Rademacher-type exact formula and higher order Turán inequalities for $r$-colored $\ell$-regular partitions
Archit Agarwal, Meghali Garg, Bibekananda Maji
TL;DR
This work derives a Rademacher-type exact formula for the $r$-colored $\ell$-regular partition function $b_{\ell}^{(r)}(n)$ via the circle method, valid for all $r\ge1$ and $\ell\ge2$, and proves higher-order Turán inequalities using the Griffin–Ono–Rolen–Zagier framework. The main result expresses $b_{\ell}^{(r)}(n)$ as a convergent series involving Kloosterman-type coefficients, finite sums over $m$ and $s$, derivatives of Bessel functions, and the generating-data $p^{(r)}(m)$ and $a^{(r)}(s)$, with a clear asymptotic by $n$. Corollaries include Hagis’s exact formula for $\ell$-regular partitions ($r=1$), a Rademacher-type formula for $p_d^{(r)}(n)$ when $\ell=2$, and higher-order Turán inequalities for $b_{\ell}^{(r)}(n)$, plus analogous results for related combinatorial functions such as $\sigma\mathrm{mex}(n)$ and $\sigma\overline{\mathrm{mex}}(n)$. The results yield precise finite-sum representations and sharp asymptotics, enhancing understanding of colored and regular partition functions and their real-rootedness properties via Jensen polynomials.
Abstract
In 1937, Rademacher refined the circle method of Hardy and Ramanujan to derive an exact convergent series for the partition function $p(n)$. In 1942, Hua derived an exact formula for the distinct part partition function, and in 1971, Hagis generalized this result to the case of $\ell$-regular partitions. More recently, Iskander, Jain, and Talvola established a Rademacher-type exact formula for the $r$-colored partition function. In this paper, we employ the circle method to obtain a Rademacher-type exact formula for $r$-colored $\ell$-regular partitions for any $r \in \mathbb{N}$ and $\ell \geq 2$. As an application, we derive higher order Turán inequalities for the $r$-colored $\ell$-regular partition function using a result of Griffin, Ono, Rolen, and Zagier. Furthermore, as additional consequences, we establish Rademacher-type exact formulas and higher order Turán inequalities for the $r$-colored distinct part partition function and for the sum of minimal excludants over ordinary partitions and overpartitions.