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Abdulhafeez A. Abdulsalam

Publications and source records attributed to Abdulhafeez A. Abdulsalam.

8 recordsLinked to original sources
arXiv2026

Signed Generalized Stirling Polynomials, Nested Sums, and Hyperbolic Secant Integral Identities

We study signed generalized Stirling polynomials $P_k(m,x)$ arising in closed forms for Malmsten-type hyperbolic secant integrals. Their product structure is used to prove recurrences, gamma--polygamma formulas for $P_{m-s}(m,x)$, a central vanishing identity, a finite approximation to $\cosh πx$, and a limit formula for $π$. We identify these polynomials as signed residues of the equal-period Barnes multiple zeta function, derive their reflection formula, and obtain finite parity-cancellation and Stirling cycle-number identities. We also compare the centered polynomials with a classical Meixner--Pollaczek orthogonal family. We evaluate finite nested sums built from the sequence $χ_n$. Fixing the lower bounds turns these sums into coefficient-counting problems: common lower bounds give binomial coefficients and staircase bounds give Catalan numbers. Combining these counts with known formulas for $χ_j$ yields explicit hyperbolic-secant integral evaluations involving Catalan's constant, zeta values, and polygamma values. A Wolfram Language package accompanies the formulas.

math.CO↗
arXiv2026

New identities for the Laplace transform and their applications

In this paper, we begin by applying the Laplace transform to derive closed forms for several challenging integrals that seem nearly impossible to evaluate. By utilizing the solution to the Pythagorean equation $a^2 + b^2 = c^2$, these closed forms become even more intriguing. This approach allows us to provide new integral representations for the error function. Additionally, by leveraging an identity we derived for the inverse Laplace transform and applying a result based on Srivastava and Yürekli's identity, we provide a closed form for a nontrivial generalized integral.

math.GM↗
arXiv2026

Well-poised basic q-Taylor expansions with complementary remainders and a two-basis kernel

We prove a nonterminating well-poised basic $q$-Taylor expansion with complementary remainders for a two-basis infinite-product kernel implicitly proposed by the second author in \cite[Sec.~5]{Schlosser2008}. The well-poised parameter $c$ gives the rational $p=0$ basis, while the elliptic nome $p$ is a separate deformation; the infinite expansions treated here are specific to the basic case. We compute the two Taylor coefficient families and show that each one-family Taylor remainder tends to the complementary basis contribution. The proof uses the well-poised Cooper formula, Jackson's terminating ${}_8ϕ_7$ summation, Rogers' ${}_6ϕ_5$ summation, and theta interpolation, but not Bailey's nonterminating ${}_8ϕ_7$ summation, which is recovered as a consequence. We also record two quadratic one-family examples and discuss a multi-kernel outlook.

math.CA↗
arXiv2024

A new approach to evaluating Malmsten's integral and related integrals

This paper discusses generalizations of logarithmic and hyperbolic integrals. A new proof for an integral presented by Vardi and several other integrals in relation to known mathematical constants are discovered. We introduce the signed generalized Stirling polynomials of the first kind from the generalized Stirling polynomials of the first kind, and we give new expressions for the signed generalized Stirling polynomials of the first kind in terms of the Stirling cycle numbers and complete Bell polynomials. We establish the role of the signed generalized Stirling polynomials of the first kind and complete Bell polynomials in generalizing Malmsten's integral for all natural powers of the hyperbolic secant function, and we derive a reduction formula for the integral sequence. We give expressions for new integral sequences, which possess similar properties with Malmsten's integral, in terms of the signed generalized Stirling polynomials of the first kind, and we discover identities and a functional equation for the signed generalized Stirling polynomials of the first kind.

math.CA↗
arXiv2024

New closed forms for a dilogarithmic integral, related integrals, and series

In this study, we present a new closed form for the generalized integral $$\int_0^1 \frac{\mathrm{Li}_2(z) \ln(1+az)}{z}\, \mathrm{d}z,$$ where $a \in \mathbb{C} \setminus(-\infty, -1)$ and $\mathrm{Li}_2(z)$ is the dilogarithm function. This generalization is achieved by leveraging our established findings in conjunction with Vălean's results. Furthermore, we provide explicit closed forms for associated integrals, prove a transformation formula for double infinite series, expressing them as the sum of the square of an infinite series and another infinite series. We utilize this relationship to derive a novel closed form for the generalized series $$\sum_{k=1}^\infty \frac{ ζ\left(m, \frac{rk-s}{r}\right) }{(rk-s)^m},$$ for $\Re(m) > 1$, $r, s \in \mathbb{C}$, where $r \neq 0$, $rk \neq s$, for any positive integer $k$, and $ζ(s, z)$ denotes the Hurwitz zeta function. Utilizing Hermite's integral representation for $ζ(s, z)$, we derive a family of integrals from this series.

math.CA↗
arXiv2024

Closed form for $\sum_{k=1}^n k^p$ through the Hermite integral representation of the Hurwitz zeta function

Sums of positive integer powers have captivated the attention of mathematicians since ancient times. Over the centuries, mathematicians from diverse backgrounds have provided expressions for the sum of positive integer powers of the first $n$ positive integers. In this paper, we contribute to this endeavour by deriving Bernoulli's formula for $\sum_{k=1}^n k^p,$ for all positive integers $n$ and nonegative integers $p$, through the utilization of the Hermite integral representation of the Hurwitz zeta function.

math.GM↗
arXiv2023

New closed forms for a class of digamma series and integrals

The pursuit of closed forms for infinite series has long been a focal point of research. In this paper, we contribute to this endeavor by presenting closed forms for the class of digamma series: \[\sum_{k=1}^\infty \frac{ψ\left(\frac{2k+2n+5}{4}\right) - ψ\left(\frac{2k+2n+3}{4}\right)}{(2k + α)^2},\] \[\sum_{k=1}^\infty (-1)^k \frac{ψ\left(\frac{2k+2n+5}{4}\right) - ψ\left(\frac{2k+2n+3}{4}\right)}{(2k + α)^2},\] for all non-negative integers $α$ and $n$. In addition to providing closed forms for these series, we unveil new identities for various generalized digamma series in the elegant form $a_0 + a_1 π+ a_2 π^2 + a_3 π^3$, where $a_0, \ldots, a_3$ are real-valued constants determined by our formulas. Furthermore, we present ten definite integrals over the interval $(0, 1)$ that have not been previously studied in the literature and appear to be nearly impossible to evaluate. Combining these series and integrals can lead to the discovery of even more new results. Our findings contribute to the study of closed forms for infinite series and integrals, offering novel results and potential avenues for further exploration.

math.CA↗