Multiple Series Representations of N -fold Mellin-Barnes Integrals

On the evaluation of the Appell $F_2$ double hypergeometric function

“…Multiple MB integrals are in fact one of the possible starting points for the study of hypergeometric functions of several variables [9]. Therefore, by the study of appropriate classes of multiple MB integrals, the method of [5] opens promising horizons in hypergeometric functions theory and, in particular, for the determination of the analytic continuations of many classes of hypergeometric series, whatever the number of their variables is, in terms of others multivariable hypergeometric series. Obviously, a large number of applications can follow in physics, as already shown in the recent works [10] and [11] in the context of the study of Feynman integrals in quantum field theory.…”

“…Although many new results can be obtained from the powerful method developed in [5], it cannot, alone, fully solve the difficult problem of finding the relevant set of transformations of a multiple hypergeometric function that will allow to evaluate it for all values of its variables. Indeed, we have mentioned above some possible difficulties in the derivation of the convergence regions of the new series representations obtained from transformation theory.…”

“…Moreover, even if the latter can be obtained, the evaluation of the MB integral in general shows that "white regions" (as called in [12]) appear in the multivariable space, where none of the obtained analytic continuations converges. Although some manipulations of the MB integral can lead to transformations and, thus, to other formulas (in addition to those obtained by a direct application of [5]), it is not clear that a systematic approach for these manipulations can be found. Therefore, in order to fully solve the problem of evaluating multivariable hypergeometric functions for all values of their variables, it may be necessary to rely on alternative approaches, in order to complete the results obtained from the MB approach.…”

“…In Section 2, we briefly list some of the well-known properties of the Appell F 2 function. In Section 3, we perform a first analytic continuation study of the Appell F 2 series from the Mellin-Barnes approach [5], which is completed in Section 4 following Olsson's method. This analysis, which yields 11 series representations of F 2 , can be extended with the use of the three Euler transformations of F 2 , allowing us to obtain a total set of 43 linear transformations of F 2 , out of which 17 are needed, in addition to the usual series definition of F 2 , to cover the (x, y) space of the F 2 variables for real values of the latter, with the exception of a few points.…”

Massive One-loop Conformal Feynman Integrals and Quadratic Transformations of Multiple Hypergeometric Series

“…In [1], we have presented the first systematic computational method allowing us to derive sets of series representations of N -fold Mellin-Barnes (MB) integrals, in the case where N is a given positive integer. Our method was described at a general level and we have given only one simple twofold example as an application to provide a quick understanding of the method and of our notation.…”