arXiv subjects

Nabin Kumar Meher

Publications and source records attributed to Nabin Kumar Meher.

10 recordsLinked to original sources

arXiv2026

Moments of the Crank Statistic for $t$-Core Partitions and Overpartitions

Recently, Kang, Kim, and Lee \cite{Kang2026} developed a unified moment-trace framework for symmetric partition statistics using complete Bell polynomials and their inversion formula. In this paper, we apply this framework to crank statistics for $t$-core partitions and overpartitions. For $t\in\{5,7,11,17,19\}$, we show that the normalized even crank moment generating functions for $t$-core partitions admit partition-trace representations in terms of the functions $D^{(t)}_{2s}(\tau)$, together with suitable Bernoulli-number shifts. We also establish inverse trace formulas that recover $D^{(t)}_{2s}(\tau)$ from the corresponding normalized even crank moments. For overpartitions, we obtain analogous trace and inverse-trace identities for the normalized even moments associated with the first and second residual crank generating functions. As applications, we use complete Bell polynomials and their inversion formula to obtain explicit expressions for the $t$-core partition numbers and overpartitions number in terms of sums involving divisor function.

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arXiv2025

On the number of irreducible representations of $\so(5)$

Let $d(n)$ be the divisor function and it is well known that $\sum_{1\leq n \leq x}d(n) = x\log x+(2\gamma-1)x +\mathcal{O}\left(x^{\theta+\epsilon}\right)$ where $\gamma$ is the Euler constant, $\epsilon>0$ and $1/4<\theta<1/3$. In this paper, we obtain an asymptotic formula for the number of irreducible representations of $\mathfrak{so}(5)$. More precisely, the irreducible representations of the Lie algebra $\mathfrak{so}(5)$ are a family of representations of dimension $jk(j+k)(j+2k)/6$ for $j, k\in \mathbb{N}_{0}$ and suppose that $\varrho_{\mathfrak{so}(5)}(n)$ is the number of irreducible $\mathfrak{so}(5)$ representations of dimension $n$. We obtain an asymptotic formula for the summatory function $\sum_{1\leq n \leq x}\varrho_{\mathfrak{so}(5)}(n)$.

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arXiv2024

Distribution and congruences of $(u,v)$-regular bipartitions

Let $B_{u,v}(n)$ denote the number of $(u,v)$-regular bipartitions of $n$. In this article, we prove that $B_{p,m}(n)$ is always almost divisible by $p,$ where $p\geq 5$ is a prime number and $m=p_1^{α_1} p_2^{α_2}\cdots p_r^{α_r}, $ where $α_i \geq 0$ and $p_i \geq 5$ be distinct primes with $\gcd(p,m)=1$ . Further, we obtain an infinities families of congruences modulo $3$ for $B_{3,7}(n),$ $B_{3,5}(n)$ and $B_{3,2}(n)$ by using Hecke eigenform theory and a result of Newman \cite{Newmann1959}. Furthermore, we get many infinite families of congruences modulo $7$, $11$ and $13$ respectively for $B_{2,7}(n)$, $B_{2,11}(n)$ and $B_{2,13}(n),$ by employing an identity of Newman \cite{Newmann1959}. In addition, we prove infinite families of congruences modulo $2$ for $B_{4,3}(n)$, $B_{8,3}(n)$ and $B_{4,5}(n)$ by applying another result of Newman \cite{Newmann1962}.

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arXiv2024

Arithmetic density and congruences of $\ell$-regular bipartitions $II$

Let $ B_{\ell}(n)$ denote the number of $\ell-$regular bipartitions of $n.$ In 2013, Lin \cite{Lin2013} proved a density result for $B_4(n).$ He showed that for any positive integer $k,$ $B_4(n)$ is almost always divisible by $2^k.$ In this article, we improved his result. We prove that $B_{2^αm}(n)$ and $B_{3^αm}(n)$ are almost always divisible by arbitrary power of $2$ and $3$ respectively. Further, we obtain an infinities families of congruences and multiplicative formulae for $B_2(n)$ and $B_4(n)$ by using Hecke eigenform theory. Next, by using a result of Ono and Taguchi on nilpotency of Hecke operator, we also find an infinite families of congruences modulo arbitrary power of $2$ satisfied by $B_{2^α}(n).$

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arXiv2024

Arithmetic density and congruences of $\ell$-regular bipartitions

Let $ B_{\ell}(n)$ denote the number of $\ell$-regular bipartitions of $n.$ In this article, we prove that $ B_{\ell}(n)$ is always almost divisible by $p_i^j$ if $p_i^{2a_i}\geq \ell,$ where $j$ is a fixed positive integer and $\ell=p_1^{a_1}p_2^{a_2}\ldots p_m^{a_m},$ where $p_i$ are prime numbers $\geq 5.$ Further, we obtain an infinities families of congruences for $B_3(n)$ and $B_5(n)$ by using Hecke eigen form theory and a result of Newman \cite{Newmann1959}. Furthermore, by applying Radu and Seller's approach, we obtain an algorithm from which we get several congruences for $B_{p}(n)$, where $p$ is a prime number.

