The coupling coefficients with six parameters and the generalized hypergeometric functions

In this study, the Gaunt coefficients, Clebsch–Gordan coefficients, and the Wigner 3j and 6j symbols are expressed as the product of generalized hypergeometric functions with unit argument and a normalization coefficient. By exploiting the symmetry properties of generalized hypergeometric functions, these functions are transformed into numerically computable forms, and the normalization coefficients are fully expressed in terms of binomial coefficients. New mathematical expressions, in the form of a series of products of three Gaunt coefficients, are presented, which can be used to verify the accuracy of numerical calculations. An algorithm has been developed to compute binomial coefficients and generalized hypergeometric functions using recurrence relations, eliminating the need for factorial functions. Utilizing this algorithm and the derived analytical expressions, the Gaunt_CG_3j_and_6j Mathematica program, which numerically calculates the Gaunt coefficients, Clebsch–Gordan coefficients, and the Wigner 3j and 6j symbols, was written without relying on Mathematica’s built-in functions. The program can be easily adapted to other programming languages and run on all versions of Mathematica.本研究中，高恩特系数（Gaunt coefficients）、克莱布希-高登系数（Clebsch–Gordan coefficients）以及威格纳3j和6j符号（Wigner 3j and 6j symbols）可表示为单位宗量广义超几何函数（generalized hypergeometric functions）与归一化系数的乘积。通过利用广义超几何函数的对称性性质，这类函数被转化为可数值计算的形式，且归一化系数可完全通过二项式系数（binomial coefficients）表达。本文提出了一系列由三个高恩特系数乘积构成的新型数学表达式，可用于验证数值计算的精度。 本研究开发了一种基于递推关系（recurrence relations）计算二项式系数与广义超几何函数的算法，无需使用阶乘函数（factorial functions）。依托该算法与本文推导得到的解析表达式，我们编写了Gaunt_CG_3j_and_6j Mathematica程序，可直接数值计算高恩特系数、克莱布希-高登系数以及威格纳3j和6j符号，且未调用Mathematica的内置函数。该程序可便捷移植至其他编程语言，且可在所有版本的Mathematica中运行。