Department of Mathematics and Statistics, San Diego State University

Seminars on Differential and Difference Equations,

and Dynamical Systems

Some new asymptotics for Whittaker's confluent hypergeometric function.

Mark Dunster, Mathematics and Statistics, SDSU

Recently Andreas Strombergsson of Bristol University, UK, asked me about the asymptotics of the W Whittaker confluent hypergeometric function. This W function can be expressed in terms of the famous U confluent hypergeometric function. The desired asymptotic results are required for obtaining bounds on certain sums of the Fourier coefficients of Maass Wave eigenfunctions, and also for estimates on the equidistribution of non-closed horocycles on non-compact surfaces. The W function has two parameters tex2html_wrap_inline16 and tex2html_wrap_inline18 , along with a dependent variable x. The asymptotics he asked for are non-trivial! Namely, he wants tex2html_wrap_inline16 purely imaginary with tex2html_wrap_inline24 , uniformly for tex2html_wrap_inline26 as well as for tex2html_wrap_inline28 . I show how such results can be obtained using a general asymptotic theory from O.D.E.s that I obtained at SDSU in 1990. In this application it is necessary to take x into the complex plane.

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