Recurrence relations for the coefficients of the confluent and Gauss hypergeometric functions in the complex plane
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| Natura: | Preprint |
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2026
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| _version_ | 1866909990953943040 |
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| author | Xu, Zi-Qiao Mao, Zhong-Xuan Tian, Jing-Feng |
| author_facet | Xu, Zi-Qiao Mao, Zhong-Xuan Tian, Jing-Feng |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_10040 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Recurrence relations for the coefficients of the confluent and Gauss hypergeometric functions in the complex plane Xu, Zi-Qiao Mao, Zhong-Xuan Tian, Jing-Feng Complex Variables Classical Analysis and ODEs 33C15, 33C05, 11B37, 41A05 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. |
| title | Recurrence relations for the coefficients of the confluent and Gauss hypergeometric functions in the complex plane |
| topic | Complex Variables Classical Analysis and ODEs 33C15, 33C05, 11B37, 41A05 |
| url | https://arxiv.org/abs/2601.10040 |