Recurrence relations and applications for the Maclaurin coefficients of squared and cubic hypergeometric functions
Zhong-Xuan Mao, Jing-Feng Tian
TL;DR
The paper establishes second- and third-order linear recurrence relations for the Maclaurin coefficients of the square and cube of the Gauss hypergeometric function, $F^2(a,b;c;z)$ and $F^3(a,b;c;z)$, in the complex domain. By deriving explicit rational coefficient formulas for these recurrences, the authors provide a unified framework to obtain series representations for $F^2$ and $F^3$ and extend the method to zero-balanced hypergeometric functions, elliptic integrals, and classical orthogonal polynomials. As applications, they prove a monotonicity result for the ratio $F^2(a,b;c;x)/F(2a,2b;2c;x)$ and present a new proof of Clausen's formula, connecting a square hypergeometric function to a ${}_3F_2$ expression. The results offer practical tools for analyzing the analytic properties of a broad class of special functions in the complex domain and illuminate their interrelations via recurrences and contiguous relations.
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.