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arXiv2023

Analytic continuation of $\ell$-generalized Fibonacci zeta function

In this paper, for any positive integer $\ell\geq2,$ we define $\ell$-generalized Fibonacci zeta function. We then study its analytic continuation to the whole complex plane $\mathbb{C}.$ Further, we compute a possible list of singularities and residues of the function at these simple poles. Moreover, we deduce that the special values of $\ell$-generalized Fibonacci zeta function at negative integer arguments are rational.

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arXiv2023

Arithmetic density and congruences of $t$-core partitions

A partition of $n$ is called a $t$-core partition if none of its hook number is divisible by $t.$ In 2019, Hirschhorn and Sellers \cite{Hirs2019} obtained a parity result for $3$-core partition function $a_3(n)$. Recently, both authors \cite{MeherJindal2022} proved density results for $a_3(n)$, wherein we proved that $a_3(n)$ is almost always divisible by arbitrary power of $2$ and $3.$ In this article, we prove that for a non-negative integer $α,$ $a_{3^α m}(n)$ is almost always divisible by arbitrary power of $2$ and $3.$ Further, we prove that $a_{t}(n)$ is almost always divisible by arbitrary power of $p_i^j,$ where $j$ is a fixed positive integer and $t= p_1^{a_1}p_2^{a_2}\ldots p_m^{a_m}$ with primes $p_i \geq 5.$ Furthermore, by employing Radu and Seller's approach, we obtain an algorithm and we give alternate proofs of several congruences modulo $3$ and $5$ for $a_{p}(n)$, where $p$ is prime number. Our results also generalizes the results in \cite{radu2011a}.

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arXiv2023

Infinite families of congruences for $2$ and $13$-core partitions

A partition of $n$ is called a $t$-core partition if none of its hook number is divisible by $t.$ In 2019, Hirschhorn and Sellers \cite{Hirs2019} obtained a parity result for $3$-core partition function $a_3(n)$. Motivated by this result, both the authors \cite{MeherJindal2022} recently proved that for a non-negative integer $α,$ $a_{3^α m}(n)$ is almost always divisible by arbitrary power of $2$ and $3$ and $a_{t}(n)$ is almost always divisible by arbitrary power of $p_i^j,$ where $j$ is a fixed positive integer and $t= p_1^{a_1}p_2^{a_2}\ldots p_m^{a_m}$ with primes $p_i \geq 5.$ In this article, by using Hecke eigenform theory, we obtain infinite families of congruences and multiplicative identities for $a_2(n)$ and $a_{13}(n)$ modulo $2$ which generalizes some results of Das \cite{Das2016}.

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arXiv2020

Cullen numbers in sums of terms of recurrence sequence

Let $(U_n)_{n\geq 0}$ be a fixed linear recurrence sequence of integers with order at least two, and for any positive integer $\ell$, let $\ell \cdot 2^{\ell} + 1$ be a Cullen number. Recently in \cite{bmt}, generalized Cullen numbers in terms of linear recurrence sequence $(U_n)_{n\geq 0}$ under certain weak assumptions has been studied. However, there is an error in their proof. In this paper, we generalize their work, as well as our result fixes their error. In particular, for a given polynomial $Q(x) \in \mathbb{Z}[x]$ we consider the Diophantine equation $U_{n_1} + \cdots + U_{n_k} = \ell \cdot x^{\ell} + Q(x)$, and prove effective finiteness result. Furthermore, we demonstrate our method by an example.

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arXiv2020

Multiple Lucas Dirichlet series associated to additive and Dirichlet characters

In this article, we obtain the analytic continuation of the multiple shifted Lucas zeta function, multiple Lucas $L$-function associated to Dirichlet characters and additive characters. We then compute a complete list of exact singularities and residues of these functions at these poles. Further, we show the rationality of the multiple Lucas $L$-functions associated with quadratic characters at negative integer arguments.

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