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” Whittaker functions for and . The integrals obtained are seen to resemble very closely known integrals for the “class-one” Whittaker functions on these groups. It is expected that this correspondence between expressions for different types of Whittaker functions will carry over to groups of arbitrary rank.
Dans cet article nous obtenons, à l’aide du théorème de la sommation de Gauss pour les séries hypergéométriques, une expression intégrale très simple pour l’inverse de la fonction beta d’Euler. Cette expression est semblable dans sa forme à plusieurs intégrales bien connues de la fonction beta proprement dite.
Nous appliquons ensuite notre nouvelle formule aux fonctions de Whittaker pour les groupes . Ces dernières sont des fonctions spéciales que l’on retrouve dans la théorie de Fourier appliquée aux formes automorphes du groupe linéaire général. Plus précisément, nous déduisons des représentations intégrales explicites pour les fonctions “fondamentales” de Whittaker pour les cas et . Les intégrales ainsi obtenues sont très semblables aux intégrales connues des fonctions “classe-première” de Whittaker sur ces groupes. Nous croyons que cette correspondance entre les expressions des différents types de fonctions de Whittaker, pourra s’étendre aux groupes de rang arbitraire.
@article{AIF_1994__44_1_93_0, author = {Stade, Eric}, title = {The reciprocal of the beta function and $GL(n, {\mathbb {R}})$ {Whittaker} functions}, journal = {Annales de l'Institut Fourier}, pages = {93--108}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {44}, number = {1}, year = {1994}, doi = {10.5802/aif.1390}, mrnumber = {96h:11046}, zbl = {0801.33001}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.1390/} }
TY - JOUR AU - Stade, Eric TI - The reciprocal of the beta function and $GL(n, {\mathbb {R}})$ Whittaker functions JO - Annales de l'Institut Fourier PY - 1994 SP - 93 EP - 108 VL - 44 IS - 1 PB - Institut Fourier PP - Grenoble UR - http://archive.numdam.org/articles/10.5802/aif.1390/ DO - 10.5802/aif.1390 LA - en ID - AIF_1994__44_1_93_0 ER -
%0 Journal Article %A Stade, Eric %T The reciprocal of the beta function and $GL(n, {\mathbb {R}})$ Whittaker functions %J Annales de l'Institut Fourier %D 1994 %P 93-108 %V 44 %N 1 %I Institut Fourier %C Grenoble %U http://archive.numdam.org/articles/10.5802/aif.1390/ %R 10.5802/aif.1390 %G en %F AIF_1994__44_1_93_0
Stade, Eric. The reciprocal of the beta function and $GL(n, {\mathbb {R}})$ Whittaker functions. Annales de l'Institut Fourier, Volume 44 (1994) no. 1, pp. 93-108. doi : 10.5802/aif.1390. http://archive.numdam.org/articles/10.5802/aif.1390/
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