Eric Stade

ArchimedeanL-factors onGL(n) ×GL(n) and generalized Barnes integrals

The Rankin-Selberg method associates, to each local factorL(s, πv × πv′) of an automorphicL-function onGL(n) ×GL(n), a certain local integral of Whittaker functions for πv andv′. In this paper we show that, if ν is archimedean, and πv andv′ are spherical principal series representations with trivial central character, then the localL-factor and local integral are, in fact, equal. This result verifies a conjecture of Bump, which predicts that the archimedean situation should, in the present context, parallel the nonarchimedean one.We also derive, as prerequisite to the above result, some identities for generalized Barnes integrals. In particular, we deduce a new transformation formula for certain single Barnes integrals, and a multiple-integral analog of the classical Barnes’ Lemma.

Generalized discrete Fourier transforms: the discrete Fourier-Riccati-Bessel transform

Abstract We present and demonstrate a discrete Fourier-Riccati-Bessel transform that incorporates Neumann boundary conditions. We also suggest that this methodology, which originated with H.F. Johnson, can be applied to other discrete Fourier-type transforms with different boundary conditions. This project was originally motivated by the development of the generalized Fourier-grid R -matrix scattering method. We present the transform itself, its derivation, analysis of the approximations involved, and some applications.

Generalized Fourier-grid R-matrix theory; a discrete Fourier-Riccati-Bessel transform approach

The authors present the latest developments in the Fourier-grid R-matrix theory of scattering. These developments are based on the generalized Fourier-grid formalism and use a new type of extended discrete Fourier transform: the discrete Fourier-Riccati-Bessel transform. They apply this new R-matrix approach to problems of potential scattering, to demonstrate how this method reduces computational effort by incorporating centrifugal effects into the representation. As this technique is quite new, they have hopes to broaden the formalism to many types of problems.

Coxeter group actions on 4F3(1) hypergeometric series

In this paper we investigate a certain linear combination

Relations among complementary and supplementary pairings of Saalschützian {}_4F_3(1) series

K(\vec{x})=K(a;b,c,d;e,f,g)

Mellin transforms of GL(n, R) whittaker functions

of two Saalschutzian hypergeometric series of type 4F3(1). We first show that

New formulas for Whittaker functions on GL(n,R)

K(\vec{x})

Hypergeometric series and Euler factors at infinity for L-functions on GL(3,R)×GL(3,R)

is invariant under the action of a certain matrix group GK, isomorphic to the symmetric group S6, acting on the affine hyperplane V={(a,b,c,d,e,f,g)∈ℂ7:e+f+g−a−b−c−d=1}. We further develop an algebra of three-term relations for K(a;b,c,d;e,f,g). We show that, for any three elements μ1,μ2,μ3 of a certain matrix group MK, isomorphic to the Coxeter group W(D6) (of order 23040) and containing the above group GK, there is a relation among

Mellin transforms of Whittaker functions onGL(4,ℝ) andGL(4,ℂ)

K(\mu_{1}\vec{x})

Hypergeometric Series, a Barnes-Type Lemma, and Whittaker Functions

,