# The reciprocal of the beta function and $GL(n,\mathbb{R})$ Whittaker functions

Annales de l'institut Fourier (1994)

- Volume: 44, Issue: 1, page 93-108
- ISSN: 0373-0956

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topStade, Eric. "The reciprocal of the beta function and $GL(n, {\mathbb {R}})$ Whittaker functions." Annales de l'institut Fourier 44.1 (1994): 93-108. <http://eudml.org/doc/75062>.

@article{Stade1994,

abstract = {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 $GL(n, \{\Bbb R\})$ 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 $GL(3,\{\Bbb R\})$ and $GL(4, \{\Bbb R\})$. 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.},

author = {Stade, Eric},

journal = {Annales de l'institut Fourier},

keywords = {automorphic forms; beta function; Whittaker functions},

language = {eng},

number = {1},

pages = {93-108},

publisher = {Association des Annales de l'Institut Fourier},

title = {The reciprocal of the beta function and $GL(n, \{\mathbb \{R\}\})$ Whittaker functions},

url = {http://eudml.org/doc/75062},

volume = {44},

year = {1994},

}

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

PB - Association des Annales de l'Institut Fourier

VL - 44

IS - 1

SP - 93

EP - 108

AB - 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 $GL(n, {\Bbb R})$ 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 $GL(3,{\Bbb R})$ and $GL(4, {\Bbb R})$. 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.

LA - eng

KW - automorphic forms; beta function; Whittaker functions

UR - http://eudml.org/doc/75062

ER -

## References

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- [15] E. STADE, On explicit integral formulas for GL(n, R)-Whittaker functions, Duke Mathematical Journal, 60-2 (1990), 313-362. Zbl0731.11027
- [16] E. STADE, GL(4, R)-Whittaker functions and 4F3(1) hypergeometric series, Trans. Amer. Math. Soc., 336-1 (1993), 253-264. Zbl0786.11027MR93e:22018
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