arXiv subjects

Alexander Berkovich

Publications and source records attributed to Alexander Berkovich.

At least 37 records · Page 2Linked to original sources

arXiv2016

New Weighted Partition Theorems with the Emphasis on the Smallest Part of Partitions

We use the $q$-binomial theorem, the $q$-Gauss sum, and the ${}_2ϕ_1 \rightarrow {}_2ϕ_2$ transformation of Jackson to discover and prove many new weighted partition identities. These identities involve unrestricted partitions, overpartitions, and partitions with distinct even parts. Smallest part of the partitions plays an important role in our analysis. This work was motivated in part by the research of Krishna Alladi.

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arXiv2016

Variation on a theme of Nathan Fine. New weighted partition identities

We utilize false theta function results of Nathan Fine to discover three new partition identities involving weights. These relations connect Göllnitz--Gordon type partitions and partitions with distinct odd parts, partitions into distinct parts and ordinary partitions, and partitions with distinct odd parts where the smallest positive integer that is not a part of the partition is odd and ordinary partitions subject to some initial conditions, respectively. Some of our weights involve new partition statistics, one is defined as the number of different odd parts of a partition larger than or equal to a given value and another one is defined as the number of different even parts larger than the first integer that is not a part of the partition. Dedicated to our friend, Krishna Alladi, on his 60th birthday.

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arXiv2016

On partitions with fixed number of even-indexed and odd-indexed odd parts

This article is an extensive study of partitions with fixed number of odd and even-indexed odd parts. We use these partitions to generalize recent results of C. Savage and A. Sills. Moreover, we derive explicit formulas for generating functions for partitions with bounds on the largest part, the number of parts and with a fixed value of BG-rank or with a fixed value of alternating sum of parts. We extend the work of C. Boulet, and as a result, obtain a four-variable generalization of Gaussian binomial coefficients. In addition we provide combinatorial interpretation of the Berkovich-Warnaar identity for Rogers-Szego polynomials.

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arXiv2015

A New Companion to Capparelli's Identities

We discuss a new companion to Capparelli's identities. Capparelli's identities for m=1,2 state that the number of partitions of $n$ into distinct parts not congruent to m, -m modulo $6$ is equal to the number of partitions of n into distinct parts not equal to m, where the difference between parts is greater than or equal to 4, unless consecutive parts are either both consecutive multiples of 3 or add up to to a multiple of 6. In this paper we show that the set of partitions of n into distinct parts where the odd-indexed parts are not congruent to m modulo 3, the even-indexed parts are not congruent to -m modulo 3, and 3l+1 and 3l+2 do not appear together as consecutive parts for any integer l has the same number of elements as the above mentioned Capparelli's partitions of n. In this study we also extend the work of Alladi, Andrews and Gordon by providing a complete set of generating functions for the refined Capparelli partitions, and conjecture some combinatorial inequalities.

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arXiv2014

On The Gauss EYPHKA Theorem And Some Allied Inequalities

We use the 1907 Hurwitz formula along with the Jacobi triple product identity to understand representation properties of two JP (Jones-Pall) forms of Kaplansky: 9x^2+ 16y^2 +36z^2 + 16yz+ 4xz + 8xy and 9x^2+ 17y^2 +32z^2 -8yz+ 8xz + 6xy. We also discuss three nontrivial analogues of the Gauss EYPHKA theorem. The technique used can be applied to all known spinor regular ternary quadratic forms.

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arXiv2014

Essentially Unique Representations by Certain Ternary Quadratic Forms

In this paper we generalize the idea of "essentially unique" representations by ternary quadratic forms. We employ the Siegel formula, along with the complete classification of imaginary quadratic fields of class number less than or equal to 8, to deduce the set of integers which are represented in essentially one way by a given form which is alone in its genus. We consider a variety of forms which illustrate how this method applies to any of the 794 ternary quadratic forms which are alone in their genus. As a consequence, we resolve some conjectures of Kaplansky regarding unique representation by the forms $x^2 +y^2 +3z^2$, $x^2 +3y^2 +3z^2$, and $x^2 +2y^2 +3z^2$.

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arXiv2013

A partition inequality involving products of two $q$-Pochhammer symbols

We use an injection method to prove a new class of partition inequalities involving certain $q$-products with two to four finitization parameters. Our new theorems are a substantial generalization of work by Andrews and of previous work by Berkovich and Grizzell. We also briefly discuss how our products might relate to lecture hall partitions.

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arXiv2013

Binary Quadratic forms and the Fourier coefficients of certain weight 1 eta-quotients

We state and prove an identity which represents the most general eta-products of weight 1 by binary quadratic forms. We discuss the utility of binary quadratic forms in finding a multiplicative completion for certain eta-quotients. We then derive explicit formulas for the Fourier coefficients of certain eta-quotients of weight 1 and level 47, 71, 135,648 1024, and 1872.

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arXiv2013

On the class of dominant and subordinate products

In this paper we provide proofs of two new theorems that provide a broad class of partition inequalities and that illustrate a naïve version of Andrews' anti-telescoping technique quite well. These new theorems also put to rest any notion that including parts of size 1 is somehow necessary in order to have a valid irreducible partition inequality. In addition, we prove (as a lemma to one of the theorems) a rather nontrivial class of rational functions of three variables has entirely nonnegative power series coefficients.

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arXiv2012

On Rogers-Ramanujan functions, binary quadratic forms and eta-quotients

In a handwritten manuscript published with his lost notebook, Ramanujan stated without proofs forty identities for the Rogers-Ramanujan functions. We observe that the functions that appear in Ramanujan's identities can be obtained from a Hecke action on a certain family of eta products. We establish further Hecke-type relations for these functions involving binary quadratic forms. Our observations enable us to find new identities for the Rogers-Ramanujan functions and also to use such identities in turn to find identities involving binary quadratic forms.

