Rademacher-type exact formula and higher order Tur\'{a}n inequalities for $r$-colored $\ell$-regular partitions

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 Tur\'{a}n 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 Tur\'{a}n inequalities for the $r$-colored distinct part partition function and for the sum of minimal excludants over ordinary partitions and overpartitions.