## 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.

### MSC:

 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
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### References:

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