We give qualitative and quantitative improvements on all the best previously known irrationality results for dilogarithms of positive rational numbers. We obtain such improvements by applying our permutation group method to the diophantine study of double integrals of rational functions related to the dilogarithm.
In this paper we derive, using the Gauss summation theorem for hypergeometric series, a simple integral expression for the reciprocal of Euler’s beta function. This expression is similar in form to several well-known integrals for the beta function itself.We then apply our new formula to the study of Whittaker functions, which are special functions that arise in the Fourier theory for automorphic forms on the general linear group. Specifically, we deduce explicit integral representations of “fundamental”...
Let be an odd prime, and let be an integer not divisible by . When is a positive integer with and is an th power residue modulo , we determine the value of the product , where In particular, if with , then
In this paper we solve three open problems and a conjecture related to the calculations of some classes of multiple series posed by Furdui in [1].