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Russelle Guadalupe

Publications and source records attributed to Russelle Guadalupe.

At least 19 recordsLinked to original sources
arXiv2026

A note on congruences modulo 7 and 11 for two restricted partition functions

For an integer $c\geq 1$, let $a_c(n)$ count the number of generalized cubic partitions of $n$, which are partitions of $n$ whose even parts may appear in $c$ different colors, and $d_c(n)$ count the number of partitions obtained by adding the links of the $c$-elongated plane partition diamonds of length $n$. We prove in this note infinite families of congruences modulo $7$ and $11$ for $a_c(n)$ and $d_c(n)$ by employing elementary $q$-series techniques. These results generalize particular congruences modulo $7$ and $11$ for $a_c(n)$ and $d_c(n)$ recently found by Dockery, and Baruah, Das, and Talukdar, respectively, using modular forms.

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arXiv2026

Linear identities for partition pairs with 4-cores

We determine an infinite family of linear identities for the number $A_4(n)$ of partition pairs of $n$ with $4$-cores by employing elementary $q$-series techniques and certain $3$-dissection formulas. We then discover an infinite family of congruences for $A_4(n)$ as a consequence of these linear identities.

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arXiv2026

Gosper-type Lambert series identities of level 14

We derive two Gosper-type Lambert series identities of level $14$ which involve the $q$-constant $Π_q$ using a special case of Bailey's $_6ψ_6$ summation formula and certain propeties of $η$-quotients and generalized $η$-quotients on the congruence subgroup $Γ_0(14)$.

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arXiv2026

Linear identities for partition pairs with $5$-cores

We prove an infinite family of linear identities for the number $A_5(n)$ of partition pairs of $n$ with $5$-cores by using certain theta function identities involving the Ramanujan's parameter $k(q)$ due to Cooper, and Lee and Park. Consequently, we deduce an infinite family of congruences for $A_5(n)$ using these linear identities.

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arXiv2026

Congruences for an analogue of Lin's partition function

We study certain arithmetic properties of an analogue $B(n)$ of Lin's restricted partition function that counts the number of partition triples $π=(π_1,π_2,π_3)$ of $n$ such that $π_1$ and $π_2$ comprise distinct odd parts and $π_3$ consists of parts divisible by $4$. With the help of elementary $q$-series techniques and modular functions, we establish Ramanujan-type congruences modulo $2,3,5,7$, and $9$ for certain sums involving $B(n)$.

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arXiv2026

Remarks on a certain restricted partition function of Lin

Let $b(n)$ be the number of partition triples $π=(π_1,π_2,π_3)$ of $n$ such that $π_1$ consists of distinct odd parts, and $π_2$ and $π_3$ consist of parts divisible by $4$. Utilizing modular forms, Lin obtained the generating functions for $b(3n+1)$ and $b(3n+2)$, which yields the congruence $b(3n+2)\equiv 0\pmod{3}$ for all $n\geq 0$. We provide in this note elementary proofs of these generating functions by employing $q$-series manipulations and dissection formulas. We also establish infinite families of internal congruences modulo $3$ for $b(n)$.

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arXiv2025

Modularity of certain products of the Rogers-Ramanujan continued fraction

We study the modularity of the functions of the form $r(τ)^ar(2τ)^b$, where $a$ and $b$ are integers with $(a,b)\neq (0,0)$ and $r(τ)$ is the Rogers-Ramanujan continued fraction, which may be considered as companions to the Ramanujan's function $k(τ)=r(τ)r(2τ)^2$. In particular, we show that under some condition on $a$ and $b$, there are finitely many such functions generating the field of all modular functions on the congruence subgroup $Γ_1(10)$. Furthermore, we establish certain arithmetic properties of the function $l(τ)=r(2τ)/r(τ)^2$, which can be used to evaluate these products. We employ the methods of Lee and Park, and some properties of $η$-quotients and generalized $η$-quotients to prove our results.

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arXiv2025

A note on congruences for the difference between even cranks and odd cranks

Recently, Amdeberhan and Merca proved some arithmetic properties of the crank parity function $C(n)$ defined as the difference between the number of partitions of $n$ with even cranks and those with odd cranks and the sequence $a(n)$ whose generating function is the reciprocal of that of $C(n)$. The function $C(n)$ was first studied by Choi, Kang, and Lovejoy. In this note, we give new elementary proofs of some of their main results and extend them. In particular, we establish Ramanujan-type congruences modulo $5$ and $25$ for certain finite sums involving $C(n)$ and $a(n)$. Our proofs employ the results of Cooper, Hirschhorn, and Lewis, and certain identities involving the Rogers-Ramanujan continued fraction $R(q)$ due to Chern and Tang.

