Recurrence relations for the coefficients of the confluent and Gauss hypergeometric functions in the complex plane

Abstract

For $a,b,c,z,p,\theta \in \mathbb{C}$, 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 $|z|<\frac{1}{\theta }$, $|\arg (1-\theta z)|<\pi$, 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 $|z|<1$, $|\arg (1-\theta z)|<\pi$. In this paper, we prove that the coefficients $u_n$ and $v_n$ for $n\ge 0$ 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.