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Rupam Barman

Publications and source records attributed to Rupam Barman.

At least 55 records · Page 3Linked to original sources
arXiv2021

Eta-quotients and divisibility of certain partition functions by powers of primes

Andrews' $(k, i)$-singular overpartition function $\overline{C}_{k, i}(n)$ counts the number of overpartitions of $n$ in which no part is divisible by $k$ and only parts $\equiv \pm i\pmod{k}$ may be overlined. In recent times, divisibility of $\overline{C}_{3\ell, \ell}(n)$, $\overline{C}_{4\ell, \ell}(n)$ and $\overline{C}_{6\ell, \ell}(n)$ by $2$ and $3$ are studied for certain values of $\ell$. In this article, we study divisibility of $\overline{C}_{3\ell, \ell}(n)$, $\overline{C}_{4\ell, \ell}(n)$ and $\overline{C}_{6\ell, \ell}(n)$ by primes $p\geq 5$. For all positive integer $\ell$ and prime divisors $p\geq 5$ of $\ell$, we prove that $\overline{C}_{3\ell, \ell}(n)$, $\overline{C}_{4\ell, \ell}(n)$ and $\overline{C}_{6\ell, \ell}(n)$ are almost always divisible by arbitrary powers of $p$. For $s\in \{3, 4, 6\}$, we next show that the set of those $n$ for which $\overline{C}_{s\cdot\ell, \ell}(n) \not\equiv 0\pmod{p_i^k}$ is infinite, where $k$ is a positive integer satisfying $p_i^{k-1}\geq \ell$. We further improve a result of Gordon and Ono on divisibility of $\ell$-regular partitions by powers of certain primes. We also improve a result of Ray and Chakraborty on divisibility of $\ell$-regular overpartitions by powers of certain primes.

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arXiv2021

Higher order moments of generalized quadratic Gauss sums weighted by $L$-functions

The main purpose of this paper is to study higher order moments of the generalized quadratic Gauss sums weighted by $L$-functions using estimates for character sums and analytic methods. We find asymptotic formulas for three character sums which arise naturally in the study of higher order moments of the generalized quadratic Gauss sums. We then use these character sum estimates to find asymptotic formulas for the $\nth{6}$ and $\nth{8}$ order moments of the generalized quadratic Gauss sums weighted by $L$-functions. Our asymptotic formulas satisfy a conjecture of Wenpeng Zhang.

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arXiv2020

On Mex-related partition functions of Andrews and Newman

The minimal excludant, or "mex" function, on a set $S$ of positive integers is the least positive integer not in $S$. In a recent paper, Andrews and Newman extended the mex-function to integer partitions and found numerous surprising partition identities connected with these functions. Very recently, da Silva and Sellers present parity considerations of one of the families of functions Andrews and Newman studied, namely $p_{t,t}(n)$, and provide complete parity characterizations of $p_{1,1}(n)$ and $p_{3,3}(n)$. In this article, we study the parity of $p_{t,t}(n)$ when $t=2^α, 3\cdot 2^α$ for all $α\geq 1$. We prove that $p_{2^α,2^α}(n)$ and $p_{3\cdot2^α, 3\cdot2^α}(n)$ are almost always even for all $α\geq 1$. Using a result of Ono and Taguchi on nilpotency of Hecke operators, we also find infinite families of congruences modulo $2$ satisfied by $p_{2^α,2^α}(n)$ and $p_{3\cdot2^α, 3\cdot2^α}(n)$ for all $α\geq 1$.

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arXiv2020

Mex-related partitions and relations to ordinary partition and singular overpartitions

In a recent paper, Andrews and Newman introduced certain families of partition functions using the minimal excludant or "mex" function. In this article, we study two of the families of functions Andrews and Newman introduced, namely $p_{t,t}(n)$ and $p_{2t,t}(n)$. We establish identities connecting the ordinary partition function $p(n)$ to $p_{t,t}(n)$ and $p_{2t,t}(n)$ for all $t\geq 1$. Using these identities, we prove that the Ramanujan's famous congruences for $p(n)$ are also satisfied by $p_{t,t}(n)$ and $p_{2t,t}(n)$ for infinitely many values of $t$. Very recently, da Silva and Sellers provide complete parity characterizations of $p_{1,1}(n)$ and $p_{3,3}(n)$. We prove that $p_{t,t}(n)\equiv \overline{C}_{4t,t}(n) \pmod{2}$ for all $n\geq 0$ and $t\geq 1$, where $\overline{C}_{4t,t}(n)$ is the Andrews' singular overpartition function. Using this congruence, the parity characterization of $p_{1,1}(n)$ given by da Silva and Sellers follows from that of $\overline{C}_{4,1}(n)$. We also give elementary proofs of certain congruences already proved by da Silva and Sellers.

