特殊函数在数学物理、数论、组合论、渐近分析、微分方程、数值计算等领域有广泛而重要的应用。圆周率、Euler-Mascheroni常数、Catalan常数等数学常数，可以看作是特殊函数的特殊取值，在数学和物理上应用广泛。超几何级数方法是组合数学中常用的方法，但在特殊函数以及特殊数学常数方面的应用还需进一步探索。本项目旨在以超几何级数求和与变换公式为基础，一方面研究圆周率、Euler-Mascheroni常数、Catalan常数等特殊数学常数和Riemann zeta函数及其相关特殊函数的级数展开式。另一方面研究Legendre、Jacobi、Bessel、Hermite以及Laguerre等特殊函数的多重超几何级数表示，并以此研究这些特殊函数的相关性质。本项目既能够促进特殊函数以及特殊数学常数的发展，又可以拓广超几何级数方法的应用范围，为特殊函数论以及特殊数学常数的研究提供新的研究视角。

Special functions play important roles in mathematical physics, number theory, combinatorics, asymptotic analysis, differential equation, numerical computation, and so on. Mathematical constants, such as pi, Euler-Mascheroni constant and Catalan constant, are all special values of some special functions and widely used in mathematics and physics. Hypergeometric series method is a commonly used method in combinatorial mathematics, but its applications in special functions and special mathematical constants still need further exploration. In this project, based on the summation and transformation formulae of hypergeometric series, on the one hand, we will study the series expansion formulae for some mathematical constants such as pi, Euler-Mascheroni constant and Catalan constant, and some special functions such as Riemann zeta function and related functions. On the other hand, we will study the representations of multiple hypergeometric series for different special functions, such as Legendre, Jacobi, Bessel, Hermite and Laguerre functions, and investigate the related properties of these special functions. This project will not only promote the development of special functions and special mathematical constants, but also extend the application range of hypergeometric series method, and provide a new research perspective for special functions and special mathematical constants.