arXiv subjects

Rupam Barman

Publications and source records attributed to Rupam Barman.

At least 19 recordsLinked to original sources

arXiv2026

Explicit hypergeometric modularity of certain weight two and four Hecke eigenforms

Recently, Allen et al. developed the Explicit Hypergeometric Modularity Method (EHMM) that establishes the modularity of a large class of hypergeometric Galois representations in dimensions two and three. Motivated by this framework, we construct two explicit families of eta-quotients, which we call the $\mathbb{K}_4$ and $\mathbb{K}_5$ functions, from the hypergeometric background. These $\mathbb{K}_4$ and $\mathbb{K}_5$ functions are constructed using the theory of weight $1/2$ Jacobi theta functions and their cubic analogues, respectively. Using these constructions, we then express the Fourier coefficients of certain Hecke eigenforms of weight two and four in terms of finite field period functions. As an application, we obtain new identities relating the Fourier coefficients of modular forms to special values of the finite field Appell series $F_1^p$ and $F_2^p$.

math.NT↗

arXiv2026

Arithmetic Aspects of Number Fields Generated by Polynomial Families

Let $f(x)=(x^{k}+c)^{m}-ax^{n}\in\mathbb{Z}[x]$ be an irreducible polynomial over $\mathbb{Q}$, where $k,m,n\in\mathbb{N}$ with $km>n$, and let $K=\mathbb{Q}(\theta)$, where $\theta$ is a root of $f(x)$. We investigate the arithmetic properties of the number fields that arise from this family. We first obtain an explicit formula for the discriminant of $f(x)$. Using this formula, we establish necessary and sufficient conditions for the monogeneity of $f(x)$, expressed in terms of the prime divisors of $a$ and $c$ and the parameters $k,m,n$. This yields infinite families of monogenic polynomials of arbitrary degree, including families with a non-square-free discriminant. Building on these results, we extend our algebraic characterization to composite polynomials, establishing some explicit conditions for the monogeneity of the composition of $f(x)$ with an arbitrary polynomial $g(x)$. From an analytic point of view, we derive asymptotic estimates for the number of monogenic polynomials in these families under natural assumptions. We further study non-monogeneity via the field index $i(K)$ and, for each prime $p$, provide sufficient conditions ensuring $\nu_p(i(K))=1$, yielding partial progress toward a problem of Narkiewicz. We also highlight a connection with a class of differential equations naturally associated with $f(x)$. As an application, we determine the conditions under which the splitting field of $f(x)$ has a full symmetric Galois group. Several explicit examples illustrate our results.

math.NT↗

arXiv2026

On the discriminant and index of a certain class of polynomials

Let $f(x) = (x^{2}+1)^{n} - a x^{n} \in \mathbb{Z}[x]$ and assume $f(x)$ is irreducible. Let $\theta$ be a root of $f(x)$, set $K= \mathbb{Q}(\theta)$, and denote by $\mathbb{Z}_{K}$ the ring of integers of $K$. The index of $f$, denoted $\operatorname{ind}(f)$, is the index of $\mathbb{Z}[\theta]$ in $\mathbb{Z}_{K}$. A polynomial $f(x)$ is said to be monogenic if $\operatorname{ind}(f) = 1$. In this article, we explicitly compute the discriminant of the polynomial $f(x)$, and then derive necessary and sufficient conditions on the parameters $a$ and $n$ for $f(x)$ to be monogenic. Furthermore, we provide a complete description of the primes that divide $\operatorname{ind}(f)$.

math.NT↗

arXiv2026

$J$-generalization of the Rogers-Ramanujan-Gordon identities via commutative algebra

The Rogers-Ramanujan-Gordon identities generalize the classical partition identities discovered independently by L. J. Rogers and S. Ramanujan. In 2021, Afsharijoo provided a commutative algebra proof of the Rogers-Ramanujan-Gordon identities. Building on the Afsharijoo's approach, we present a commutative algebra proof of a broader family of identities introduced by Coulson \textit{et al.}, which includes the Rogers-Ramanujan-Gordon identities as a special case. In the proof, we relate the generating functions associated with these identities to the Hilbert-Poincar\'e series of suitably constructed graded algebras.