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arXiv2020

Certain product formulas and values of Gaussian hypergeometric series

In this article we find finite field analogues of certain product formulas satisfied by the classical hypergeometric series. We express product of two ${_2}F_1$-Gaussian hypergeometric series as ${_4}F_3$- and ${_3}F_2$-Gaussian hypergeometric series. We use properties of Gauss and Jacobi sums and our earlier works on finite field Appell series to deduce these product formulas satisfied by the Gaussian hypergeometric series. We then use these transformations to evaluate explicitly some special values of ${_4}F_3$- and ${_3}F_2$-Gaussian hypergeometric series. By counting points on CM elliptic curves over finite fields, Ono found certain special values of ${_2}F_1$- and ${_3}F_2$-Gaussian hypergeometric series containing trivial and quadratic characters as parameters. Later, Evans and Greene found special values of certain ${_3}F_2$-Gaussian hypergeometric series containing arbitrary characters as parameters from where some of the values obtained by Ono follow as special cases. We show that some of the results of Evans and Greene follow from our product formulas including a finite field analogue of the classical Clausen's identity.

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arXiv2020

On Andrews' integer partitions with even parts below odd parts

Recently, Andrews defined a partition function $\mathcal{EO}(n)$ which counts the number of partitions of $n$ in which every even part is less than each odd part. He also defined a partition function $\overline{\mathcal{EO}}(n)$ which counts the number of partitions of $n$ enumerated by $\mathcal{EO}(n)$ in which only the largest even part appears an odd number of times. Andrews proposed to undertake a more extensive investigation of the properties of $\overline{\mathcal{EO}}(n)$. In this article, we prove infinite families of congruences for $\overline{\mathcal{EO}}(n)$. We next study parity properties of $\overline{\mathcal{EO}}(n)$. We prove that there are infinitely many integers $N$ in every arithmetic progression for which $\overline{\mathcal{EO}}(N)$ is even; and that there are infinitely many integers $M$ in every arithmetic progression for which $\overline{\mathcal{EO}}(M)$ is odd so long as there is at least one. Very recently, Uncu has treated a different subset of the partitions enumerated by $\mathcal{EO}(n)$. We prove that Uncu's partition function is divisible by $2^k$ for almost all $k$. We use arithmetic properties of modular forms and Hecke eigenforms to prove our results.

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arXiv2019

Divisibility of Andrews' Singular Overpartitions by Powers of 2 and 3

Andrews introduced the partition function $\overline{C}_{k, i}(n)$, called singular overpartition, which counts the number of overpartitions of $n$ in which no part is divisible by $k$ and only parts $\equiv \pm i\pmod{k}$ may be overlined. He also proved that $\overline{C}_{3, 1}(9n+3)$ and $\overline{C}_{3, 1}(9n+6)$ are divisible by $3$ for $n\geq 0$. Recently Aricheta proved that for an infinite family of $k$, $\overline{C}_{3k, k}(n)$ is almost always even. In this paper, we prove that for any positive integer $k$, $\overline{C}_{3, 1}(n)$ is almost always divisible by $2^k$ and $3^k.$

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arXiv2018

Arithmetic properties of cubic and overcubic partition pairs

Let $b(n)$ denote the number of cubic partition pairs of $n$. We give affirmative answer to a conjecture of Lin, namely, we prove that $$b(49n+37)\equiv 0 \pmod{49}.$$ We also prove two congruences modulo $256$ satisfied by $\overline{b}(n)$, the number of overcubic partition pairs of $n$. Let $\overline{a}(n)$ denote the number of overcubic partition of $n$. For a fixed positive integer $k$, we further show that $\overline{b}(n)$ and $\overline{a}(n)$ are divisible by $2^k$ for almost all $n$. We use arithmetic properties of modular forms to prove our results.

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arXiv2018

Supercongruences for truncated hypergeometric series and p-adic gamma function

We prove three more general supercongruences between truncated hypergeometric series and $p$-adic Gamma function from which some known supercongruences follow. A supercongruence conjectured by Rodriguez-Villegas and proved by E. Mortenson using the theory of finite field hypergeometric series follows from one of our more general supercongruences. We also prove a supercongruence for ${_7}F_6$ truncated hypergeometric series which is similar to a supercongruence proved by L. Long and R. Ramakrishna.

