Algorithms for the Multiple Gamma and Related Functions

Carnegie Mellon University, Pittsburgh PA

Investigators

Abstract

ABSTRACT Adamchik, Victor 0204003 Carnegie Mellon U Great progress has been made in recent years in the construction of software for numeric and symbolic computation of transcendental functions. The algorithms have been implemented in the most widely used symbolic algebra systems such as Maple and Mathemat-ica. Recently, the National Institute of Standards and Technology started a project of creating the Digital Library of Mathematical Functions (http://dlmf.nist.gov/). The goal of the project is to collect all recent scientific results regarding theory and computational algorithms of the well-known special functions and make them available to the public in electronic form. How-ever, enormous gaps remain for "new" transcendental functions. One of such functions is the multiple gamma (also called Barnes) Gn-function. The Gn function, defined by a recurrence-functional equation as a generalization of the Euler gamma function, was originally intro-duced (but in different forms) by Kinkelin, Glaisher, and Barnes around 1900. Today, due to the pioneer work of Peter Sarnak of Princeton University, the interest to the Barnes function is revived. Sarnak has been pushing the idea that zeros of certain "zeta functions" (L-functions) can be understood in terms of the distribution of eigenvalues from classes of random matrices. It has been conjectured that the limiting distribution of the non-trivial zeros of the Riemann zeta function is the same as that of the eigenphases of matrices in the CUE (the circulat uni-tary ensemble) It has been shown in works by Mehta, Sarnak, Conrey, Keating, Snaith that the Barnes function naturally appeares there as a closed representation for statistical averages over CUE of N x N unitary matrices (as well as some other classical compact groups), when