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THE ACTION OF HECKE OPERATORS ON HYPERGEOMETRIC FUNCTIONS

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  21 September 2010

VICTOR H. MOLL ,
SINAI ROBINS  and
KIRK SOODHALTER
VICTOR H. MOLL*
Affiliation:
Department of Mathematics, Tulane University, New Orleans, LA 70118, USA (email: [email protected])
SINAI ROBINS
Affiliation:
Division of Mathematical Sciences, Nanyang Technological University, SPMS-MAS-03-01, 21 Nanyang Link, Singapore 637371, Singapore (email: [email protected])
KIRK SOODHALTER
Affiliation:
Department of Mathematics, Temple University, Philadelphia, PA 19122, USA (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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We study the action of the Hecke operators Un 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 {na:n∈ℕ,a∈ℤ}, and that the polylogarithms play an important role in the study of the eigenfunctions of the Hecke operators Un 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.

Keywords

Hecke operatorshypergeometric functionseigenvaluescompletely multiplicative

MSC classification

Secondary: 11F25: Hecke-Petersson operators, differential operators (one variable)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 89 , Issue 1 , August 2010 , pp. 51 - 74
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

The work of the first author was partially funded by NSF-DMS 0713836. The work of the second author was partially funded by the Singapore SUG grant given by Nanyang Technological University.

References

[1]Andrews, G. E., Askey, R. and Roy, R., Special Functions, Encyclopedia of Mathematics and its Applications, 71 (Cambridge University Press, Cambridge, 1999).CrossRefGoogle Scholar
[2]Apostol, T., Modular Functions and Dirichlet Series in Number Theory, 2nd edn, Graduate Texts in Mathematics, 41 (Springer, New York, 1990).CrossRefGoogle Scholar
[3]Bennett, C. D. and Mosteig, E., ‘On the collection of integers that index the fixed points of maps on the space of rational functions’, in: Tapas in Experimental Mathematics, Contemporary Mathematics, 457 (American Mathematical Society, Providence, RI, 2008), pp. 53–67.Google Scholar
[4]Boros, G., Little, J., Moll, V., Mosteig, E. and Stanley, R., ‘A map on the space of rational functions’, Rocky Mountain J. Math. 35(6) (2005), 1861–1880.Google Scholar
[5]Gil, J. and Robins, S., ‘Hecke operators on rational functions. I’, Forum Math. 17(4) (2005), 519–554.CrossRefGoogle Scholar
[6]Mosteig, E., ‘Fixed points of maps on the space of rational functions’, Online J. Anal. Comb. 1 (2006), Article 2, 9 p.Google Scholar
[7]Stanley, R. P., Enumerative Combinatorics, Vol. 1, Cambridge Studies in Advanced Mathematics, 49 (Cambridge University Press, Cambridge, 1997).Google Scholar