Efficient Algorithms for Computing the Parameter Derivatives of k-hypergeometric Functions and Their Extensions to Other Special Functions

摘要

The k-hypergeometric functions are de-fined aspFq(a, k, b, s; z) =∞Pn=0(a1)n,k1 (a2)n,k2···(ap)n,kp zn (b1)n,s1 (b2)n,s2···(bq)n,sq n! ,where (x)n,k = x(x + k)(x + 2k)· · ·(x + (n 1)k) isthe Pochhammer k-symbol. In this paper, efficientrecursive algorithms for computing the parameterderivatives of the k-hypergeometric functions are developed. As the generalized hypergeometric functionsare special cases of this function and many specialfunctions can be expressed in terms of the generalizedhypergeometric functions, the algorithms can also beextended to computing the parameter derivatives ofthe hypergeometric functions and many other specialfunctions. The Bessel functions and modified Besselfunctions are presented as examples of such an application. Theoretical analysis is worked out, some computation using Mathematica is performed, and datais provided to show the advantages of our algorithms.