engleski

Modular forms, hypergeometric functions and congruences

Using the theory of Stienstra and Beukers (Math. Ann., 271: 269-304, 1985), we prove that the numbers A(3)(n) and A(3)(n - 1) - 16A(3)(n - 2) satisfy three term congruence relations similar to those satisfied by Apery numbers. Moreover, for k >= 3 and p prime, we prove divisibility by p of some simple linear combinations of the numbers Ak(n), for n is an element of N. To obtain this, we study the arithmetic properties of the Fourier coefficients of certain holomorphic and weakly holomorphic modular forms.

modular forms ; hypergeometric functions ; congruences

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