ACTA ARITHMETICA 158.1 (2013) Quadratic forms and a product-to-sum formula by Kenneth S. Williams (Ottawa) 1. Introduction. The set of positive integers is denoted by N and the set of nonnegative integers by N 0 so that N 0 = N ∪{0}. The domain of all integers is denoted by Z and the field of complex numbers by C. Throughout this paper q ∈ C is taken to satisfy |q| < 1. For such q we define (1.1) E k = E k (q) := Y n∈N (1 - q kn ), k ∈ N. We note for later use that replacing q by -q in (1.1) gives (1.2) E k (-q)=    E 3 2k E k E 4k if k is odd, E k if k is even. If f (q)= ∑ ∞ n=0 f n q n we write [f (q)] n = f n , n ∈ N 0 . Scattered throughout the mathematical literature there are a number of results of the form (1.3) [q a E a 1 m 1 ··· E a ‘ m ‘ ] n = X (x 1 ,...,xm)∈Z m Q(x 1 ,...,xm)=n P (x 1 ,...,x m ), n ∈ N 0 , where a ∈ N 0 , ‘ ∈ N, m 1 ,...,m ‘ ∈ N with m 1 < ··· <m ‘ , a 1 ,...,a ‘ ∈ Z \{0}, m ∈ N, P is a polynomial in x 1 ,...,x m with rational coefficients and Q is a positive-definite, diagonal, quadratic form in x 1 ,...,x m with integral coefficients. For example it is a classical result of Klein and Fricke 2010 Mathematics Subject Classification : Primary 11E25; Secondary 11F20, 11F25. Key words and phrases : quadratic forms, theta functions, Eisenstein series, infinite prod- ucts, product-to-sum formulae. DOI: 10.4064/aa158-1-5 [79] c  Instytut Matematyczny PAN, 2013