Finite differences, Clebsch–Gordan coefficients, and hypergeometric functions

A generalization of the theory of angular momenta is proposed. The generating representation is a representation of finite generalized hypergeometric series by means of operators of finite differences and symbolic powers. A number of new relations are obtained. These generalize the concept of coupling (addition) of angular momenta, in particular, the expression of the Racah coefficients as a sum of products of two Clebsch–Gordan coefficients. The efficiency of the method of finite differences is demonstrated and a study is made of difference differentiation and integration of the Clebsch–Gordan coefficients and $j$-symbols with respect to the angular momenta and their projections. The formulas obtained by this method yield directly numerical values of the $j$-symbols and the other quantities in the theory of angular momenta.