𝐺𝐿(4,𝑅)-Whittaker functions and ₄𝐹₃(1) hypergeometric series

Stade, Eric

doi:10.1090/S0002-9947-1993-1102226-1

$\textrm {GL}(4,\textbf {R})$-Whittaker functions and ${}_ 4F_ 3(1)$ hypergeometric series
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Trans. Amer. Math. Soc. 336 (1993), 253-264 Request permission

Abstract:

In this paper we consider spaces of ${\text {GL}}(4,\mathbb {R})$-Whittaker functions, which are special functions that arise in the study of ${\text {GL}}(4,\mathbb {R})$ automorphic forms. Our main result is to determine explicitly the series expansion for a ${\text {GL}}(4,\mathbb {R})$-Whittaker function that is "fundamental," in that it may be used to generate a basis for the space of all ${\text {GL}}(4,\mathbb {R})$-Whittaker functions of fixed eigenvalues. The series that we find in the case of ${\text {GL}}(4,\mathbb {R})$ is particularly interesting in that its coefficients are not merely ratios of Gamma functions, as they are in the lower-rank cases. Rather, these coefficients are themselves certain series— namely, they are finite hypergeometric series of unit argument. We suspect that this is a fair indication of what will happen in the general case of ${\text {GL}}(n,\mathbb {R})$.

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18

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