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Mohamed El Bachraoui

Publications and source records attributed to Mohamed El Bachraoui.

At least 19 recordsLinked to original sources
arXiv2026

Solutions for Hecke Sum Questions of Banerjee and Bringmann

We previously conjectured a Hecke-type identity for a series arising from two-color partitions. Banerjee and Bringmann proved the conjecture using modular methods and asked for a direct proof using Bailey pairs. They also suggested investigating the odd part of the series. We answer both questions by applying Bailey's transform to a Bailey pair of Andrews and a conjugate Bailey pair of Lovejoy. Our main theorem gives formulas for the even and odd parts of a series $P_a(q)$ depending on a parameter $a$. The specialization $a=1$ gives the conjectured formula and a companion formula for the odd part.

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arXiv2026

The Corners of Ramanujan: Partial- and False-Theta Companions to Third-Order Mock Theta Functions

Most of the traditional third-order mock theta functions occur naturally as specialized values of partition-rank generating functions. The same finite rank kernels also occur in the ordinary and odd unimodal-rank series studied previously in the literature. We organize a parallel collection of corner series by retaining these kernels and replacing their quadratic core weights by linear peak weights. Their root-of-unity specializations are governed by partial and false theta identities recorded in Ramanujan's Lost Notebook; in one exceptional case the corner companion is itself a mock theta function. The classical rank series and Ramanujan identities are used as inputs. Our new results include exact two- and three-color partition identities with sparse theta discrepancies and an oriented three-dimensional corner with a distinct tower. In both color cases, positive decompositions separate the colored companion into copies of the relevant odd-corner series and a corresponding decorated even-corner series. Thus both sparse discrepancies become signed differences between natural corner families. Burson's boxed modular involution provides a common rank-refined model for the two corrections, whose surviving fixed points produce the exceptional quadratic sequences. For the side rank of the three-dimensional corners we obtain an exact formula at $z=-1$, a strict parity bias, quartic and cubic root-of-unity rank-difference identities, and exact equidistribution modulo $3$ on the progression $6n+5$. The cubic identity also gives exact congruence formulas on the progressions $6n+2$ and $6n+3$. We further classify the associated binary triangular support modulo every integer and derive general congruence families.

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arXiv2026

A $q$-microscope for mock-theta denominator patterns

We give a systematic creative-microscoping treatment of truncated basic hypergeometric sums attached to three classical third-order mock theta denominator patterns, represented by $ϕ(q)$, $ω(q)$, and $ν(q)$. The main conceptual point is that these denominator patterns naturally admit exact finite theta evaluations at $a=q^{\pm n}$, microscopic $q$-supercongruences, and cyclotomic/$p$-adic consequences within a single framework. For \[ S(q,a):=\sum_{n\ge 0}\frac{q^{2n+1}(aq,q/a;q^2)_n}{(-q^2;q^2)_n}, \] we prove, for odd $n$, a finite theta evaluation at $a=q^{\pm n}$, equivalently a microscopic $q$-supercongruence. We prove an analogous finite theta evaluation for the $ν$-type denominator \[ N_m(q,a):=\sum_{j=0}^{m} q^{2j+1}\frac{(aq,q/a;q^2)_j}{(-q;q^2)_{j+1}}. \] We then introduce the one-parameter denominator family \[ \T(q;b):=\sum_{n\ge 0} q^{2n}\frac{(q;q^2)_n^2}{(q^2;q^2)_n(b;q^2)_n}, \] whose specializations include the $ω$-type shifted denominator ($b=q^3$) and the two-color family ($b=q^{2k}$). For its truncated $a$-extension we obtain an exact evaluation at $a=q^{\pm n}$, deduce microscopic congruences modulo $(a-q^n)(1-aq^n)$, and derive cyclotomic consequences, including zero congruences in the $q^{2k}$-subfamily and nontrivial $p$-adic versions for the cases $b=q$ and $b=q^3$.

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arXiv2026

On a two-color partition series and its companions

We study the two-color distinct-part series \(S_1(q)\), equivalently Andrews' generating function \(v_d(q)\) for strictly concave compositions, and its odd and even companions \(T_o(q)\) and \(T_e(q)\). We determine the coefficients of \(S_1(q)\) modulo \(4\) and obtain a complete criterion for the resulting Ramanujan-type progressions. For the even companion, we give a direct overpartition interpretation of its coefficients and show that two natural partition families are each counted by half of those coefficients. For the eta-normalized odd companion \(C(q)=(q;q)_\infty T_o(q)\), we prove a quintic self-similarity, derive exact vanishing relations and infinite sign changes for its coefficients, and show that \(c(n)\) can be nonzero only when \(24n+28\) is represented by \(x^2+3y^2\).

