

Remark 4.1.

Since for $q\to 1^{-},$ the (2DqMLAP) ${}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x,y)$ reduce to the 2D-modified-Laguerre-Appell polynomials ${}_f{\mathcal{A}}_n^{(m,\lambda )}(x,y)$ such that $${\mathcal{A}}_n(f_1^{(m,\lambda )}(x),y){=}_f{\mathcal{A}}_n^{(m,\lambda )}(x,y),$$ thus, taking $q\to 1^{-}$ in equation (4.1)(4.1) $${\mathcal{A}}_q\left(t\right)\frac{1}{{(1-t)}_q^m}{e}_q\left(\lambda \mathit{\text{xt}}\right)E_q\left(\mathit{\text{yt}}\right)=\sum _{n=0}^{\infty }{}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x,y)\frac{t^n}{{\left[n\right]}_q!},0<\left|q\right|<1,\left|\lambda \mathit{\text{xt}}\right|<\frac{1}{1-q},$$(4.1) , we get (4.5) $$\mathcal{A}\left(t\right)\frac{1}{{(1-t)}^m}e\left(\lambda \mathit{\text{xt}}\right)E\left(\mathit{\text{yt}}\right)=\sum _{n=0}^{\infty }{}_f{\mathcal{A}}_n^{(m,\lambda )}(x,y)\frac{t^n}{n!},$$(4.5) which is the generating function for the 2D-modified-Laguerre-Appell polynomials (2DMLAP) ${}_f{\mathcal{A}}_n^{(m,\lambda )}(x,y).$

For $q\to 1^{-},$ the series definition (4.2)(4.2) $${}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x,y)=\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q q^{\left({}_2^k\right)} y^k{}_f{\mathcal{A}}_{n-k,q}^{(m,\lambda )}(x).$$(4.2) of (2DqMLAP) ${}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x,y),$ gives the series definition of ${}_f{\mathcal{A}}_n^{(m,\lambda )}(x,y)$ as (4.6) $${}_f{\mathcal{A}}_n^{(m,\lambda )}(x,y)=\sum _{k=0}^n\left[\begin{array}{c}n\\ k\end{array}\right]y^k{}_f{\mathcal{A}}_{n-k}^{(m,\lambda )}(x).$$(4.6)

Certain members belonging to the 2Dq-Appell family are listed in Table 2. There exists a new class of special polynomials belonging to the 2Dq-modified-Laguerre-Appell family corresponding to each member belonging to the 2D q-Appell family. The generating functions and series definitions for the corresponding members belonging to the 2D q-modified-Laguerre-Appell family can be obtained by making suitable choices for the function ${\mathcal{A}}_q\left(t\right)$ in equations (4.1)(4.1) $${\mathcal{A}}_q\left(t\right)\frac{1}{{(1-t)}_q^m}{e}_q\left(\lambda \mathit{\text{xt}}\right)E_q\left(\mathit{\text{yt}}\right)=\sum _{n=0}^{\infty }{}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x,y)\frac{t^n}{{\left[n\right]}_q!},0<\left|q\right|<1,\left|\lambda \mathit{\text{xt}}\right|<\frac{1}{1-q},$$(4.1) and (4.2)(4.2) $${}_f{\mathcal{A}}_{n,q}^{(m,\lambda )}(x,y)=\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q q^{\left({}_2^k\right)} y^k{}_f{\mathcal{A}}_{n-k,q}^{(m,\lambda )}(x).$$(4.2) . The resultant members of the 2D q-modified-Laguerre Appell family along with their generating functions and series definitions are listed in Table 3.

Table 3. Some members of 2D q-modified-Laguerre-Appell polynomials.

