Modular forms, hypergeometric functions and congruences

作者：Kazalicki Matija^*

来源：Ramanujan Journal, 2014, 34(1): 1-9.

DOI：10.1007/s11139-013-9477-z

• Modular forms
• Hypergeometric functions
• Congruences
• Apery numbers

Let %26lt;br%26gt;[GRAPHICS] %26lt;br%26gt;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 %26gt;= 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.

• 出版日期2014-5