General Mathematics
We introduce a family of deformed Bailey pairs whose $q$-series, which converge in a two-step limit ($q \to 1^-$ followed by $n \to \infty$) to Dirichlet $L$-functions scaled by $1/\sqrt{\pi}$. This construction generalizes to arbitrary…
The Riemann Hypothesis states that within the strip region of the complex plane $0 < {\rm Re}(s) < 1$, the Riemann $\xi(s)$ function has zeros only on the critical line ${\rm Re}(s) = \frac{1}{2}$ and none elsewhere. To prove the Riemann…
We construct a unified analytic framework connecting Bernoulli numbers, zeta-regularization, and Fredholm determinants associated with trigonometric selector kernels. Starting from the Bernoulli-Stirling algebra, Euler-Maclaurin corrections…
If a (weighted) spherical design is defined as an integration (cubature) rule for a unitarily invariant space P of polynomials (on the sphere), then any unitary image of it is also such a spherical design. It therefore follows that such…
Diffusion within porous media, such as biological tissues, exhibits departures from conventional Fick's laws, which could result in space-fractional diffusion. The paper considers a reaction-diffusion system with two spatial compartments --…
In this manuscript, we present the complete monotonicity of functions defined in terms of the poly-double gamma function \begin{align*} \psi_2^{(n)}(x) = (-1)^{n+1} n! \sum_{k=0}^{\infty} \dfrac{(1+k)}{(x+k)^{n+1}}, \quad x > 0, \ n\geq 2.…
In this manuscript, various properties of the Ramanujan integral $I_R(x)$, defined as \begin{align*} I_R(x) = \int_0^\infty e^{-xt} \dfrac{dt}{t(\pi^2 + \log^2 t)}, \quad x>0, \end{align*} are investigated, including its monotonicity,…
In this paper, we introduce a new family of Szasz-Mirakyan-Durrmeyer operators defined on the half-line [0,\infty), constructed using Laguerre-type kernels. We discuss in detail the algebraic structure and analytical properties of these…
We extend the classical results of Stanislaw Golab, on the values of pi in arbitrary normed planes, to asymmetric norms where the unit ball has one axis of symmetry. First, we characterize the values of $\pi_B$ for different families of…
In the work "Generalized approximation spaces generation from $\mathbb{I}_{j}$-neighborhoods and ideals with application to Chikungunya disease, published in \emph{AIMS Mathematics}, \textbf{9}(4) (2024), 10050$-$10077," Al-Shami and Hosny…
This article provides a proof of the famous \textit{Prime Number Theorem} by establishing an analogous statement of the same in terms of the second \textit{Chebyshev Function} $\psi(x)$. We shall be extensively using complex analytic…
In this paper we consider the question of smoothness of slowly varying functions satisfying the modern definition that, in the last two decades, gained prevalence in the applications concerning function spaces and interpolation. We show,…
In this manuscript, we consider the Riemann zeta function $\zeta$, defined through the Abel summation formula. We present a simple analytical method based on a complex differential equation. The aim is to propose a new analytical approach,…
The Gregorian calendar -- first established for daily use on Friday, October 15th, 1582 by Pope Gregory XIII in Catholic countries -- is presently the most pervasive calendar in the world. As such, algorithms for performing various…
In this paper, we show some results about the gap between a prime number and its consecutive prime number for large enough prime numbers. We show that the gap between a prime number $p_n$ and its consecutive prime number is not larger than…
For some smooth special case of generalized $\varphi-$divergences as well as of new divergences (called scaled shift divergences), we derive approximations of the omnipresent (weighted) $\ell_{1}-$distance and (weighted) $\ell_{1}-$norm.
We propose two new proofs of the Pythagorean theorem via area rearrangement arguments starting from very simple geometric configurations. The constructions depend on an angular parameter, each choice of which yields a proof. For specific…
In this paper, the exact solutions of certain non-linear differential equations defined on a fractal subset of the real line are presented. Particular attention is paid to the Riccati-type fractal differential equation, for which a…
Quasiorders $\varrho\subseteq A^{2}$ have the property that an operation $f:A^{n}\to A$ preserves $\varrho$ if and only if each (unary) translation obtained from $f$ is an endomorphism of $\rho$. Generalized quasiorders $\rho\subseteq A^{m}…
This study investigates a generalisation of the Pythagorean theorem to the lengths of conic arcs constructed symmetrically on the sides of a right triangle. It is demonstrated that the theorem remains valid whenever the conic eccentricity…