math.CO↗

arXiv2026

On the G\"ollnitz-Gordon-Andrews identities via commutative algebra

The G\"ollnitz-Gordon-Andrews identities generalize the classical partition identities discovered independently by H. G\"ollnitz and B. Gordon. These are Rogers-Ramanujan-type identities involving generating functions of partitions satisfying certain kinds of difference conditions on the one hand and infinite periodic products on the other. In 2021, Afsharijoo provided a commutative algebra proof of the Rogers-Ramanujan-Gordon identities. Building on Afsharijoo's approach, we investigate the G\"ollnitz-Gordon-Andrews identities using techniques from commutative algebra. More generally, we establish a broader family of identities, of which the G\"ollnitz-Gordon-Andrews identities arise as special cases. Our approach interprets the associated generating functions in terms of Hilbert-Poincar\'e series of suitably constructed graded algebras, providing the first commutative algebra framework for these identities.

math.CO↗

arXiv2026

On Monogeneity of reciprocal polynomials

Let $\mathbb{Z}_K$ denote the ring of integers of the number field $K = \mathbb{Q}(\theta)$, where $\theta$ is a root of the monic irreducible polynomial $f(x) \in \mathbb{Z}[x]$. We say that $f(x)$ is monogenic if $\mathbb{Z}_K = \mathbb{Z}[\theta]$. A polynomial $f(x) \in \mathbb{Z}[x]$ is called reciprocal if $f(x) = x^{\operatorname{deg}(f)} f(1/x)$. In this article, we derive sufficient conditions for the monogeneity of even degree reciprocal polynomials. By employing properties of the discriminant of reciprocal polynomials, we partially prove a conjecture proposed by Jones in $2021$. Furthermore, we establish a lower bound on the number of certain sextic monogenic reciprocal polynomials.

math.NT↗

arXiv2025

On the Monogenity of Polynomials with Non-Squarefree Discriminants

In 2012, for any integer $n \ge 2$, Kedlaya constructed an infinite class of monic irreducible polynomials of degree $n$ with integer coefficients having squarefree discriminants. Such polynomials are necessarily monogenic. Further, by extending Kedlaya's approach, for any odd prime $q$, Jones constructed a class of degree $q$ polynomials with non-squarefree discriminants. In this article, using a similar method provided by Jones, we present another infinite class of monogenic polynomials of degree $q$ with non-squarefree discriminants, where $q$ is a prime of the form $ q = q_0 + q_1 - 1 $, with $ q_0 $ and $ q_1 $ being prime numbers. In addition to this we present a class of non-monogenic polynomials whose coefficients are Sterling numbers of the first kind.

math.NT↗

arXiv2024

Generalized twisted Edwards curves over finite fields and hypergeometric functions

Let $\mathbb{F}_q$ be a finite field with $q$ elements. For $a,b,c,d,e,f \in \mathbb{F}_q^{\times}$, denote by $C_{a,b,c,d,e,f}$ the family of algebraic curves over $\mathbb{F}_q$ given by the affine equation \begin{align*} C_{a,b,c,d,e,f}:ay^2+bx^2+cxy=d+ex^2y^2+fx^3y. \end{align*} The family of generalized twisted Edwards curves is a subfamily of $C_{a,b,c,d,e,f}$. Let $\#C_{a,b,c,d,e,f}(\mathbb{F}_q)$ denote the number of points on $C_{a,b,c,d,e,f}$ over $\mathbb{F}_q$. In this article, we find certain expressions for $\#C_{a,b,c,d,e,f}(\mathbb{F}_q)$ when $af=ce$. If $c^2-4ab\neq 0$, we express $\#C_{a,b,c,d,e,f}(\mathbb{F}_q)$ in terms of a $p$-adic hypergeometric function $\mathbb{G}(x)$ whose values are explicitly known for all $x\in \mathbb{F}_q$. Next, if $c^2-4ab=0$, we express $\#C_{a,b,c,d,e,f}(\mathbb{F}_q)$ in terms of another $p$-adic hypergeometric function and then relate it to the traces of Frobenius endomorphisms of a family of elliptic curves. Furthermore, using the known values of the hypergeometric functions, we deduce some nice formulas for $\#C_{a,b,c,d,e,f}(\mathbb{F}_q)$.

math.NT↗

arXiv2024

On hook length biases in $t$-regular partitions

Let $t\geq2$ and $k\geq1$ be integers. A $t$-regular partition of a positive integer $n$ is a partition of $n$ such that none of its parts is divisible by $t$. Let $b_{t,k}(n)$ denote the number of hooks of length $k$ in all the $t$-regular partitions of $n$. Recently, the first and the third authors proved that $b_{3,2}(n)\geq b_{2,2}(n)$ for all $n\geq 4$, and conjectured that $b_{t+1,2}(n)\geq b_{t,2}(n)$ for all $t\geq 3$ and $n\geq 0$. In this paper, we prove that the conjecture is true for $t=3$.

