arXiv subjects

S. N. Fathima

Publications and source records attributed to S. N. Fathima.

11 recordsLinked to original sources

arXiv2026

Overcolored Partition $k$-tuples Restricted by Parity of the Parts

In this paper, we study the combinatorial object $\bar{b}^k_{r,s}(n)$ which counts the overcolored partition $k$-tuples wherein both even and odd parts are colored with $r$ and $s$ colors, respectively. We extend results of Chacon and Sellers for several families of $r,s$ and $k$. We also establish divisibility properties for $\bar{b}^k_{r,s}(n)$ modulo prime $p$ and conclude with new congruences modulo powers of $2$. The techniques involved to obtain our results rely on theta function identities and modular forms.

math.NT↗

arXiv2026

Generalization of Ramanujan's Continued Fractions for Even Order

In this paper, we derive three generalized continued fractions of any even order $k$ with the aid of a general continued fraction identity of Ramanujan and we establish general theta function identities for these continued fractions. As an application of continued fraction of order seventy-six, we obtain partition theoretic identities and some vanishing coefficient results.

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arXiv2026

Arithmetic Properties for $k$-Color Analogue of Simultaneously $s$-Regular and $t$-Distinct Partitions

In this article, we discuss general generating functions for partitions of $n$, simultaneously $s$-regular and $t$-distinct in 3-colors. In addition, we obtain infinite families of congruences modulo powers of 3 for specific values of $(\ell,t)$. For instance, for positive integers $n$ and $k$, we have \begin{align*} \sum_{n=o}^{\infty}RD_3^{3,3}\left(3^kn+\frac{3^k+1}{2}\right)q^n\equiv0 \pmod{3^{k+1}}. \end{align*}

math.CO↗

arXiv2026

Arithmetic Properties of Overcolored Odd Partitions

Let $\bar{a}_s(n)$ denote the number of partitions of $n$, wherein each odd part is multicolored (atmost $s\ge 1$ colors) and the first appearance of parts may be overlined. In this paper, we establish new families of congruences modulo powers of $2$ satisfied by $\bar{a}_s(n)$ for infinitely many $s$. Our approach builds upon generating function manipulations, Hecke eigenform theory and results of Newman.

math.NT↗

arXiv2026

On Ramanujan's $q$-Continued Fractions of Order Thirty-Four and Sixty-Eight

We derived $q$-continued fractions $X_i(q)$ of order thirty-four and continued fractions $Y_i(q)$ of order sixty-eight from a general continued fraction identity of Ramanujan, where $i=1,2,3,4,5,6,7$ and $8$. We established some theta-function identities, and one has been proved for the continued fractions $X_i(q)$ and $Y_i(q)$. Furthermore, we obtained results on vanishing coefficients arising from these continued fractions and their reciprocals. As an application of the theta-function identities for $Y_i(q)$, we derived certain color partition identities.

math.NT↗

arXiv2026

Overcolored Partition Restricted by Parity of the Parts

Very recently, Thejitha, Sellers, and Fathima defined the function $a_{r,s}(n)$, which enumerates the number of multicolored partitions of $n$, wherein both even parts and odd parts may appear in one of $r$-colors and $s$-colors, respectively, for fixed $r,s\ge 1$. In this paper, we extend the concept to overpartitions.

math.CO↗

arXiv2026

Arithmetic Properties of Colored Partitions Restricted by Parity of the Parts

Let $a_{r,s}(n)$ denote the number of mutlicolored partitions of $n$, wherein both even parts and odd parts may appear in one of $r$-colors and $s$-colors, respectively, for fixed $r,s\ge 1$. The paper aims to study arithmetic properties satisfied by $a_{r,s}(n)$, using elementary generating function manipulations and classical $q$-series techniques.

math.CO↗

arXiv2026

Additional Congruences for generalized Color Partitions of Hirschhorn and Sellers

Let $a_k(n)$ denote the number of partitions of $n$ wherein even parts come in only one color, while the odd parts may be ``colored" with one of $k$ colors, for fixed $k$. In this note, we find some congruences for $a_k(n)$ in the spirit of Ramanujan's congruences. We prove a number of results for $a_k(n)$ modulo powers of $2$ for infinitely many values of $k$. Our approach is truly elementary, relying on generating function manipulations, theta functions and $q$-dissection techniques. We then close by demonstrating an infinite family of congruences modulo 11 which is proven using a result of Ahlgren.

math.NT↗

arXiv2025

Arithmetic Properties of Partitions with 1-colored Even Parts and r-colored Odd Parts

Recently, Hirschhorn and Sellers defined the partition function $a_r(n)$, which counts the number of partitions of $n$ wherein even parts come in only one color, while the odd parts may appear in one of $r$-colors for fixed $r\ge1$. The aim of this paper is to prove several new infinite families of congruences modulo 3 and 5 by employing a result of Newman and theory of modular forms.

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arXiv2025

On recent Partition function of Kaur and Rana

Recently, Kaur and Rana introduced the partition function denoted by $ρ(n)$, where the largest part $λ$ appears exactly once, and the remaining parts constitute a partition of $λ$. In this paper, we establish new generating functions for certain variants of $ρ(n)$. Further, we obtain a linear recurrence relation for our new generating function.

math.CO↗

arXiv2025

Vanishing Coefficients of q^{5n+r} and q^{7n+r} in Certain Infinite q-series Expansions

Motivated by the recent work of several authors on vanishing coefficients of the arithmetic progression in certain $q$-series expansion, we study some variants of these $q$-series and prove some comparable results. For instance, if $\sum_{n=0}^{\infty}c_1(n)q^n=\left(\pm q^2,\pm q^3; q^5\right)_\infty^2 \left( q, q^{14}; q^{15}\right)_\infty$, then $c_1(5n+3)=0$.

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