[论文解读] Recurrence relations and applications for the Maclaurin coefficients of squared and cubic hypergeometric functions

Zhong-Xuan Mao, Jing-Feng Tian|arXiv (Cornell University)|Jan 14, 2026

Mathematical functions and polynomials被引用 0

一句话总结

论文在复域中导出 Gauss 超几何函数 F^2(a,b;c;z) 和 F^3(a,b;c;z) 的 Maclaurin 系数的二阶和三阶递推关系，并将其应用于椭圆积分和经典多项式，同时给出单调性以及 Clausen 公式的新证明等结果。

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 $-c otin \mathbb{N} \cup \{0\}$) 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.

研究动机与目标

• Motivate the study of squared and cubic hypergeometric functions through their Maclaurin coefficients.
• Derive explicit second- and third-order recurrence relations for these coefficients in the complex domain.
• Demonstrate the applicability of these recurrences to zero-balanced hypergeometric functions, elliptic integrals, and classical orthogonal polynomials.
• Provide applications including monotonicity results and a new proof of Clausen’s formula.

提出的方法

• Use Cauchy product to represent F^2 and F^3 and derive recurrence relations for coefficients.
• Transform the problem to polynomial identities by equating coefficients via hypergeometric equations.
• Compute and manipulate derivatives of F to obtain relations among u_n and u_{n±1} (Theorem 2.1).
• Obtain explicit recurrence coefficients alpha_0(n), alpha_1(n) for u_n in F^2 and beta_0(n), beta_1(n), beta_2(n) for v_n in F^3 (Theorem 3.1).
• Specialize parameters to derive recurrences for square/cube of elliptic integrals and classical polynomials (Corollaries 2.x, 3.x).
• Apply recurrence framework to monotonicity of a hypergeometric expression and provide a new proof of Clausen’s formula.]

实验结果

研究问题

• RQ1 What recurrence relations do the Maclaurin coefficients of F^2(a,b;c;z) and F^3(a,b;c;z) satisfy in the complex domain?
• RQ2 How can these recurrences be used to obtain coefficient sequences for special functions such as elliptic integrals and classical polynomials?
• RQ3 Can these recurrences yield insights into monotonicity properties and classical hypergeometric identities (e.g., Clausen’s formula)?

主要发现

• F^2(a,b;c;z) has coefficients u_n satisfying a second-order linear recurrence: u_{n+1}=alpha_0(n)u_n+alpha_1(n)u_{n-1} with explicit alpha_0(n), alpha_1(n).
• F^3(a,b;c;z) has coefficients v_n satisfying a third-order linear recurrence: v_{n+1}=beta_0(n)v_n+beta_1(n)v_{n-1}+beta_2(n)v_{n-2} with explicit beta_0(n), beta_1(n), beta_2(n).
• Special choices yield recurrences for squares/cubes of K(z), E(z), Chebyshev, Legendre, Gegenbauer, Jacobi polynomials, and their shifted/related forms.
• Applications show monotonicity of a function involving F^2 and F and provide a new proof of Clausen’s formula.
• The results extend recurrence-based coefficient analysis to the complex domain and connect to several classical special functions.