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

The k-hypergeometric functions are defined as F-p(q)(a, k, b, s; z) = Sigma(infinity )(n=0)(a(1))n,k(1) (a(2))n,k(2) ... (a(p))(n),k(p)(zn)/(b(1))(n),(s1) (b(2))(n),(s2) ... (bq)(n,sq)(n!), where (x)(n,k) = x(x + k)(x + 2k) ... (x + (n - 1)k) is the Pochhammer k-symbol. In this paper, efficient recursive algorithms for computing the parameter derivatives of the k-hypergeometric functions are developed. As the generalized hypergeometric functions are special cases of this function and many special functions can be expressed in terms of the generalized hypergeometric functions, the algorithms can also be extended to computing the parameter derivatives of the hypergeometric functions and many other special functions. The Bessel functions and modified Bessel functions are presented as examples of such an application. Theoretical analysis is worked out, some computation using Mathematica is performed, and data is provided to show the advantages of our algorithms.