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arXiv2026

Weighted partitions with interval restrictions: exact formulas and a bivariate master identity

Let $a_2''(n)$ and $b_2''(n)$ be the signed partition functions introduced by Andrews and El Bachraoui for interval-restricted partitions whose parts greater than $1$ are controlled by the smallest even part and by the number of ones. We prove two conjectures for these functions. The first gives the generating function for $a_2''(n)$ as an elementary rational term plus a false theta series with periodic signs; the second asserts that the companion coefficients $b_2''(n)$ take only the values $-1,0,1,2$. The central structural result introduces an auxiliary variable $z$ recording the number of non-compulsory parts greater than $1$. We obtain closed forms for the two resulting generating functions and prove the master identity $(1+q^2)\mathcal B(z,q)-(1+q)\mathcal A(z,q)=-q^4/(1-q^3)$ using both analytic and combinatorial techniques. At $z=-1$, this identity, together with a Rogers--Fine evaluation, gives the false theta formula for $a_2''(n)$ and an explicit generating function for $b_2''(n)$. The latter formula implies the asserted coefficient range and leads to an exact coefficient description of $b_2''(n)$. We also include a direct Heine--Rogers--Fine proof of the false theta formula, ordinary and fixed-refinement consequences of the master identity, and the resulting quantum modular interpretation.

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arXiv2026

Two-color partitions with evens in one color

We consider sequences counting integer partitions in two colors (red and blue) in which the even parts occur only in blue color. We focus on subsequences defined by constraints on the parity and color of the summands. We establish formulas for our sequences and deduce identities of integer partitions.

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arXiv2026

A gamma function in two variables

We introduce a gamma function $\Ga(x,z)$ in two complex variables which extends the classical gamma function $\Ga(z)$ in the sense that $\lim_{x\to 1}\Ga(x,z)=\Ga(z)$. We will show that many properties which $\Ga(z)$ enjoys extend in a natural way to the function $\Ga(x,z)$. Among other things we shall provide functional equations, a multiplication formula, and analogues of the Stirling formula with asymptotic estimates as consequences.

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arXiv2026

Proofs for Andrews' Conjectures 5 and 6 on $v_1(q)$

Folsom, Males, Rolen, and Storzer recently proved Andrews' Conjecture~4 for the coefficients of \[ v_1(q)=\sum_{n\ge 0}\frac{q^{n(n+1)/2}}{(-q^2;q^2)_n}=\sum_{n\ge 0}V_1(n)q^n. \] They also proved a refined density-one version of Andrews' Conjecture~3. In this paper we prove Andrews' Conjectures~5 and~6. Our proof relies on an investigation of the simple zeros of the trigonometric factor in the Folsom--Males--Rolen--Storzer asymptotic and showing that the relevant quadratic sequence stays a positive distance from the integers infinitely often. The argument is unconditional.

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arXiv2026

On congruence conjectures of Andrews and Bachraoui

Andrews and the third author recently studied congruences for certain restricted two-color partitions. They made two conjectures for Ramanujan-type congruences and a vanishing identity for the limiting sequence. In this paper, we settle these conjectures by relating the corresponding generating function to modular forms and mock theta functions.

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arXiv2026

Congruences for two-color partitions with odd smallest part

For a fixed positive integer $k$, let $C(k,n)$ denote the number of two-color partitions of $n$ with odd smallest part and restrictions on even parts, and let $C_k(q)$ be its generating function. We show that $C(1,n)\equiv d(2n-1)\pmod{4}$ and obtain congruences modulo $2$ and $4$ for $C(k,n)$ when $k=2,3$. Using $q$-series methods we derive closed formulas for $C_k(q)$ in terms of eta-quotients and formulate Ramanujan-type congruences for the limiting sequence arising from $\lim_{k\to\infty} C_k(q)$.

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arXiv2025

Certain positive $q$-series and inequalities for two-color partitions

We consider some $q$-series which depend on a pair of positive integers $(k,m)$. While positivity of these series holds for the first few values of $(k,m)$, the situation is quite unclear for other values of $(k,m)$. In addition, our series generate the number of certain two-color integer partitions weighted by $(-1)^j$ where $j$ is the number of even parts. Therefore, inequalities involving these partitions will be deduced from the positivity of their generating functions.

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arXiv2025

Positive Weighted Partitions Generated by Double Series

We investigate some weighted integer partitions whose generating functions are double-series. We will establish closed formulas for these $q$-double series and deduce that their coefficients are non-negative. This leads to inequalities among integer partitions.

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arXiv2025

On a formula of the $q$-series $_{2k+4}ϕ_{2k+3}$ and its applications

In this paper we apply a formula of the very-well poised $_{2k+4}ϕ_{2k+3}$ to write a $k$-tuple sum of $q$-series as a linear combination of terms wherein each term is a product of expressions of the form $\frac{1}{(qy, qy^{-1};q)_\infty}$. As an application, we shall express a variety of sums and double sums of $q$-series as linear combinations of infinite products. Our formulas are motivated by their connection to overpartition pairs.

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arXiv2021

$N$-tuple sum analogues for Ramanujan-type congruences

In this paper, we prove supercongruence relations for truncated $N$-tuple sums of basic hypergeometric series. As an application, we give double, triple, and quadruple sum analogs of some Ramanujan-type supercongruences.

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arXiv2021

On the vanishing of some mock theta functions at odd roots of unity

We consider the problem of whether or not certain mock theta functions vanish at the roots of unity with an odd order. We prove for any such function $f(q)$ that there exists a constant $C>0$ such that for any odd integer $n>C$ the function $f(q)$ does vanish at the primitive $n$-th roots of unity. This leads us to conjecture that $f(q)$ does not vanish at the primitive $n$-th roots of unity for any odd positive integer $n$.

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arXiv2020

Positive alternating sums of integer partitions

We combine an extended version of Bailey's transform with an identity of Bressoud and with some identities of Berkovich and Warnaar to prove a variety of positivity results for alternating sums involving partition functions.

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