Infinite product representations of some q-series

For integers a and b (not both 0) we define the integers c ( a , b ; n ) ( n = 0 , 1 , 2 , … ) by ∑ n = 0 ∞ c ( a , b ; n ) q n = ∏ n = 1 ∞ 1 - q n a ( 1 - q 2 n ) b ( | q | < 1 ) . These integers include the numbers t k ( n ) = c ( - k , 2 k ; n ) , which count the number of representations of n as a sum of k triangular numbers, and the numbers ( - 1 ) n r k ( n ) = c ( 2 k , - k ; n ) , where r k ( n ) counts the number of representations of n as a sum of k squares. A computer search was carried out for integers a and b , satisfying - 24 ≤ a , b ≤ 24 , such that at least one of the sums 0.1 ∑ n = 0 ∞ c ( a , b ; 3 n + j ) q n , j = 0 , 1 , 2 , is either zero or can be expressed as a nonzero constant multiple of the product of a power of q and a single infinite product of factors involving powers of 1 - q rn with r ∈ { 1 , 2 , 3 , 4 , 6 , 8 , 12 , 24 } for all powers of q up to q 1000 . A total of 84 such candidate identities involving 56 pairs of integers ( a , b ) all satisfying a ≡ b ( mod 3 ) were found and proved in a uniform manner. The proof of these identities is extended to establish general formulas for the sums (0.1). These formulas are used to determine formulas for the sums ∑ n = 0 ∞ t k ( 3 n + j ) q n , ∑ n = 0 ∞ r k ( 3 n + j ) q n , j = 0 , 1 , 2 .