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arXiv2012

Races among products

We will revisit a 1987 question of Rabbi Ehrenpreis. Among many things, we will provide an elementary injective proof that P_1(L,y,n)>=P_2(L,y,n) for any L,n>0 and any odd y>1 . Here, P_1(L,y,n) denotes the number of partitions of n into parts congruent to 1, y+2, or 2y mod 2(y+1) with the largest part not exceeding 2(y+1)L-2 and P_2(L,y,n) denotes the number of partitions of n into parts congruent to 2, y, or 2y+1 mod 2(y+1) with the largest part not exceeding 2(y+1)L-1.

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arXiv2012

On representation of an integer as a sum by X^2+Y^2+Z^2 and the modular equations of degree 3 and 5

I discuss a variety of results involving s(n), the number of representations of n as a sum of three squares. One of my objectives is to reveal numerous interesting connections between the properties of this function and certain modular equations of degree 3 and 5. In particular, I show that s(25n)=(6-(-n|5))s(n)-5s(n/25) follows easily from the well known Ramanujan modular equation of degree 5. Moreover, I establish new relations between s(n) and h(n), g(n), the number of representations of $n$ by the ternary quadratic forms 2x^2+2y^2+2z^2-yz+zx+xy and x^2+y^2+3z^2+xy, respectively. I propose an interesting identity for s(p^2n)- p s(n) with p being an odd prime. This identity makes nontrivial use of the ternary quadratic forms with discriminants p^2, 16p^2.

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arXiv2011

A proof of the S-genus identities for ternary quadratic forms

In this paper we prove the main conjectures of Berkovich and Jagy about weighted averages of representation numbers over an S-genus of ternary lattices (defined below) for any odd squarefree S \in N. We do this by reformulating them in terms of local quantities using the Siegel-Weil and Conway-Sloane formulas, and then proving the necessary local identities. We conclude by conjecturing generalized formulas valid over certain totally real number fields as a direction for future work.

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arXiv2011

On representation of an integer as the sum of three squares and the ternary quadratic forms with the discriminants p^2, 16p^2

Let s(n) be the number of representations of n as the sum of three squares. We prove a remarkable new identity for s(p^2n)- ps(n) with p being an odd prime. This identity makes nontrivial use of ternary quadratic forms with discriminants p^2 and 16p^2. These forms are related by Watson's transformations. To prove this identity we employ the Siegel--Weil and the Smith--Minkowski product formulas.

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arXiv2009

Ternary Quadratic Forms, Modular Equations and Certain Positivity Conjectures

We show that many of Ramanujan's modular equations of degree 3 can be interpreted in terms of integral ternary quadratic forms. This way we establish that for any n in N |{n= x(x+1)/2 + y^2 +z^2 : x,y,z in Z}| >= |{n= x(x+1)/2 + 3y^2 +3z^2: x,y,zin Z}|, just to mention one among many similar positive results of this type. In particular, we prove the recent conjecture of H. Yesilyurt and the first author stating that for any n in N |{n= x(x+1)/2 + y^2 +z^2 : x,y,z in Z}| >= |{n= x(x+1)/2 + 7y^2 + 7z^2: x,y,z in Z}|. We prove a variety of identities for certain ternary forms with discriminants 144,400, 784,3600 by converting these into identities for appropriate eta- quotients. In the process we discover and prove a few new modular equations of degree 5 and 7. For any square free odd integer S with prime factorization p_1.....p_r, we define the S-genus as a union of 2^r specially selected genera of ternary quadratic forms, all with discriminant 16 S^2. This notion of S-genus arises naturally in the course of our investigation. It entails an interesting injection from genera of binary quadratic forms with discriminant -8 S to genera of ternary quadratic forms with discriminant 16 S^2.

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arXiv2008

The GBG-Rank and t-Cores I. Counting and 4-Cores

Let r_j(π,s) denote the number of cells, colored j, in the s-residue diagram of partition π. The GBG-rank of πmod s is defined as r_0+r_1*w_s+r_2*w_s^2+...+r_(s-1)*w_s^(s-1), where w_s=exp(2*Π*I/s). We will prove that for (s,t)=1, v(s,t) <= binomial(s+t,s)/(s+t), where v(s,t) denotes a number of distinct values that GBG-rank mod s of t-core may assume. The above inequality becomes an equality when s is prime or when s is composite and t<=2p_s, where p_s is a smallest prime divisor of s. We will show that the generating functions for 4-cores with the prescribed values of GBG-rank mod 3 are all eta-products.

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arXiv2008

On the representations of integers by the sextenary quadratic form x^2+y^2+z^2+ 7s^2+7t^2+ 7u^2 and 7-cores

In this paper we derive an explicit formula for the number of representations of an integer by the sextenary form x^2+y^2+z^2+ 7s^2+7t^2+ 7u^2. We establish the following intriguing inequalities 2b(n)>=a_7(n)>=b(n) for n not equal to 0,2,6,16. Here a_7(n) is the number of partitions of n that are 7-cores and b(n) is the number of representations of n+2 by the sextenary form (x ^2+ y ^2+z ^2+ 7s ^2 + 7t ^2+ 7u^2)/8 with x,y,z,s,t and u being odd.

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arXiv2008

The tri-pentagonal number theorem and related identities

I revisit an automated proof of Andrews' pentagonal number theorem found by Riese. I uncover a simple polynomial identity hidden behind his proof. I explain how to use this identity to prove Andrews' result along with a variety of new formulas of similar type. I reveal an interesting relation between the tri-pentagonal theorem and items (19), (20), (94), (98) on the celebrated Slater list. Finally, I establish a new infinite family of multiple series identities.

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