Aritram Dhar

University of Florida | UF · Department of Mathematics

Analogous to Andrews' and Newman's discovery and work on the minimal excludant or "mex" of partitions, we define four new classes of minimal excludants for overpartitions and unearth relations to certain functions due to Ramanujan.

Motivated by recent research of Krattenthaler and Wang, we propose five new "Borwein-type" conjectures modulo 3 and two new "Borwein-type" conejctures modulo 5.

In 1797, Pfaff gave a simple proof of a ${}_3F_2$ hypergeometric series which was much later reproved by Andrews in 1996. In the same paper, Andrews also proved other well-known hypergeometric identities using Pfaff's method. In this paper, we prove a number of terminating $q$-hypergeometric series-product identities using Pfaff's method thereby pr...

In this paper, we conjecture an extension to Bressoud's 1996 generalization of Borwein's famous 1990 conjecture. We then state two infinite hierarchies of non-negative $q$-series identities which are interesting and elegant examples of our proposed conjecture and Bressoud's generalized conjecture respectively. Finally, using certain positivity-pres...

In 2009, Berkovich and Garvan introduced a new partition statistic called the GBG-rank modulo $t$ which is a generalization of the well-known BG-rank. In this paper, we use the Littlewood decomposition of partitions to study partitions with bounded largest part and fixed integral value of GBG-rank modulo primes. As a consequence, we obtain new eleg...

In this paper, we provide proofs of two [Formula: see text] summation formulas of Bailey using a [Formula: see text] identity of Carlitz. We show that in the limiting case, the two [Formula: see text] identities give rise to two [Formula: see text] summation formulas of Bailey. Finally, we prove the two [Formula: see text] identities using a techni...

The BG-rank BG($\pi$) of an integer partition $\pi$ is defined as $$\text{BG}(\pi) := i-j$$ where $i$ is the number of odd-indexed odd parts and $j$ is the number of even-indexed odd parts of $\pi$. In a recent work, Fu and Tang ask for a direct combinatorial proof of the following identity of Berkovich and Uncu $$B_{2N+\nu}(k,q)=q^{2k^2-k}\left[\b...

In a recent pioneering work, Andrews and Newman defined an extended function $p_{A,a}(n)$ of their minimal excludant or "mex" of a partition function. By considering the special cases $p_{k,k}(n)$ and $p_{2k,k}(n)$, they unearthed connections to the rank and crank of partitions and some restricted partitions. In this paper, we build on their work a...

In this paper, we provide proofs of two ${}_5\psi_5$ summation formulas of Bailey using a ${}_5\phi_4$ identity of Carlitz. We show that in the limiting case, the two ${}_5\psi_5$ identities give rise to two ${}_3\psi_3$ summation formulas of Bailey. Finally, we prove the two ${}_3\psi_3$ identities using a technique initially used by Ismail to pro...

In a recent pioneering work, Andrews and Newman defined an extended function $p_{A,a}(n)$ of their minimal excludant or "mex" of a partition function. By considering the special cases $p_{k,k}(n)$ and $p_{2k,k}(n)$, they unearthed connections to the rank and crank of partitions and some restricted partitions. In this paper, we build on their work a...

In this paper, we first provide an analytic and a bijective proof of a formula stated by Vladeta Jovovic in the OEIS sequence A117989. We also provide a bijective proof of another interesting result stated by him on the same page concerning integer partitions with fixed differences between the largest and smallest parts.