On the critical case of the Weber-Schafheitlin integral and a certain generalization. (English) Zbl 1015.33004

The integral under consideration is \[ F(s)= \int^\infty_0 x^{s-1} {_0F_1} [-;1+\mu; -a^2x^2]{_1F_2}[\alpha; \beta,1+ \nu;-b^2] \] for the case \(a=b\). (The case \(a\neq b\) was treated earlier by the author and H. M. Srivastava [J. Aust. Math. Soc., Ser. B 40, No. 2, 222–237 (1998; Zbl 0915.33006)]. For \(\alpha=\beta\) the integral becomes the Weber-Schafheitlin discontinuous integral. The derivations avoid contour integrals ahd residue calculus. First for \(\alpha=\beta\), under convergence conditions, if \(\mu-\nu\) is an odd integer then the value of the integral is proportional to a ratio of gamma functions. Then, if \(\mu-\nu+\alpha-\beta\) is an odd integer and \(\alpha\) or \(\mu\) is such that the Mellin transform exists, then the evaluation stops at \(a{_3F_2}\) at \(+1\). Another representation for \(F(s)\) is obtained which leads to an evaluation in terms of gamma functions and \(_6F_5\) at \(-1\). From this, several summation formulas for hypergeometric functions are obtained.


33C20 Generalized hypergeometric series, \({}_pF_q\)
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
44A20 Integral transforms of special functions


Zbl 0915.33006
Full Text: DOI


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