On the relation between Gegenbauer polynomials and the Ferrers function of the first kind

Published

March 30, 2022

Author(s)

Roberto S. Costas-Santos, Howard Cohl

Abstract

Using the direct relation between the Gegenbauer polynomials C_n^λ(x) and the Ferrers function of the first kind P_μ^ν(x), we compute interrelations between certain Jacobi polynomials, Meixner polynomials, and Ferrers functions of the first and second kind. We then compute Rodrigues-type, standard integral orthogonality and Sobolev orthogonality relations for Ferrers functions of the first and second kinds. In the remainder of the paper using the relation between Gegenbauer polynomials and the Ferrers function of the first kind we derive connection and linearization relations, some definite integral and series expansions, Christoffel–Darboux summation formulas, Poisson kernel and infinite series closure relations (Dirac delta distribution expansions).

Citation

Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

Volume

48

Issue

3

Pub Type

Journals

Download Paper

https://doi.org/10.1007/s10476-022-0123-0

Local Download

Keywords

Ferrers functions, Gegenbauer polynomials, orthogonal polynomials

Citation

Costas-Santos, R. and Cohl, H. (2022), On the relation between Gegenbauer polynomials and the Ferrers function of the first kind, Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), [online], https://doi.org/10.1007/s10476-022-0123-0, https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=932710 (Accessed October 14, 2025)