math.CO↗

arXiv2024

Recursive Formulas for MacMahon and Ramanujan $q$-series

In the present work, we extend current research in a nearly-forgotten but newly revived topic, initiated by P. A. MacMahon, on a generalized notion which relates the divisor sums to the theory of integer partitions and two infinite families of $q$-series by Ramanujan. Our main emphasis will be on explicit representations for a variety of $q$-series, studied primarily by MacMahon and Ramanujan, with an eye towards their modular properties and their proper place in the ring of quasimodular forms of level one and level two.

math.NT↗

arXiv2024

Hook length inequalities for $t$-regular partitions in the $t$-aspect

Let $t\geq2$ and $k\geq1$ be integers. A $t$-regular partition of a positive integer $n$ is a partition of $n$ such that none of its parts is divisible by $t$. Let $b_{t,k}(n)$ denote the number of hooks of length $k$ in all the $t$-regular partitions of $n$. In this article, we prove some inequalities for $b_{t,k}(n)$ for fixed values of $k$. We prove that for any $t\geq2$, $b_{t+1,1}(n)\geq b_{t,1}(n)$, for all $n\geq0$. We also prove that $b_{3,2}(n)\geq b_{2,2}(n)$ for all $n>3$, and $b_{3,3}(n)\geq b_{2,3}(n)$ for all $n\geq0$. Finally, we state some problems for future works.

math.CO↗

arXiv2024

$p$-Adic quotient sets: linear recurrence sequences with reducible characteristic polynomials

Let $(x_n)_{n\geq0}$ be a linear recurrence sequence of order $k\geq2$ satisfying $$x_n=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}$$ for all integers $n\geq k$, where $a_1,\dots,a_k,x_0,\dots, x_{k-1}\in \mathbb{Z},$ with $a_k\neq0$. In 2017, Sanna posed an open question to classify primes $p$ for which the quotient set of $(x_n)_{n\geq0}$ is dense in $\mathbb{Q}_p$. In a recent paper, we showed that if the characteristic polynomial of the recurrence sequence has a root $\pm α$, where $α$ is a Pisot number and if $p$ is a prime such that the characteristic polynomial of the recurrence sequence is irreducible in $\mathbb{Q}_p$, then the quotient set of $(x_n)_{n\geq 0}$ is dense in $\mathbb{Q}_p$. In this article, we answer the problem for certain linear recurrence sequences whose characteristic polynomials are reducible over $\mathbb{Q}$.

math.NT↗

arXiv2024

Hook length biases in ordinary and $t$-regular partitions

In this article, we study hook lengths of ordinary partitions and $t$-regular partitions. We establish hook length biases for the ordinary partitions and motivated by them we find a few interesting hook length biases in $2$-regular partitions. For a positive integer $k$, let $p_{(k)}(n)$ denote the number of hooks of length $k$ in all the partitions of $n$. We prove that $p_{(k)}(n)\geq p_{(k+1)}(n)$ for all $n\geq0$ and $n\ne k+1$; and $p_{(k)}(k+1)- p_{(k+1)}(k+1)=-1$ for $k\geq 2$. For integers $t\geq2$ and $k\geq1$, let $b_{t,k}(n)$ denote the number of hooks of length $k$ in all the $t$-regular partitions of $n$. We find generating functions of $b_{t,k}(n)$ for certain values of $t$ and $k$. Exploring hook length biases for $b_{t,k}(n)$, we observe that in certain cases biases are opposite to the biases for ordinary partitions. We prove that $b_{2,2}(n)\geq b_{2,1}(n)$ for all $n>4$, whereas $b_{2,2}(n)\geq b_{2,3}(n)$ for all $n\geq 0$. We also propose some conjectures on biases among $b_{t,k}(n)$.

math.CO↗

arXiv2024

$p$-Adic hypergeometric functions and certain weight three newforms

For an odd prime $p$ and a positive integer $n$, let ${_n}G_n[\cdots]_p$ denote McCarthy's $p$-adic hypergeometric function. In this article, we prove $p$-adic analogue of certain classical hypergeometric identities and using these identities we express the $p$-th Fourier coefficient of certain weight three newforms in terms of special values of ${_3}G_3[\cdots]_p$. Rodriguez-Villegas conjectured certain supercongruences between values of truncated hypergeometric series and the $p$-th Fourier coefficients of these newforms. As a consequence of our main results, we obtain another proof of these supercongruences which were earlier proved by Mortenson and Sun.

