Integrals involving product of Bessel functions and generalized hypergeometric functions.
The method of brackets is a collection of heuristic rules, some of which have being made rigorous, that provide a flexible, direct method for the evaluation of definite integrals. The present work uses this method to establish classical formulas due to Frullani which provide values of a specific family of integrals. Some generalizations are established.
Integrals of logarithmic and hypergeometric functions are intrinsically connected with Euler sums. In this paper we explore many relations and explicitly derive closed form representations of integrals of logarithmic, hypergeometric functions and the Lerch phi transcendent in terms of zeta functions and sums of alternating harmonic numbers.
MSC 2010: 35J05, 33C10, 45D05
Les valeurs aux entiers pairs (strictement positifs) de la fonction de Riemann sont transcendantes, car ce sont des multiples rationnels de puissances de . En revanche, on sait très peu de choses sur la nature arithmétique des , pour entier. Apéry a démontré en 1978 que est irrationnel. Rivoal a prouvé en 2000 qu’une infinité de sont irrationnels, mais sans pouvoir en exhiber aucun autre que . Il existe plusieurs points de vue sur la preuve d’Apéry ; celui des séries hypergéométriques...