Modular forms, hypergeometric functions and congruences

Abstract

Using the theory of Stienstra and Beukers [9], we prove various elementary congruences for the numbersTo obtain that, we study the arithmetic properties of Fourier coefficients of certain (weakly holomorphic) modular forms.

Congruences for sporadic sequences and modular forms for non-congruence subgroups

“…Recently, Osburn and Straub [11] proved similar congruences for all but one of the six Zagier's sporadic sequences (three cases were already known to be true by the work of Stienstra and Beukers) and conjectured the congruence for the sixth sequence F (n) (which is a solution of recursion determined by triple (17,6,72). In this paper we prove that remaining congruence.…”

“…satisfies a three-term Atkin and Swinnerton-Dyer congruence relation with respect to f (τ ) for all primes p > 3 (see Proposition 2). The similar idea was used previously by the author [6] in proving three term congruence relations for some multinomial sums by employing Atkin and Swinnerton-Dyer congruence relations satisfied by the Fourier coefficients of certain weakly holomorphic modular forms (but for congruence subgroups).…”

“…Inspired by Beukers [3], Zagier [18] performed a computer search on first 100 million triples (A, B, C) ∈ Z 3 and found that the recursive relation generalizing b n (n + 1)u n+1 − (An 2 + An + B)u n + Cn 2 u n−1 = 0, 1 with the initial conditions u −1 = 0 and u 0 = 1 has (non-degenerate i.e. C(A 2 − 4C) = 0) integral solution for only six more triples (whose solutions are so called sporadic sequences) (0, 0, −16), (7, 2, −8), (9,3,27), (10,3,9), (12,4,32) and (17,6,72).…”

“…

In 1979, in the course of the proof of the irrationality of ζ(2) Robert Apéry introduced numbers bn = n k=0 n k 2 n+k k that are, surprisingly, integral solutions of recursive relations (n + 1) 2 un+1 − (11n 2 + 11n + 3)un − n 2 un−1 = 0.Zagier performed a computer search on first 100 million triples (A, B, C) ∈ Z 3 and found that the recursive relation generalizing bnwith the initial conditions u−1 = 0 and u0 = 1 has (non-degenerate i.e. C(A 2 − 4C) = 0)integral solution for only six more triples (whose solutions are so called sporadic sequences) .Stienstra and Beukers showed that for the prime p ≥ 5Recently, Osburn and Straub proved similar congruences for all but one of the six Zagier's sporadic sequences (three cases were already known to be true by the work of Stienstra and Beukers) and conjectured the congruence for the sixth sequence (which is a solution of recursion determined by triple (17,6, 72).In this paper we prove that remaining congruence by studying Atkin and Swinnerton-Dyer congruences between Fourier coefficients of certain cusp form for non-congurence subgroup.

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Asymptotic expansions of the hypergeometric function with two large parameters—application to the partition function of a lattice gas in a field of traps

“…The above discussion provides just one of the many instances in which the Gauss hypergeometric function -hereafter abbreviated as the HGF -appears in various fields of physics and mathematics [27][28][29][30][31][32][33][34][35][36]. In many of the physical applications of the HGF F a,b c ; z , some of its parameters (a, b and/or c) adopt very large values [1-3, 5-8, 10-12, 14, 19, 21-23, 25, 27, 28].…”