S. No	${\mathcal{A}}_q(t)$	Generating functions	Series definitions	Polynomials
I	${\mathcal{A}}_q(t)=\frac{t}{e_q(t)-1}$	$\frac{t}{e_q(t)-1}\frac{1}{{(1-t)}_q^m}{e}_q(\lambda \mathit{\text{xt}})E_q(\mathit{\text{yt}})$	${}_f{\mathfrak{B}}_{n,q}^{(m,\lambda )}(x,y)={\sum }_{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q$	The 2D q-modified-Laguerre-Bernoulli Polynomials
		$={\sum }_{n=0}^{\infty }{}_f{\mathfrak{B}}_{n,q}^{(m,\lambda )}(x,y)\frac{t^n}{{\left[n\right]}_q!},$	$q^{({}_2^k)} y^k{}_f{\mathfrak{B}}_{n-k,q}^{(m,\lambda )}(x)$
		$\left|t\right|<2\pi ,\left|\lambda \mathit{\text{xt}}\right|<\frac{1}{1-q},$
II	${\mathcal{A}}_q(t)=\frac{{[2]}_q}{e_q(t)+1}$	$\frac{{[2]}_q}{e_q(t)+1}\frac{1}{{(1-t)}_q^m}{e}_q(\lambda \mathit{\text{xt}})E_q(\mathit{\text{yt}})$	${}_f{\mathcal{E}}_{n-k,q}^{(m,\lambda )}(x,y)={\sum }_{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q$	The 2D q-modified-Laguerre-Euler polynomials
		$={\sum }_{n=0}^{\infty }{}_f{\mathcal{E}}_{n,q}^{(m,\lambda )}(x,y)\frac{t^n}{{\left[n\right]}_q!},$	$q^{({}_2^k)} y^k{}_f{\mathcal{E}}_{n-k,q}^{(m,\lambda )}(x)$
		$\left|t\right|<\pi ,\left|\lambda \mathit{\text{xt}}\right|<\frac{1}{1-q}$

5. Graphical representations

Withe the help of Mathematica software, we plot some graphs of q-modified-Laguerre-Bernoulli polynomials ${}_f{\mathcal{B}}_{n,q}^{(m,\lambda )}(x),$ q-modified-Laguerre-Euler polynomials ${}_f{\mathcal{E}}_{n,q}^{(m,\lambda )}(x),$ 2D q-modified-Laguerre-Bernoulli polynomials ${}_f{\mathcal{B}}_{n,q}^{(m,\lambda )}(x,y)$ and 2D q-modified-Laguerre-Euler polynomials ${}_f{\mathcal{E}}_{n,q}^{(m,\lambda )}(x,y).$

Taking $q=\frac{1}{2},$ in the determinant definitions (3.5)(3.5) $${}_f{\mathfrak{B}}_{n,q}^{(m,\lambda )}(x)={(-1)}^n\left|\begin{array}{cccccc}1& f_{1,q}^{(m,\lambda )}(x)& f_{2,q}^{(m,\lambda )}(x)& \cdots & f_{n-1,q}^{(m,\lambda )}(x)& f_{n,q}^{(m,\lambda )}(x)\\ & & & & & \\ 1& \frac{1}{{\left[2\right]}_q}& \frac{1}{{\left[3\right]}_q}& \cdots & \frac{1}{{\left[n\right]}_q}& \frac{1}{{[n+1]}_q}\\ & & & & & \\ 0& 1& {\left[\begin{array}{c}2\\ 1\end{array}\right]}_q\frac{1}{{\left[2\right]}_q}& \cdots & {\left[\begin{array}{c}n-1\\ k\end{array}\right]}_q\frac{1}{{[n-1]}_q}& {\left[\begin{array}{c}n\\ 1\end{array}\right]}_q\frac{1}{{\left[n\right]}_q}\\ & & & & & \\ 0& 0& 1& \cdots & {\left[\begin{array}{c}n-1\\ 2\end{array}\right]}_q\frac{1}{{[n-2]}_q}& {\left[\begin{array}{c}n\\ 2\end{array}\right]}_q\frac{1}{{[n-1]}_q}\\ & & & & & \\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ & & & & & \\ 0& 0& 0& \dots & 1& {\left[\begin{array}{c}n\\ n-1\end{array}\right]}_q\frac{1}{{\left[2\right]}_q}\end{array}\right|$$(3.5) and (3.10)(3.10) $${}_f{\mathcal{E}}_{n,q}^{(m,\lambda )}(x)={(-1)}^n\left|\begin{array}{cccccc}1& f_{1,q}^{(m,\lambda )}(x)& f_{2,q}^{(m,\lambda )}(x)& \cdots & f_{n-1,q}^{(m,\lambda )}(x)& f_{n,q}^{(m,\lambda )}(x)\\ & & & & & \\ 1& \frac{1}{2}& \frac{1}{2}& \cdots & \frac{1}{2}& \frac{1}{2}\\ & & & & & \\ 0& 1& {\left[\begin{array}{c}2\\ 1\end{array}\right]}_q\frac{1}{2}& \cdots & {\left[\begin{array}{c}n-1\\ 1\end{array}\right]}_q\frac{1}{2}& {\left[\begin{array}{c}n\\ 1\end{array}\right]}_q\frac{1}{2}\\ & & & & & \\ 0& 0& 1& \cdots & {\left[\begin{array}{c}n-1\\ 2\end{array}\right]}_q\frac{1}{2}& {\left[\begin{array}{c}n\\ 2\end{array}\right]}_q\frac{1}{2}\\ & & & & & \\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ & & & & & \\ 0& 0& 0& \dots & 1& {\left[\begin{array}{c}n\\ n-1\end{array}\right]}_q\frac{1}{2}\\ & & & & & \end{array}\right|,$$(3.10) , we get the results mentioned in Table 4 for n = 0,1,2,3.

