Research
Research
Research
Research
1,905+ open-access research outputs.
Starting with Ramanujan's famous taxicab problem, we can study the solvability of the equations $p^n+q^n=r^n+s^n$ and, more generally, $p_1^{k_1}+\dots+p_m^{k_m}=0$ among polynomials.…
Let $G$ be an $n$-vertex graph with Laplacian eigenvalues $0=\lambda_1(G)\le \lambda_2(G)\le\cdots\le \lambda_n(G)$. Motivated by the Alon-Boppana bound and the Ramanujan phenomenon for regular graphs…
We investigate standing waves for the energy critical Schr\"odinger system with three waves interaction arising as a model for the Raman amplification in a plasma. Several results are proved: simultan…
We give an independent eta-product derivation of the level-8 Apery limit lim B_n^{(8)}/s_n = (7/32) zeta(3), where s_n = sum_{k=0}^n C(n,k)^2 C(2k,n)^2 and B_n^{(8)} is the rational companion sequence…
We present a complete formalization, in the Lean interactive theorem prover with the Mathlib library, of the Ramanujan--Nagell theorem: the only integer solutions to the Diophantine equation $x^2 + 7 …
Recently, Andrews and Bachraoui investigated congruences for certain restricted two-color partitions. They made two conjectures for Ramanujan type congruences and a vanishing identity for the limiting…
We introduce the study of \textit{randomly oriented divisor graphs}. For each $\rho \in [0,1]$, the randomly oriented divisor graph $\mathcal{D}_\rho(N)$ is obtained from the divisor graph on $\{1, 2,…
In this paper, we study the Koshliakov zeta function $\eta_p(s)$, whose theory appears to be more involved than that of its counterpart $\zeta_p(s)$, owing to the fact that its defining series is not …
In $2012$, Guillera and Zudilin established the following two supercongruences involving truncated Ramanujan-type series: for any odd prime $p>2$, \begin{align*} \sum_{n=0}^{p-1}\frac{(\frac{1}{2})_n(…
We prove that the ratio $B_n/D_n$ of the Ap\'ery-like sequence $B_n$ to the Domb numbers $D_n$ converges to $(7/24)\zeta(3)$, and that $\sum_{n=1}^{\infty} 64^n/(n^3 D_n D_{n-1}) = (56/3)\zeta(3)$. As…
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 limitin…
Building off work of Farenick and Rahaman, we extend the definition of the density space and the Bures metric to the setting of non-unital C*-algebras equipped with a faithful trace and prove that the…
Using generalized binomial coefficient identities and some results of John Dougall, we derive some families of series involving the cubes of Catalan numbers. We also establish a family of series conta…
By employing the classical tools from the theory of $q$-series and theta functions, new fascinating identities on different continued fractions can be achieved. In this article, we use the product exp…
We construct Igusa stacks for all Shimura varieties of abelian type and derive consequences for the cohomology of these Shimura varieties. As an application, we prove that the Fargues--Scholze local L…
Recently, the second author [Ramanujan J. 2026] introduced and proved a $q$-series identity that appears to provide the first known $q$-analogue of an evaluation for a ${}_{2}F_{1}$-series known as …
In 1919, Ramanujan discovered his famous congruences for the partition function. Not too long after, Freeman Dyson conjectured a combinatorial statistic existed that explained the three congruences, w…
The study of integer partitions and their congruences dates back to 1919 when Ramanujan discovered his famous congruences for the partition function, $p(n)$. Since then, many other kinds of partition …
We construct an infinite family of 6-regular graphs $\{G_n\}_{n\ge 3}$ by taking $n$ copies of the Petersen graph and wiring corresponding vertices according to an $n$-cycle permutation. Each $G_n$ ha…
Slater's list of Rogers-Ramanujan type identities consists of 130 series-product identities whose analytic proofs rely primarily on Bailey pair techniques. Although these identities play an important …
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