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New summation formula for $_3F_2(\frac{1}{2})$ and a Kummer-type II transformation of $_2F_2(x)$. (English) Zbl 1146.33002
The Kummer type I transformation for the confluent hypergeometric function ${}_1F_1(a;b;x)$ takes the form $$e^{-x}{}_1F_1(a;b;x)={}_1F_1(b-a;b;x).$$ Recently, this was extended to the ${}_2F_2$ function of the form ${}_2F_2(a,1+d;b,d;x)$. The Kummer type II transformation is $$e^{-x/2}{}_1F_1(a;2a+1;x)={}_0F_1(a+1/2;(x/4)^2).$$ 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 $e^{-x/2}{}_2F_2(a,1+d;2a+1,d;x)$ can be expressed in terms of two ${}_0F_1$ functions. This is achieved by means of a summation formula obtained by the authors for $${}_3F_2(a,b,c+1;c,{\frac{1}{2}} a+{\frac{1}{2}} b+1;{\frac{1}{2}}).$$

33C20Generalized hypergeometric series, ${}_pF_q$
33B15Gamma, beta and polygamma functions