Abstract
It is a well-known fact that the function f Q(τ)= ∑ n=0 ∞ P(nQ+l) exp 2πiτ n+ 24l−1 24Q has the behavior of a modular function of level (“Stufe”) Q and dimension 1 2 , if Q is an integrr and l satisfies the condition 24 l ≡ 1 ( Q) and 0 ≤ l < Q. In paper we show that these conditions are also necessary. Furthermore, the zeros and poles (including main parts) in all rational points are determined. Several applications are given. For q = 5, 7 or 13 we get well-known results with very simple proof. For q = 17 we get a new identity, and a congruence relation between the number of partitions of partitions and the number of representations of quaternary quadratic forms of discriminant 17 and 17 3, respectively.
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