On the critical case of the Weber–Schafheitlin integral and a certain generalization

When the orders of the product of two Bessel functions differ by an odd integer, we show that the critical case (defined in the sequel as the supercritical case) of the discontinuous integral of Weber and Schafheitlin is proportional to a ratio of products of gamma functions. The derivation we give is elementary in the sense that it avoids the use of contour integration and the calculus of residues. We consider also a generalization of the later integral by essentially replacing one of the Bessel functions by the hypergeometric function 1F2. As a byproduct of this investigation we deduce Whippleʹs transformation formula for a well-poised 6F5(−1) and allied results.