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It is known that the numbers which occur in Apéry's proof of the irrationality of ζ(2) have many interesting congruence properties while the associated generating function satisfies a second order differential equation. We prove supercongruences for a generalization of numbers which arise in Beukers' and Zagier's study of integral solutions of Apéry-like… (More)
We prove two congruences for the coefficients of power series expansions in t of modular forms where t is a modular function. As a result, we settle two recent conjectures of Chan, Cooper and Sica. Additionally, we provide tables of congruences for numbers which appear in similar power series expansions and in the study of integral solutions of Apéry-like… (More)
Following Rankin's method, D. Zagier computed the n-th Rankin-Cohen bracket of a modular form g of weight k 1 with the Eisenstein series of weight k 2 and then computed the inner product of this Rankin-Cohen bracket with a cusp form f of weight k = k 1 + k 2 + 2n and showed that this inner product gives, upto a constant, the special value of the… (More)
Using the relationship between Jacobi forms of half-integral weight and vector valued modular forms, we obtain the number of components which determine the given Jacobi form of index p, p 2 or pq, where p and q are odd primes.
Using techniques due to Coster, we prove a supercongruence for a generalization of the Domb numbers. This extends a recent result of Chan, Cooper and Sica and confirms a con-jectural supercongruence for numbers which are coefficients in one of Zagier's seven " sporadic " solutions to second order Apéry-like differential equations.
We prove two supercongruences for the coefficients of power series expansions in t of modular forms where t is a modular function. As a result, we settle two recent conjectures of Chan, Cooper and Sica. Additionally, we provide a table of supercongruences for numbers which appear in similar power series expansions and in the study of integral solutions of… (More)
We prove two-term supercongruences for generalizations of recently discovered sporadic sequences of Cooper. We also discuss recent progress and future directions concerning other types of supercon-gruences.
We use Rankin-Cohen brackets for modular forms and quasimodular forms to give a different proof of the results obtained by D. Lanphier  and D. Niebur  on the van der Pol type identities for the Ramanujan's tau function τ (n). We also obtain new identities for τ (n) and as consequences we obtain convolution sums and congruence relations involving the… (More)