The action of Hecke operators on hypergeometric functions (bibtex)

by Victor H. Moll, Sinai Robins, Kirk Soodhalter

Abstract:

We study the action of the Hecke operators \(\lbrace U_n \rbrace_1^\infty\) on the set of hypergeometric functions, as well as on formal power series. We show that the spectrum of these operators on the set of hypergeometric functions is the set \(\lbrace na:n\in \mathbb N , a\in \mathbb Z \rbrace\), and that the polylogarithms play an important role in the study of the eigenfunctions of the Hecke operators \(\lbrace U_n \rbrace_1^\infty\) on the set of hypergeometric functions. As a corollary of our results on simultaneous eigenfunctions, we also obtain an apparently unrelated result regarding the behavior of completely multiplicative hypergeometric coefficients.

Reference:

The action of Hecke operators on hypergeometric functions (Victor H. Moll, Sinai Robins, Kirk Soodhalter), In J. Aust. Math. Soc., volume 89, 2010.

Bibtex Entry:

@ARTICLE{MRS.2010, author = {Moll, Victor H. and Robins, Sinai and Soodhalter, Kirk}, title = {The action of {H}ecke operators on hypergeometric functions}, journal = {J. Aust. Math. Soc.}, year = {2010}, volume = {89}, pages = {51--74}, number = {1}, abstract = {We study the action of the Hecke operators \(\lbrace U_{n} \rbrace_{1}^{\\infty}\) on the set of hypergeometric functions, as well as on formal power series. We show that the spectrum of these operators on the set of hypergeometric functions is the set \(\lbrace na:n\\in \mathbb N , a\\in \mathbb Z \rbrace\), and that the polylogarithms play an important role in the study of the eigenfunctions of the Hecke operators \(\lbrace U_{n} \rbrace_{1}^{\\infty}\) on the set of hypergeometric functions. As a corollary of our results on simultaneous eigenfunctions, we also obtain an apparently unrelated result regarding the behavior of completely multiplicative hypergeometric coefficients.}, doi = {10.1017/S1446788710001461}, fjournal = {Journal of the Australian Mathematical Society}, issn = {1446-7887}, keywords = {paper}, mrclass = {11F25 (33C05 33C70)}, mrnumber = {2727063 (2011m:11086)}, mrreviewer = {Thomas R. Shemanske}, owner = {kirk}, timestamp = {2012.08.16}, url = {http://arxiv.org/abs/1005.2946} }

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