Table 4. The first four expressions of ${}_f{\mathfrak{B}}_{n,1/4}^{(2,1/3)}(x)$ and ${}_f{\mathcal{E}}_{n,1/4}^{(2,1/3)}(x).$

n	0	1	2	3
${}_f{\mathfrak{B}}_{n,1/4}^{(2,1/3)}(x)$	1	$\frac{1}{3}x$	$\frac{4}{45}{x}^2+\frac{11}{12}x-\frac{697}{336}$	$\frac{1}{2835}{x}^3+\frac{165}{540}{x}^2+\frac{191}{240}x+\frac{19161120}{36556800}$
${}_f{\mathcal{E}}_{n,1/4}^{(2,1/3)}(x)$	1	$\frac{1}{3}x+\frac{3}{4}$	$\frac{4}{45}{x}^2+\frac{1}{12}x+\frac{1}{16}$	$\frac{1}{2835}{x}^3-\frac{59}{900}{x}^2+\frac{945}{2400}x+\frac{706}{320}$

Now, with the help of Mathematica and using (3.5)(3.5) $${}_f{\mathfrak{B}}_{n,q}^{(m,\lambda )}(x)={(-1)}^n\left|\begin{array}{cccccc}1& f_{1,q}^{(m,\lambda )}(x)& f_{2,q}^{(m,\lambda )}(x)& \cdots & f_{n-1,q}^{(m,\lambda )}(x)& f_{n,q}^{(m,\lambda )}(x)\\ & & & & & \\ 1& \frac{1}{{\left[2\right]}_q}& \frac{1}{{\left[3\right]}_q}& \cdots & \frac{1}{{\left[n\right]}_q}& \frac{1}{{[n+1]}_q}\\ & & & & & \\ 0& 1& {\left[\begin{array}{c}2\\ 1\end{array}\right]}_q\frac{1}{{\left[2\right]}_q}& \cdots & {\left[\begin{array}{c}n-1\\ k\end{array}\right]}_q\frac{1}{{[n-1]}_q}& {\left[\begin{array}{c}n\\ 1\end{array}\right]}_q\frac{1}{{\left[n\right]}_q}\\ & & & & & \\ 0& 0& 1& \cdots & {\left[\begin{array}{c}n-1\\ 2\end{array}\right]}_q\frac{1}{{[n-2]}_q}& {\left[\begin{array}{c}n\\ 2\end{array}\right]}_q\frac{1}{{[n-1]}_q}\\ & & & & & \\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ & & & & & \\ 0& 0& 0& \dots & 1& {\left[\begin{array}{c}n\\ n-1\end{array}\right]}_q\frac{1}{{\left[2\right]}_q}\end{array}\right|$$(3.5) and (3.10)(3.10) $${}_f{\mathcal{E}}_{n,q}^{(m,\lambda )}(x)={(-1)}^n\left|\begin{array}{cccccc}1& f_{1,q}^{(m,\lambda )}(x)& f_{2,q}^{(m,\lambda )}(x)& \cdots & f_{n-1,q}^{(m,\lambda )}(x)& f_{n,q}^{(m,\lambda )}(x)\\ & & & & & \\ 1& \frac{1}{2}& \frac{1}{2}& \cdots & \frac{1}{2}& \frac{1}{2}\\ & & & & & \\ 0& 1& {\left[\begin{array}{c}2\\ 1\end{array}\right]}_q\frac{1}{2}& \cdots & {\left[\begin{array}{c}n-1\\ 1\end{array}\right]}_q\frac{1}{2}& {\left[\begin{array}{c}n\\ 1\end{array}\right]}_q\frac{1}{2}\\ & & & & & \\ 0& 0& 1& \cdots & {\left[\begin{array}{c}n-1\\ 2\end{array}\right]}_q\frac{1}{2}& {\left[\begin{array}{c}n\\ 2\end{array}\right]}_q\frac{1}{2}\\ & & & & & \\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ & & & & & \\ 0& 0& 0& \dots & 1& {\left[\begin{array}{c}n\\ n-1\end{array}\right]}_q\frac{1}{2}\\ & & & & & \end{array}\right|,$$(3.10) and the expressions of ${}_f{\mathfrak{B}}_{n,1/4}^{(2,1/3)}(x)$ and ${}_f{\mathcal{E}}_{n,1/4}^{(2,1/3)}(x)$ from Table 4, we get the graphs shown at Figure 1 and 2.

