Recurrence relations for the coefficients of the confluent and Gauss hypergeometric functions in the complex plane
Zi-Qiao Xu, Zhong-Xuan Mao, Jing-Feng Tian
TL;DR
The paper develops a unified framework to study the Maclaurin coefficients of the modified confluent and Gauss hypergeometric functions in the complex plane by deriving explicit 3rd-order recurrence relations for their coefficients: $u_{n+1}=\beta_0(n)u_n+\beta_1(n)u_{n-1}+\beta_2(n)u_{n-2}$ for $\mathcal{M}(z)=(1-\theta z)^p M(a;c;z)$ and $v_{n+1}=\frac{\mu_{n,p,\theta}(a,b,c)}{(n+1)(n+c)}v_n-\theta\frac{\zeta_{n,p,\theta}(a,b,c)}{(n+1)(n+c)}v_{n-1}+\theta^2\frac{\sigma_{n,p}(a,b)}{(n+1)(n+c)}v_{n-2}$ for $\mathcal{G}(z)=(1-\theta z)^p F(a,b;c;z)$, with explicit initial values and auxiliary relations. It extends these recurrences to a variety of special functions via applications, including the error function, incomplete gamma, Bessel, complete elliptic integrals, Chebyshev polynomials, and various products with logarithmic or inverse-trigonometric factors. The results provide a powerful tool for efficient coefficient computation and analytic study of hypergeometric functions in complex domains. The paper thus offers new structural insight and practical recurrence formulas for a wide class of special functions.
Abstract
For $a,b,c,z,p, θ\in \mathbb{C}$, where $\mathbb{C}$ is the complex plane, $-c\notin \mathbb{N\cup }\left\{ 0\right\} $, let \begin{equation*} \mathcal{M}\left( z\right) =\left( 1-θz\right) ^{p}M\left(a;c;z\right) =\sum_{n=0}^{\infty }u_{n}z^{n}, \end{equation*} where $|z| <\frac{1}θ$, $|\arg (1-θz)| < π$, and let \begin{equation*} \mathcal{G}\left( z\right) =(1-θz) ^{p}F(a,b;c;z) =\sum_{n=0}^{\infty }v_{n} z^{n}, \end{equation*} where $|z| < 1$, $|\arg (1-θz)| < π$. In this paper, we prove that the coefficients $u_{n}$ and $v_{n}$ for $n\geq 0$ satisfy a 3-order recurrence relation. These offer a new way to study confluent hypergeometric function $M(a;c;z)$ and Gauss hypergeometric function $F(a,b;c;z)$. And we provide other special functions' recurrence relations of their coefficients, such as error function, Bessel function, incomplete gamma function, complete elliptic integral and Chebyshev polynomials.