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New summation formula for 3 F 2 (1 2) and a Kummer-type II transformation of 2 F 2 (x). (English) Zbl 1146.33002

The Kummer type I transformation for the confluent hypergeometric function 1 F 1 (a;b;x) takes the form

e -x 1 F 1 (a;b;x)= 1 F 1 (b-a;b;x)·

Recently, this was extended to the 2 F 2 function of the form 2 F 2 (a,1+d;b,d;x). The Kummer type II transformation is

e -x/2 1 F 1 (a;2a+1;x)= 0 F 1 (a+1/2;(x/4) 2 )·

The aim of this note is to establish the extension of the type II transformation on the lines of the above-mentioned extension of the type I transformation. The result shows that e -x/2 2 F 2 (a,1+d;2a+1,d;x) can be expressed in terms of two 0 F 1 functions. This is achieved by means of a summation formula obtained by the authors for

3 F 2 (a,b,c+1;c,1 2a+1 2b+1;1 2)·


MSC:
33C20Generalized hypergeometric series, p F q
33B15Gamma, beta and polygamma functions