Binary quadratic forms and the Fourier coefficients of certain weight 1 eta -quotients

Abstract

We state and prove an identity which represents the most general eta-products of weight 1 by binary quadratic forms. We discuss the utility of binary quadratic forms in finding a multiplicative completion for certain eta-quotients. We then derive explicit formulas for the Fourier coefficients of certain eta-quotients of weight 1 and level 47, 71, 135, 648, 1024, and 1872.

A note on the parity of broken 7-diamond partitions

“…In this paper we find more congruences for ∆ 7 (n) by using a result of Berkovich and Patane [3]. The following theorem was proved in [1].…”

“…To prove Theorem 1.2 we will heavily depend on the following result of Berkovich and Patane [3]. For convenience we record it here.…”

On a generalized identity connecting theta series associated with discriminants $\varDelta $ and $\varDelta p^2$

“…The class group for ∆ = −3 consists of the single reduced form (1, 1, 1). The class group for discriminant ∆p 2 = −3 · 7 2 = 147 consists of the two reduced forms (1,1,37) and (3,3,13). The forms in (3.1) (counting repetition) consist of (1, 1, 37) union the forms listed in Table 1.…”

“…Let ∆ = −23 and p = 3. The class group and genus structure for the relevant discriminants is given by (1, 1, 52), (4, 1, 13), (4, −1, 13) +1 +1 (8,7,8), (2,1,26), (2, −1, 26) +1 −1 .…”

“…Theorem 7.1 gives the total number of representations by all forms of a given genus of discriminant −207. To find (a, b, c; n) for any particular form of discriminant −207, one can employ the techniques of [1].…”