Figure 1. Graph of ${}_f{\mathfrak{B}}_{n,1/4}^{(2,1/3)}(x).$

Figure 2. Graph of ${}_f{\mathcal{E}}_{n,1/4}^{(2,1/3)}(x).$

Furthermore, setting $m=2,$ $\lambda =\frac{1}{3}$ and $q=\frac{1}{4}$ in the series definitions given in Table 3, we have (5.1) $${}_f{\mathfrak{B}}_{3,1/4}^{(2,1/3)}(x,y)=\frac{1}{2835}{x}^3+\frac{1}{64}{y}^3+\frac{165}{540}{x}^2+\frac{84}{720}{x}^{2}y+\frac{21}{192}x{y}^2+\frac{231}{192}\mathit{\text{xy}}+\frac{191}{240}x-\frac{14637}{86376}y+\frac{19161120}{36558800}$$(5.1) (5.2) $${}_f{\mathcal{E}}_{3,1/4}^{(2,1/3)}(x,y)=\frac{1}{2835}{x}^3+\frac{1}{64}{y}^3-\frac{59}{900}{x}^2+\frac{84}{720}{x}^{2}y+\frac{21}{192}x{y}^2+\frac{21}{192}\mathit{\text{xy}}+\frac{945}{240}x-\frac{21}{256}y+\frac{706}{320}.$$(5.2)

In view of equations (5.1)(5.1) $${}_f{\mathfrak{B}}_{3,1/4}^{(2,1/3)}(x,y)=\frac{1}{2835}{x}^3+\frac{1}{64}{y}^3+\frac{165}{540}{x}^2+\frac{84}{720}{x}^{2}y+\frac{21}{192}x{y}^2+\frac{231}{192}\mathit{\text{xy}}+\frac{191}{240}x-\frac{14637}{86376}y+\frac{19161120}{36558800}$$(5.1) and (5.2)(5.2) $${}_f{\mathcal{E}}_{3,1/4}^{(2,1/3)}(x,y)=\frac{1}{2835}{x}^3+\frac{1}{64}{y}^3-\frac{59}{900}{x}^2+\frac{84}{720}{x}^{2}y+\frac{21}{192}x{y}^2+\frac{21}{192}\mathit{\text{xy}}+\frac{945}{240}x-\frac{21}{256}y+\frac{706}{320}.$$(5.2) , we get the surface plots shown at Figure 3 and 4.

Figure 3. Surface plot of ${}_f{\mathcal{B}}_{3,1/4}^{(2,1/3)}(x,y).$

Figure 4. Surface plot of ${}_f{\mathcal{E}}_{3,1/4}^{(2,1/3)}(x,y).$

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No potential conflict of interest was reported by the authors.