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

References

• Marcellán F, Xu Y. On Sobolev orthogonal polynomials. Expo Math. 2015;33:308–352. doi: 10.1016/j.exmath.2014.10.002
   Web of Science ®Google Scholar
• Alfaro M, Marcellán F, Rezola ML. Orthogonal polynomials on Sobolev spaces: old and new directions. J Comput Appl Math. 1993;48:113–131. doi: 10.1016/0377-0427(93)90318-6
   Web of Science ®Google Scholar
• Durán AJ, De la Iglesia MD. Differential equations for discrete Jacobi-Sobolev orthogonal polynomials. J Spectr Theory. 2018;8:191–234. doi: 10.4171/jst
   Web of Science ®Google Scholar
• Everitt WN, Kwon KH, Littlejohn LL, et al. Orthogonal polynomial solutions of linear ordinary differential equations. J Comput Appl Math. 2001;133:85–109. doi: 10.1016/S0377-0427(00)00636-1
   Web of Science ®Google Scholar
• Mañas-Mañas JF, Moreno-Balcázar JJ, Wellman R. Eigenvalue problem for discrete Jacobi-Sobolev orthogonal polynomials. Mathematics. 2020;8:182,19.
   Web of Science ®Google Scholar
• Markett C. The differential equation for Jacobi-Sobolev polynomials with two linear perturbations. J Approx Theory. 2022;280:105782. doi: 10.1016/j.jat.2022.105782
   Web of Science ®Google Scholar
• Koekoek J, Koekoek R. Differential equations for generalized Jacobi polynomials. J Comput Appl Math. 2000;126:1–31. doi: 10.1016/S0377-0427(99)00338-6
   Web of Science ®Google Scholar
• Bavinck H. Differential operators having Sobolev-type Jacobi polynomials as eigenfunctions. J Comput Appl Math. 2003;151:271-–295. doi: 10.1016/S0377-0427(02)00810-5
   Web of Science ®Google Scholar
• Bavinck H. Differential operators having Sobolev-Laguerre polynomials as eigenfunctions: new developments. J Comput Appl Math. 2001;133:183–193. doi: 10.1016/S0377-0427(00)00642-7
   Web of Science ®Google Scholar
• Markett C. On the differential equation for the Laguerre-Sobolev polynomials. J Approx Theory. 2019;247:48–67. doi: 10.1016/j.jat.2019.07.007
   Web of Science ®Google Scholar
• Bavinck H, Meijer HG. Orthogonal polynomials with respect to a symmetric inner product involving derivatives. Appl Anal. 1989;33:103–117. doi: 10.1080/00036818908839864
   Google Scholar
• Bavinck H. Differential operators having Sobolev-type Gegenbauer polynomials as eigenfunctions. J Comput Appl Math. 2000;118:23–42. doi: 10.1016/S0377-0427(00)00279-X
   Web of Science ®Google Scholar
• Bavinck H, Koekoek J. Differential operators having symmetric orthogonal polynomials as eigenfunctions. J Comput Appl Math. 1999;106:369–393. doi: 10.1016/S0377-0427(99)00094-1
   Web of Science ®Google Scholar
• Ciaurri Ó, Mínguez Ceniceros J. Fourier series of Gegenbauer-Sobolev polynomials. SIGMA. 2018;14:11.
   Google Scholar
• Fejzullahu BX, Marcellán F. On convergence and divergence of Fourier expansions with respect to some Gegenbauer-Sobolev type inner product. Commun Anal Theory Contin Fract. 2008/09;16:1-–11.
   Google Scholar
• Fejzullahu BX, Marcellán F. A Cohen type inequality for Gegenbauer-Sobolev expansions. Rocky Mountain J Math. 2013;213:135–148.
   Google Scholar
• Foulquié Moreno A, Marcellán F, Osilenker BP. Estimates for polynomials with respect to some Gegenbauer-Sobolev type inner product. J Inequal Appl. 1999;3:101–119.
   Web of Science ®Google Scholar
• NIST Digital Library of Mathematical Functions. National Institute of Standards and Technology, U.S. Department of Commerce. Available from: https://dlmf.nist.gov.
   Google Scholar
• Erdélyi A, Magnus W, Oberhettinger F, et al. Higher transcendental functions. New York (NY): McGraw-Hill; 1953.
   Google Scholar
• Kwon KH, Littlejohn LL, Yoon GJ. Orthogonal polynomial solutions of spectral type differential equations: magnus' conjecture. J Approx Theory. 2001;112:189–215. doi: 10.1006/jath.2001.3586
   Web of Science ®Google Scholar
• Kwon KH, Littlejohn LL, Yoon GJ. Construction of differential operators having Bochner-Krall orthogonal polynomials as eigenfunctions. J Math Anal Appl. 2006;324:285–303. doi: 10.1016/j.jmaa.2005.11.059
   Web of Science ®Google Scholar
• Koornwinder TH. Orthogonal polynomials with weight function (1−x)α(1+x)β+Mδ(x+1)+Nδ(x−1). Canad Math Bull. 1984;27(2):205-–214. doi: 10.4153/CMB-1984-030-7
   Web of Science ®Google Scholar
• Markett C. Symmetric differential operators for Sobolev orthogonal polynomials of Laguerre- and Jacobi-type. Integral Transforms Spec Funct. 2021;32: doi: 10.1080/10652469.2020.17676135–8,568–587.
   Web of Science ®Google Scholar
• Markett C. An elementary representation of the higher-order Jacobi-type differential equation. Indag Math. 2017;28:976–991. doi: 10.1016/j.indag.2017.06.015
   Web of Science ®Google Scholar
• Markett C. New representation and factorizations of the higher-order ultrapherical-type differential equation. J Math Anal Appl. 2015;421:244–259. doi: 10.1016/j.jmaa.2014.07.001
   Web of Science ®Google Scholar