Congruences Relating Regular Partition Functions, a Genearalised Tau Function and Partition Function Weighted Composition Sums
S. Sriram, A. David Christopher
TL;DR
The paper studies the generalized tau-function $\tau_k(n)$ defined by $q\prod_{m=1}^{\infty}(1-q^m)^k=\sum_{n\ge1}\tau_k(n)q^n$ and connects it to $t$-regular partitions $R_t(n)$. It derives a binomial-coefficient partition sum representation for $\tau_k(n)$, enabling congruence relations with $R_t(n)$ and partition-weighted composition sums, and then develops parity and modular results across multiple primes and moduli via logarithmic differentiation, binomial identities, and Euler-type product expansions. The work provides explicit modulo results for $\tau_k(n)$ (and special cases like Ramanujan's $\tau(n)$) and recursive/structural relations for $R_9(n)$ and $R_p(n)$, culminating in a general theorem on weighted composition-sum divisibility that yields Ramanujan-type congruences as corollaries. Overall, it offers a cohesive framework linking regular partitions, generalized tau functions, and weighted partition sums with broad implications for arithmetic of partitions and modular forms. Key mechanisms include $q$-series manipulations, Euler's pentagonal theorem, Jacobi's triple product, and binomial-modulus techniques.
Abstract
Let $n$ and $t$ be positive integers with $t\geq 2$. Let $R_t(n)$ be the number of $t$-regular partitions of $n$. A class of functions, denoted $τ_k(n)$, is defined as follows: \[q\prod_{m=1}^{\infty}(1-q^m)^k=\sum_{n=1}^{\infty}τ_k(n)q^n, \] where $k$ is an integer. We express $τ_k(n)$ as a binomial coefficient weighted partition sum. Consequently, we obtain congruence identities that relate $τ_k(n)$, $R_t(n)$ and partition function weighted composition sums.