THE RAMANUJAN JOURNAL, Vol. 14, No. 2 (2007), 223-247.

The Atkin orthogonal polynomials for congruence subgroups of low levels

HIROYUKI TSUTSUMI
Osaka University of Health and Sport Sciences, 1-1 Asashirodai, Kumatori-cho, Sennan-gun, Osaka 590-0496, Japan
Email: tsutsumi@ouhs.ac.jp

Received January 5, 2004; Accepted March 23, 2004

Abstract. The aim of this paper is to give explicit formulae for the orthogonal functions of the Atkin inner product for the noncompact arithmetic triangular groups and to relate these functions to the supersingular elliptic curves. Also, we state the Atkin orthogonal polynomials for the classical well known group $\Gamma(2)$.

Keywords: Hypergeometric series, Atkin's orthogonal polynomials

2000 Mathematics Classification: Primary 11F03, 11F11, 11F25