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Alexander Berkovich

Publications and source records attributed to Alexander Berkovich.

At least 19 recordsLinked to original sources
arXiv2026

$\vec M$ Versions of Andrews-Gordon Identities Revisited

In this paper, we revisit the work of Berkovich and Paule on variants of the Andrews-Gordon identities. We find a generalization of their principal polynomial identity and, as a consequence, obtain new $\vec M$ versions of the Andrews-Gordon identities. More precisely, for non-negative integers $M_1\ge M_2\ge M_3\ge\ldots\ge M_\nu$, we show that a broad class of multi-sums of the form $$ \sum\limits_{\mathbf n} \frac{ q^{ N_1^2+\cdots+N_\nu^2 - M_1N_1 - M_2N_2 - \cdots - M_\nu N_\nu } }{ (q)_{n_1}(q)_{n_2}\cdots(q)_{n_\nu} } $$ can be expressed as a sum of products. Above, we use standard notations for $q$-Pochhammer symbols and $N_i = \sum\limits_{k=i}^{\nu}n_k$ for $1\le i\le \nu$.

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arXiv2025

Further Applications of Cubic $q$-Binomial Transformations

Consider \begin{align*} G(N,M;\alpha,\beta,K,q) = \sum\limits_{j\in\mathbb{Z}}(-1)^jq^{\frac{1}{2}Kj((\alpha+\beta)j+\alpha-\beta)}\left[\begin{matrix}M+N\\N-Kj\end{matrix}\right]_{q}. \end{align*} In this paper, we prove the non-negativity of coefficients of some cases of $G(N,M;\alpha,\beta,K,q)$. For instance, for non-negative integers $n$ and $t$, we prove that\\ \begin{align*} G\left(n,n;\frac{4}{3}+\frac{3(3^t-1)}{2},\frac{5}{3}+\frac{3(3^t-1)}{2},3^{t+1},q\right) \end{align*} and \begin{align*} G\left(n-\frac{3^t-1}{2},n+\frac{3^t+1}{2};\frac{8}{3}+2(3^t-1),\frac{4}{3}-(3^t-1),3^{t+1},q\right)\\ \end{align*} are polynomials in $q$ with non-negative coefficients. Using cubic positivity preserving transformations of Berkovich and Warnaar and some known formulae arising from Rogers-Szeg\"{o} polynomials, we establish new identities such as\\ \begin{align*} \sum\limits_{0\le 3j\le n}\dfrac{(q^3;q^3)_{n-j-1}(1-q^{2n})q^{3j^2}}{(q;q)_{n-3j}(q^6;q^6)_{j}} = \sum\limits_{j=-\infty}^{\infty}(-1)^jq^{6j^2}{2n\brack n-3j}_q. \end{align*}

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arXiv2025

Finite analogs of partition bias related to hook length two and a variant of Sylvester's map

In this paper, we count the total number of hooks of length two in all odd partitions of $n$ and all distinct partitions of $n$ with a bound on the largest part of the partitions. We generalize inequalities of Ballantine, Burson, Craig, Folsom and Wen by showing there is a bias in the number of hooks of length two in all odd partitions over all distinct partitions of $n$ in presence of a bound on the largest part. To establish such a bias, we use a variant of Sylvester's map. Then, we conjecture a similar finite bias for a weighted count of hooks of length two and prove it when we remove the bound on the largest part.

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arXiv2024

New Borwein-type conjectures

Motivated by recent research of Wang and Krattenthaler, we use Maple to propose five new ``Borwein-type'' conjectures modulo $3$ and two new ``Borwein-type'' conjectures modulo $5$.

math.CO↗
arXiv2024

Extension of Bressoud's generalization of Borwein's conjecture and some exact results

In this paper, we conjecture an extension to Bressoud's 1996 generalization of Borwein's famous 1990 conjecture. We then state a few infinite hierarchies of non-negative $q$-series identities which are interesting examples of our proposed conjecture and Bressoud's generalized conjecture. Finally, using certain positivity-preserving transformations for $q$-binomial coefficients, we prove the non-negativity of the infinite families.

math.CO↗
arXiv2023

On partitions with bounded largest part and fixed integral GBG-rank modulo primes

In 2009, Berkovich and Garvan introduced a new partition statistic called the GBG-rank modulo $t$ which is a generalization of the well-known BG-rank. In this paper, we use the Littlewood decomposition of partitions to study partitions with bounded largest part and fixed integral value of GBG-rank modulo primes. As a consequence, we obtain new elegant generating function formulas for unrestricted partitions, self-conjugate partitions, and partitions whose parts repeat a finite number of times.

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arXiv2023

On the q-binomial identities involving the Legendre symbol modulo 3

I use polynomial analogue of the Jacobi triple product identity together with the Eisenstein formula for the Legendre symbol modulo 3 . to prove six identities involving the $q$-binomial coefficients. These identities are then extended to the new infinite hierarchies of q-series identities by means of the special case of Bailey's lemma. Some of the identities of Ramanujan, Slater, McLaughlin and Sills are obtained this way.

