Congruences Relating Regular Partition Functions, a Genearalised Tau Function and Partition Function Weighted Composition Sums

Let n and t be positive integers with t≥ 2. Let Rₜ(n) be the number of t-regular partitions of n. A class of functions, denoted τₖ(n), is defined as follows: \[q∏ₘ₌₁∞(1-q^m)^k=∑ₙ₌₁∞τ_k(n)q^n, \] where k is an integer. We express τₖ(n) as a binomial coefficient weighted partition sum. Consequently, we obtain congruence identities that relate τₖ(n), Rₜ(n) and partition function weighted composition sums.