math.NT↗

arXiv2024

On the $p$-adic valuation of third order linear recurrence sequences

In a recent paper, Bilu et al. studied a conjecture of Marques and Lengyel on the $p$-adic valuation of the Tribonacci sequence. In this article, we study the $p$-adic valuation of third order linear recurrence sequences by considering a generalisation of the conjecture of Marques and Lengyel for third order linear recurrence sequences. Suppose that $(x_n)$ is a third order linear recurrence sequence whose characteristic polynomial has a root $\gamma$ such that $|\gamma|>1$. We show that if there exists a prime $p$ for which the conjecture holds for $(x_n)$, then the solution set of the Diophantine equation given by $x_n=m!$ in positive integers $n,m$ is finite. We also show that the solutions can be effectively computed when the form of the conjecture is explicitly known.

math.NT↗

arXiv2023

$p$-Adic hypergeometric functions and the trace of Frobenius of elliptic curves

Let $p$ be an odd prime and $q=p^r$, $r\geq 1$. For positive integers $n$, let ${_n}G_n[\cdots]_q$ denote McCarthy's $p$-adic hypergeometric functions. In this article, we prove an identity expressing a ${_4}G_4[\cdots]_q$ hypergeometric function as a sum of two ${_2}G_2[\cdots]_q$ hypergeometric functions. This identity generalizes some known identities satisfied by the finite field hypergeometric functions. We also prove a transfomation that relates ${_{n+2}}G_{n+2}[\cdots]_q$ and ${_n}G_n[\cdots]_q$ hypergeometric functions. Next, we express the trace of Frobenius of elliptic curves in terms of special values of ${_4}G_4[\cdots]_q$ and ${_6}G_6[\cdots]_q$ hypergeometric functions. Our results extend the recent works of Tripathi and Meher on the finite field hypergeometric functions to wider classes of primes.

math.NT↗

arXiv2023

Arithmetic properties and asymptotic formulae for $σ_o\text{mex}(n)$ and $σ_e\text{mex}(n)$

The minimal excludant of an integer partition is the least positive integer missing from the partition. Let $σ_o\text{mex}(n)$ (resp., $σ_e\text{mex}(n)$) denote the sum of odd (resp., even) minimal excludants over all the partitions of $n$. Recently, Baruah et al. proved a few congruences for these partition functions modulo $4$ and $8$, and asked for asymptotic formulae for the same. In this article, we study the lacunarity of $σ_o\text{mex}(n)$ and $σ_e\text{mex}(n)$ modulo arbitrary powers of $2$ and also prove some infinite families of congruences for $σ_o\text{mex}(n)$ and $σ_e\text{mex}(n)$ modulo $4$ and $8$. We also obtain Hardy-Ramanujan type asymptotic formulae for both $σ_o\text{mex}(n)$ and $σ_e\text{mex}(n)$.

math.NT↗

arXiv2023

Hypergeometric functions for Dirichlet characters and Peisert-like graphs on $\mathbb{Z}_n$

For a prime $p\equiv 3\pmod 4$ and a positive integer $t$, let $q=p^{2t}$. The Peisert graph of order $q$ is the graph with vertex set $\mathbb{F}_q$ such that $ab$ is an edge if $a-b\in\langle g^4\rangle\cup g\langle g^4\rangle$, where $g$ is a primitive element of $\mathbb{F}_q$. In this paper, we construct a similar graph with vertex set as the commutative ring $\mathbb{Z}_n$ for suitable $n$, which we call \textit{Peisert-like} graph and denote by $G^\ast(n)$. Owing to the need for cyclicity of the group of units of $\mathbb{Z}_n$, we consider $n=p^α$ or $2p^α$, where $p\equiv 1\pmod 4$ is a prime and $α$ is a positive integer. For primes $p\equiv 1\pmod 8$, we compute the number of triangles in the graph $G^\ast(p^α)$ by evaluating certain character sums. Next, we study cliques of order 4 in $G^\ast(p^α)$. To find the number of cliques of order $4$ in $G^\ast(p^α)$, we first introduce hypergeometric functions containing Dirichlet characters as arguments, and then express the number of cliques of order $4$ in $G^\ast(p^α)$ in terms of these hypergeometric functions.

math.CO↗