Zafer Selcuk Aygin

Fourier coefficients of a class of eta quotients of weight 2

We determine the coefficients of the Fourier series of a class of eta quotients of weight 2. For example, we show that where and are Jacobi–Kronecker symbols. We prove our results using the theory of modular forms.

Extensions of Ramanujan–Mordell formula with coefficients 1 and p

Abstract We use properties of modular forms to prove the following extension of the Ramanujan–Mordell formula, z k − j z p j = p χ k − j − 1 p χ k − 1 F p ( k , j ; τ ) + p χ k − p χ k − j p χ k − 1 F p ( k , j ; p τ ) + z k A p ( k , j ; τ ) , for all 1 k ∈ N , 0 ≤ j ≤ k and p an odd prime. We obtain this result by computing the Fourier series expansions of modular forms at all cusps of Γ 0 ( 4 p ) .

Infinite products with coefficients which vanish on certain arithmetic progressions

Let q denote a complex variable with |q| < 1. For a positive integer k let Ek = Ek(q) :=∏n=1∞(1 − qkn). If f(q) =∑n=0∞f nqn we define [f(q)]n := fn for each nonnegative integer n. In this paper, we determine results of the type [E1−8E 224] 4k+3 = 0,k = 0, 1, 2, 3,….

On Eisenstein series in M2(Γ0(N)) and their applications

Abstract Let k , N ∈ N with N square-free and k > 1 . We prove an orthogonal relation and use this to compute the Fourier coefficients of the Eisenstein part of any f ( z ) ∈ M 2 k ( Γ 0 ( N ) ) in terms of sum of divisors function. In particular, if f ( z ) ∈ E 2 k ( Γ 0 ( N ) ) , then the computation will to yield to an expression for the Fourier coefficients of f ( z ) . Then we apply our main theorem to give formulas for convolution sums of the divisor function to extend the result by Ramanujan, and to eta quotients which yields to formulas for number of representations of integers by certain families of quadratic forms. At last we give essential results to derive similar results for modular forms in a more general setting.