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An extension of a Kummer’s quadratic transformation formula with an application. (English) Zbl 1298.33002
Summary: In the theory of the generalized hypergeometric functions, a remarkable amount of mathematicians’ concern has been given to develop their transformation formulas and summation identities. Here we aim at presenting a new and (potentially) useful transformation formula for hypergeometric functions $$_pF_q$$:
$(1-x)^{-a}\,_3F_2\left[\begin{matrix} \frac12a,\frac12+\frac12a-b,1+d;\\ \qquad \quad 2+a-b,d;\end{matrix} -\frac{4x}{(1-x)^2}\right]=\,_4F_3\left[\begin{matrix} a,b-1,\frac12a-\frac12A+1,\frac12 a+\frac12 A+1;\\ \qquad \;2+a-b,\frac12a-\frac12A,\frac12a+\frac12A;\end{matrix} x\right],$
where $$A^2=a^2-4f(b-1)$$ with $$f=\frac{d(a+1)}{2d+2b-a-1}$$. Further we apply the so-called beta integral method which was used systematically by Krattenthaler and Rao to the Kummer quadratic transformation formula and the above-presented transformation formula and present (presumably new) transformation formulas for hypergeometric series. Relevant connections of some special cases of the results presented here with those obtained in earlier works are also pointed out.
##### MSC:
 33B20 Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals) 33C20 Generalized hypergeometric series, $${}_pF_q$$ 33B15 Gamma, beta and polygamma functions 33C05 Classical hypergeometric functions, $${}_2F_1$$