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arXiv2018

Certain character sums and hypergeometric series

We prove two transformations for the $p$-adic hypergeometric series which can be described as $p$-adic analogues of a Kummer's linear transformation and a transformation of Clausen. We first evaluate two character sums, and then relate them to the $p$-adic hypergeometric series to deduce the transformations. We also find another transformation for the $p$-adic hypergeometric series from which many special values of the $p$-adic hypergeometric series as well as finite field hypergeometric functions are obtained.

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arXiv2018

A finite field analogue of the Appell series $F_4$

We define a function $F_4^{\ast}$ as a finite field analogue of the classical Appell series $F_4$ using Gauss sums. We establish identities for $F_4^{\ast}$ analogous to those satisfied by the classical Appell series $F_4$.

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arXiv2016

Summation identities and transformations for hypergeometric series

We find summation identities and transformations for the McCarthy's $p$-adic hypergeometric series by evaluating certain Gauss sums which appear while counting points on the family $$Z_λ: x_1^d+x_2^d=dλx_1x_2^{d-1}$$ over a finite field $\mathbb{F}_p$. A. Salerno expresses the number of points over a finite field $\mathbb{F}_p$ on the family $Z_λ$ in terms of quotients of $p$-adic gamma function under the condition that $d|p-1$. In this paper, we first express the number of points over a finite field $\mathbb{F}_p$ on the family $Z_λ$ in terms of McCarthy's $p$-adic hypergeometric series for any odd prime $p$ not dividing $d(d-1)$, and then deduce two summation identities for the $p$-adic hypergeometric series. We also find certain transformations and special values of the $p$-adic hypergeometric series. We finally find a summation identity for the Greene's finite field hypergeometric series.

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arXiv2016

Proof of a Limited Version of Mao's Partition Rank Inequality using a Theta Function Identity

Ramanujan's congruence $p(5k+4) \equiv 0 \pmod 5$ led Dyson \cite{dyson} to conjecture the existence of a measure "rank" such that $p(5k+4)$ partitions of $5k+4$ could be divided into sub-classes with equal cardinality to give a direct proof of Ramanujan's congruence. The notion of rank was extended to rank differences by Atkin and Swinnerton-Dyer \cite{atkin}, who proved Dyson's conjecture. More recently, Mao proved several equalities and inequalities, leaving some as conjectures, for rank differences for partitions modulo 10 \cite{mao10} and for $M_2$ rank differences for partitions with no repeated odd parts modulo $6$ and $10$ \cite{maom2}. Alwaise et. al. proved four of Mao's conjectured inequalities \cite{swisher}, while leaving three open. Here, we prove a limited version of one of the inequalities conjectured by Mao.

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arXiv2014

Certain Transformations for Hypergeometric series in $p$-adic setting

In \cite{mccarthy2}, McCarthy defined a function $_{n}G_{n}[\cdots]$ using the Teichmüller character of finite fields and quotients of the $p$-adic gamma function. This function extends hypergeometric functions over finite fields to the $p$-adic setting. In this paper, we give certain transformation formulas for the function $_{n}G_{n}[\cdots]$ which are not implied from the analogous hypergeometric functions over finite fields.

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arXiv2013

p-adic Gamma function and traces of Frobenius of elliptic curves

In \cite{mccarthy2}, McCarthy defined a function $_{n}G_{n}[\cdots]$ using Teichmüller character of finite fields and quotients of $p$-adic gamma function, and expressed the trace of Frobenius of elliptic curves in terms of special values of $_{2}G_{2}[\cdots]$. We establish two different expressions for the traces of Frobenius of elliptic curves in terms of the function $_{2}G_{2}[\cdots]$. As a result, we obtain two relations between special values of the function $_{2}G_{2}[\cdots]$ with different parameters.

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arXiv2013

Hyperelliptic curves over $\mathbb{F}_q$ and Gaussian hypergeometric series

Let $d\geq2$ be an integer. Denote by $E_d$ and $E'_{d}$ the hyperelliptic curves over $\mathbb{F}_q$ given by $$E_d: y^2=x^d+ax+b~~~ \text{and} ~~~E'_d: y^2=x^d+ax^{d-1}+b,$$ respectively. We explicitly find the number of $\mathbb{F}_q$-points on $E_d$ and $E'_d$ in terms of special values of ${_{d}}F_{d-1}$ and ${_{d-1}}F_{d-2}$ Gaussian hypergeometric series with characters of orders $d-1$, $d$, $2(d-1)$, $2d$, and $2d(d-1)$ as parameters. This gives a solution to a problem posed by Ken Ono \cite[p. 204]{ono2} on special values of ${_{n+1}}F_n$ Gaussian hypergeometric series for $n > 2$. We also show that the results of Lennon \cite{lennon1} and the authors \cite{BK3} on trace of Frobenius of elliptic curves follow from the main results.

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