×

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

Citations:

Zbl 0915.33006
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] W.N. Bailey, Generalized Hypergeometric Series, Hefner, New York, 1972. · Zbl 0011.02303
[2] Grosjean, C.C., Solution to SIAM review problem 97-13, SIAM rev., 40, 726-729, (1998)
[3] Luke, Y.L., Integrals of Bessel functions, (1962), McGraw-Hill New York · Zbl 0144.06902
[4] Miller, A.R., A Mellin transform: SIAM review problem 97-13, SIAM rev., 39, 515, (1997)
[5] Miller, A.R., On the Mellin transform of products of Bessel and generalized hypergeometric functions, J. comput. appl. math., 85, 271-286, (1997) · Zbl 0889.44004
[6] Miller, A.R.; Srivastava, H.M., On the Mellin transform of a product of hypergeometric functions, J. austral. math. soc. ser. B, 40, 222-237, (1998) · Zbl 0915.33006
[7] Prudnikov, A.P.; A. Brychkov, Yu.; Marichev, O.I., Integrals and series, vol. 3, (1990), Gordon and Breach New York · Zbl 0967.00503
[8] Slater, L.J., Generalized hypergeometric functions, (1966), Cambridge University Press Cambridge, UK · Zbl 0135.28101
[9] Srivastava, H.M.; Exton, H., A generalization of the weber – schafheitlin integral, J. reine angew. math., 309, 1-6, (1979) · Zbl 0393.33002
[10] Wang, Z.X.; Guo, D.R., Special functions, (1989), World Scientific Singapore · Zbl 0724.33001
[11] Watson, G.N., A treatise on the theory of Bessel functions, (1944), Cambridge University Press Cambridge, UK · Zbl 0063.08184
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.