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

TL;DR

This paper explores the mathematical relationships between Gegenbauer polynomials, Ferrers functions, and other special functions, deriving new formulas, relations, and expansions to deepen understanding of their interconnections.

Contribution

It introduces new interrelations, orthogonality, and connection formulas between Gegenbauer polynomials, Ferrers functions, and related special functions, along with asymptotic and integral expansions.

Findings

Derived Rodrigues-type and orthogonality relations for Ferrers functions.

Established connection and linearization formulas for special functions.

Presented asymptotic expansions and summation formulas for Ferrers functions.

Abstract

Using the direct relation between the Gegenbauer polynomials and the Ferrers function of the first kind, we compute interrelations between certain Jacobi polynomials, Meixner polynomials, and the Ferrers function of the first kind. We then compute Rodrigues-type and 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, some asymptotic expansions of Mehler-Heine type, Christoffel-Darboux summation formulas, and infinite series closure relations (Dirac delta distribution).