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arXiv2022

Bressoud's identities for even moduli. New companions and related positivity results

I revisit Bressoud's generalised Borwein conjecture. Making use of certain positivity-preserving transformations for q-binomial coefficients, I establish the truth of infinitely many new cases of the Bressoud conjecture. In addition, I prove new doubly-bounded refinement of the Foda-Quano identities. Finally, I discuss new companions to the Bressoud even moduli identities. In particular, all 10 mod 20 identities are derived.

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arXiv2022

On Finite Analogs of Schmidt's Problem and Its Variants

We refine Schmidt's problem and a partition identity related to 2-color partitions which we will refer to as Uncu-Andrews-Paule theorem. We will approach the problem using Boulet-Stanley weights and a formula on Rogers-Szegő polynomials by Berkovich-Warnaar, and present various Schmidt's problem alike theorems and their refinements. Our new Schmidt type results include the use of even-indexed parts' sums, alternating sum of parts, and hook lengths as well as the odd-indexed parts' sum which appears in the original Schmidt's problem. We also translate some of our Schmidt's problem alike relations to weighted partition counts with multiplicative weights in relation to Rogers-Ramanujan partitions.

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arXiv2020

Some New Positive Observations

We revisit Bressoud's generalized Borwein conjecture. Making use of new positivity-preserving transformations for q-binomial coefficients we establish the truth of infinitely many cases of the Bressoud conjecture. In addition, we prove new bounded version of Lebesgue's identity and of Euler's Pentagonal Number Theorem. Finally, we discuss new companions to Andrews-Gordon mod 21 and Bressoud mod 20 identities.

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arXiv2020

On the difference of partial theta functions

Sums of the form add((-1)^n q^(n(n-1)/2) x^n, n>=0) are called partial theta functions. In his lost notebook, Ramanujan recorded many identities for those functions. In 2003, Warnaar found an elegant formula for a sum of two partial theta functions. Subsequently, Andrews and Warnaar established a similar result for the product of two partial theta functions. In this note, I discuss the relation between the Andrews-Warnaar identity and the (1986) product formula due to Gasper and Rahman. I employ nonterminating extension of Sears-Carlitz transformation for 3ϕ_2 to provide a new elegant proof for a companion identity for the difference of two partial theta series. This difference formula first appeared in the work of Schilling-Warnaar (2002). Finally, I show that Schilling-Warnnar (2002) and Warnaar (2003) formulas are, in fact, equivalent.

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arXiv2019

Refined $q$-Trinomial Coefficients and Two Infinite Hierarchies of $q$-Series Identities

We will prove an identity involving refined $q$-trinomial coefficients. We then extend this identity to two infinite families of doubly bounded polynomial identities using transformation properties of the refined $q$-trinomials in an iterative fashion in the spirit of Bailey chains. One of these two hierarchies contains an identity which is equivalent to Capparelli's first Partition Theorem.

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arXiv2019

Polynomial Identities Implying Capparelli's Partition Theorems

We propose and recursively prove polynomial identities which imply Capparelli's partition theorems. We also find perfect companions to the results of Andrews, and Alladi, Andrews and Gordon involving $q$-trinomial coefficients. We follow Kurşungöz's ideas to provide direct combinatorial interpretations of some of our expressions. We use of the trinomial analogue of Bailey's lemma to derive new identities. These identities relate triple sums and products. A couple of new Slater type identities are also noted.

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arXiv2018

Elementary Polynomial Identities Involving $q$-Trinomial Coefficients

We use $q$-binomial theorem to prove three new polynomial identities involving $q$-trinomial coefficients. We then use summation formulas for the $q$-trinomial coefficients to convert our identities into another set of three polynomial identities, which imply Capparelli's partition theorems when the degree of the polynomial tends to infinity. This way we also obtain an interesting new result for the sum of the Capparelli's products. We finish this paper by proposing an infinite hierarchy of polynomial identities.

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arXiv2017

On some polynomials and series of Bloch-Polya Type

We will show that $(1-q)(1-q^2)\dots (1-q^m)$ is a polynomial in $q$ with coefficients from $\{-1,0,1\}$ iff $m=1,\ 2,\ 3,$ or $5$ and explore some interesting consequences of this result. We find explicit formulas for the $q$-series coefficients of $(1-q^2)(1-q^3)(1-q^4)(1-q^5)\dots$ and $(1-q^3)(1-q^4)(1-q^5)(1-q^6)\dots$. In doing so, we extend certain observations made by Sudler in 1964. We also discuss the classification of the products $(1-q)(1-q^2)\dots (1-q^m)$ and some related series with respect to their absolute largest coefficients.

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arXiv2017

Some Elementary Partition Inequalities and Their Implications

We prove various inequalities between the number of partitions with the bound on the largest part and some restrictions on occurrences of parts. We explore many interesting consequences of these partition inequalities. In particular, we show that for $L\geq 1$, the number of partitions with $l-s \leq L$ and $s=1$ is greater than the number of partitions with $l-s\leq L$ and $s>1$. Here $l$ and $s$ are the largest part and the smallest part of the partition, respectively.

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