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.