Cooper–Hirschhorn type identities for three or more squares. III

Let dollar sign rs left parenthesis n right parenthesis dollar sign rₛ(n) r s ( n ) denote the number of representations of dollar sign n dollar sign n n as a sum of dollar sign s dollar sign s s squares. Hurwitz established eleven identities expressing the generating function of dollar sign r 3 left parenthesis an plus b right parenthesis dollar sign r₃(an+b) r 3 ( a n + b ) as a simple infinite product. Cooper and Hirschhorn (Discrete Math 274 (1-3):9–24, 2004) proved that for any dollar sign k greater than or equals 0 dollar sign k≥0 k ≥ 0 , the generating functions dollar sign sigma summation n equals 0 infinity r 3 left parenthesis 32 kn right parenthesis qn dollar sign ∑ₙ₌₀^∞ r₃(3²ᵏn)qⁿ ∑ n = 0 ∞ r 3 ( 3 2 k n ) q n and dollar sign sigma summation n equals 0 infinity r 3 left parenthesis 32 k plus 1 n right parenthesis qn dollar sign