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.

Program Summary

Program title: Gaunt_CG_3j_and_6j

CPC Library link to program files: https://doi.org/10.17632/pwhry4278g.1

Licensing provisions: GPLv2

Programming language: Wolfram Language (Mathematica 9.0 or higher)

Nature of problem:

In this study, analytical expressions for the Gaunt coefficients and Wigner 6j symbols are derived as the product of generalized hypergeometric functions and a normalization coefficient. Analytical expressions for the Wigner 3j symbols in terms of the Clebsch–Gordan (CG) coefficients are given in the same form in Ref. [27]. This study introduces new analytical expressions involving sums to verify the accuracy of numerical calculations for all coupling coefficients with six parameters consolidated into a single formula. The Gaunt_CG_3j_and_6j program numerically calculates the coupling coefficients presented in this study and Ref. [27].

Solution method:

In this study, all coupling coefficients with six parameters are expressed as the product of the generalized hypergeometric function with unit argument and the “normalization” constant, which is written in terms of binomial coefficients. Therefore, the main structure of our program consists of numerical calculation of the binomial coefficients and generalized hypergeometric functions with unit argument.

In our algorithm, binomial coefficients with parameters containing negative or positive integers are calculated using the binomcag and binomcalc modules, based on Eq. (12) in Ref. [31]. The binomcalc module computes the binomial coefficients using the recurrence relation:$F\left(n,m\right)=\frac{n}{n-m}F\left(n-1,m\right),n\ge 1,m\ge 0$with the initial conditions $F\left(m,m\right)=1$ and $F\left(n,m\right)=0,form>n$.

To calculate the coupling coefficients with six parameters, generalized hypergeometric functions and binomial coefficients can be used, as demonstrated in this study, as an alternative to recurrence relations or serial expressions and factorial functions. We implemented the hypergeowhl and hipergeomcal modules in our algorithm to calculate the generalized hypergeometric functions. We used Eq. (21) in Ref. [29] to compute the generalized hypergeometric functions with unit argument.

Additional comments including Restrictions and Unusual features:

When determining CPU times, issues unrelated to our program arise from Mathematica. The command lines we use to measure CPU time are: x1=TimeUsed[];

$⋮$ x2= TimeUsed[]; time=x2-x1

We encountered two problems when determining CPU times in this manner. The first problem occurs when we try to measure the calculation time for a single coupling coefficient, where Mathematica reports this time as zero (in seconds). We calculated the average CPU time by computing 1000 or more coupling coefficients for a single quantum set to overcome this. The second problem arises from the CPU times for non-zero coupling coefficients (particularly in expressions involving sums), which are unstable and vary each time the program is run. Since these results are reproducible across multiple program executions, we take the average of the computed times.

Long write-up