arXiv subjects

Yazan Alamoudi

Publications and source records attributed to Yazan Alamoudi.

5 recordsLinked to original sources

arXiv2026

On subradically sifted sums related to Alladi's higher order duality between prime factors

In this paper, I utilize a variant of the Selberg--Delange method to find quantitative estimates of the sums \[M_{k,ω}(x,y)=\sum_{\substack{p_{1}(n)> y\\ n\leq x} } μ(n) {ω(n)-1\choose k-1},\] where $y$ can grow with $x$ but we must have $y\leq Y_0\exp(\mathscr{p}\frac{\log x}{(\log\log (x+1))^{1+ε}})$ with $Y_0,\mathscr{p},ε>0$. Moreover, I give preliminary upper bounds for the general range $1.9\leq y\leq x^{\frac{1}{k}}$. In addition, I formalize the notions of subradical and radical dominance and discuss their relevance to the analytic approach to the study of arithmetic functions. Lastly, I give a fascinating formula related to the derivatives of the gamma function and the Hankel contour, which should be relevant for those employing the Selberg--Delange method to obtain higher-order terms.

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arXiv2026

Capparelli's partition theorem as part of an infinite hierarchy: Combinatorial and Weighted Words extensions of recent work

In a recent paper, the authors introduced an infinite hierarchy of $q$-hypergeometric identities, of which the first three orders, $0$, $1$, and $2$, relate to the partition theorems of Euler, Lebesgue, and Capparelli, and stated a partition theorem at order 4 which lies beyond Capparelli's theorem. Here, we first state certain partition theorems that hold at all even orders beyond Capparelli and provide bijective proofs for these theorems. In doing so, we show that there is a fourfold infinite hierarchy of partition theorems that emanates from Capparelli's theorem, which is the base case. It is also shown that the equality of two of the four generating functions holds for all orders, odd and even. Lastly, a very general framework for the remaining two functions is constructed via the method of weighted words, encompassing all possible orders and yielding several infinite hierarchies with different dilations and translations.

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arXiv2026

On a nonnegativity conjecture of Andrews

I settle a conjecture of Andrews related to the Alladi-Schur polynomials. In addition, I give further relations and implications to two families of polynomials related to the Alladi-Schur polynomials.

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arXiv2025

A bijective proof of Andrews' refinement of the Alladi-Schur theorem

This paper gives a bijective proof of Andrews' refinement of the Alladi-Schur theorem. Moreover, it demonstrates that the bijective framework introduced here can be used to reproduce and provide a bijective account of Andrews' recursive relations for the Alladi-Schur polynomials.

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arXiv2025

Some $q$-hypergeometric identities associated with partition theorems of Lebesgue, Schur and Capparelli

Here, we establish a polynomial identity in three variables $a, b, c$, and with the degree of the polynomial given in terms of two integers $L, M$. By letting $L$ and $M$ tend to infinity, we get the 1993 Alladi-Gordon $q$-hypergeometric key-identity for the generalized Schur Theorem as well as the fundamental Lebesgue identity by two different choices of the variables. This polynomial identity provides a generalization and a unified approach to the Schur and Lebesgue theorems. We discuss other analytic identities for the Lebesgue and Schur theorems and also provide a key identity ($q$-hypergeometric) for Andrews' deep refinement of the Alladi-Schur theorem. Finally, we discuss a new infinite hierarchy of identities, the first three of which relate to the partition theorems of Euler, Lebesgue, and Capparelli, and provide their polynomial versions as well.

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