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arXiv2025

The $k$-elongated plane partition function modulo small powers of $5$

Andrews and Paule revisited combinatorial structures known as the $k$-elongated partition diamonds, which were introduced in connection with the study of the broken $k$-diamond partitions. They found the generating function for the number $d_k(n)$ of partitions obtained by summing the links of such partition diamonds of length $n$ and discovered congruences for $d_k(n)$ using modular forms. Since then, congruences for $d_k(n)$ modulo certain powers of primes have been proven via elementary means and modular forms by many authors, most recently Banerjee and Smoot who established an infinite family of congruences for $d_5(n)$ modulo powers of $5$. We extend in this paper the list of known results for $d_k(n)$ by proving infinite families of congruences for $d_k(n)$ modulo $5,25$, and $125$ using classical $q$-series manipulations and $5$-dissections.

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arXiv2025

Ramanujan's continued fractions of order $10$ as modular functions

We explore the modularity of the continued fractions $I(τ), J(τ), T_1(τ), T_2(τ)$ and $U(τ)=I(τ)/J(τ)$ of order $10$, where $I(τ)$ and $J(τ)$ are introduced by Rajkhowa and Saikia, which are special cases of certain identities of Ramanujan. In particular, we show that these fractions can be expressed in terms of an $η$-quotient $g(τ)$ that generates the field of all modular functions on the congruence subgroup $Γ_0(10)$. Consequently, we prove that modular equations for $g(τ)$ and $U(τ)$ exist at any level and derive these equations of prime levels $p\leq 11$. We also show that the continued fractions of order $10$ can be explicitly evaluated using a singular value of $g(τ)$, which under certain conditions, generates the Hilbert class field of an imaginary quadratic field. We employ the methods of Lee and Park to establish our results.

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arXiv2025

A note on congruences for generalized cubic partitions modulo primes

Recently, Amdeberhan, Sellers, and Singh introduced the notion of a generalized cubic partition function $a_c(n)$ and proved two isolated congruences via modular forms, namely, $a_3(7n+4)\equiv 0\pmod{7}$ and $a_5(11n+10)\equiv 0\pmod{11}$. In this paper, we provide another proof of these congruences by using classical $q$-series manipulations. We also give infinite families of congruences for $a_c(n)$ for primes $p\not\equiv 1\pmod{8}$.

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arXiv2024

Analogue of Ramanujan's function $k(τ)$ for the continued fraction $X(τ)$ of order six

Motivated by the recent work of Park on the analogue of the Ramanujan's function $k(τ)=r(τ)r^2(2τ)$ for the Ramanujan's cubic continued fraction, where $r(τ)$ is the Rogers-Ramanujan continued fraction, we use the methods of Lee and Park to study the modularity and arithmetic of the function $w(τ) = X(τ)X(3τ)$, which may be considered as an analogue of $k(τ)$ for the continued fraction $X(τ)$ of order six introduced by Vasuki, Bhaskar and Sharath. In particular, we show that $w(τ)$ can be written in terms of the normalized generator $u(τ)$ of the field of all modular functions on $Γ_0(18)$, and derive modular equations for $u(τ)$ of smaller prime levels. We also express $j(dτ)$ for $d\in\{1,2,3,6,9,18\}$ in terms of $u(τ)$, where $j$ is the modular $j$-invariant.

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arXiv2024

A remark on modular equations involving Rogers-Ramanujan continued fraction via $5$-dissections

In this paper, we study the $5$-dissections of certain Ramanujan's theta functions, particularly $ψ(q)ψ(q^2), φ(-q)$ and $φ(-q)φ(-q^2)$, and derive an identity for $q(q;q)_{\infty}^6/(q^5;q^5)_{\infty}^6$ in terms of certain products of the Rogers-Ramanujan continued fraction $R(q)$. Using this identity, we give another proof of the modular equation involving $R(q), R(q^2)$ and $R(q^4)$, which was recorded by Ramanujan in his lost notebook, and establish modular equations involving $R(q), R(q^2), R(q^4), R(q^8)$ and $R(q^{16})$.

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arXiv2024

A note on the exact formulas for certain $2$-color partitions

Let $p\leq 23$ be a prime and $a_p(n)$ counts the number of partitions of $n$ where parts that are multiple of $p$ come up with $2$ colors. Using a result of Sussman, we derive the exact formula for $a_p(n)$ and obtain an asymptotic formula for $\log a_p(n)$. Our results partially extend the work of Mauth, who proved the asymptotic formula for $\log a_2(n)$ conjectured by Banerjee et al.

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arXiv2023

A note on the squares of the form $\prod_{k=1}^n (2k^2+l)$ with $l$ odd

Let $l$ be a positive odd integer. Using Cilleruelo's method, we establish an explicit lower bound $N_l$ depending on $l$ such that for all $n\geq N_l$, $\prod_{k=1}^n (2k^2+l)$ is not a square. As an application, we determine all values of $n$ such that $\prod_{k=1}^n (2k^2+l)$ is a square for certain values of $l$.

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