Let ℕ, ℕ0, ℤ, ℚ, and ℂ denote the sets of positive integers, nonnegative integers, integers, rational numbers, and complex numbers, respectively. If f (q) is a complex-valued function with (Formula presented.) we define (Formula presented.) For k ∈ ℕ we define (Formula presented.) We show how modular equations of a special form can be used in conjunction with the representation numbers of certain quadratic forms to determine (Formula presented.) for certain products qrEa1 1...Eam m. For example, we show that (Formula presented.) where N denotes the odd part of the positive integer n and (Formula presented.).

Additional Metadata
Keywords Infinite products, Modular equations, Quadratic forms, Representations, Theta functions
Persistent URL dx.doi.org/10.1007/s11139-013-9503-1
Journal Ramanujan Journal
Citation
Pehlivan, L. (Lerna), & Williams, K.S. (2014). The power series expansion of certain infinite products qr∏n=1 ∞(1-qn)a1(1-q2n)a2..(1-qmn)am. Ramanujan Journal, 33(1), 23–53. doi:10.1007/s11139-013-9503-1