Recurrence relations and applications for the Maclaurin coefficients of squared and cubic hypergeometric functions

TL;DR

This paper derives recurrence relations for the Maclaurin coefficients of squared and cubed hypergeometric functions, enabling analysis of their properties and applications to classical special functions.

Contribution

It introduces new second- and third-order recurrence relations for these coefficients, extending to various classical special functions and providing new proofs and monotonicity results.

Findings

Recurrence relations for coefficients of squared and cubed hypergeometric functions.

Application to classical special functions like elliptic integrals and orthogonal polynomials.

New proof of Clausen's formula and monotonicity results.

Abstract

In this paper, we present and prove that the coefficients $u_n$ and $v_n$ in the series expansions $F^2(a,b;c;z) = \sum_{n=0}^\infty u_n z^n$ and $F^3(a,b;c;z) = \sum_{n=0}^\infty v_n z^n$ ($a,b,c,z \in \mathbb{C}$ and $-c \notin \mathbb{N} \cup \{0\}$) satisfy second- and third-order linear recurrence relations, respectively, where $F(a,b;c;x)$ denotes the Gaussian hypergeometric function and $\mathbb{C}$ is the complex plane. Our results provide recurrence relations for the Maclaurin coefficients of the squares and cubes of several classical special functions in the complex domain, including zero-balanced Gauss hypergeometric functions, elliptic integrals, as well as classical orthogonal polynomials such as Chebyshev, Legendre, Gegenbauer, and Jacobi polynomials. As applications, we first establish the monotonicity of a function involving Gauss hypergeometric functions and then present a new proof of the well-known Clausen's formula.