Integral Representation and Explicit Formula at Rational Arguments for Apostol–Tangent Polynomials

The Fourier series expansion of Apostol–tangent polynomials is derived using the Cauchy residue theorem and a complex integral over a contour. This Fourier series and the Hurwitz–Lerch zeta function are utilized to obtain the explicit formula at rational arguments of these polynomials. Using the Lipschitz summation formula, an integral representation of Apostol–tangent polynomials is also obtained.

On the generalized fractional Laplacian

The objective of this paper is, for the first time, to extend the fractional Laplacian (−△) s u(x) over the space Ck (Rn ) (which contains S(Rn ) as a proper subspace) for all s > 0 and s ≠ 1, 2, …, based on the normalization in distribution theory, Pizzetti’s formula and surface integrals in Rn . We further present two theorems showing that our extended fractional Laplacian is continuous at the end points 1, 2, … . Two illustrative examples are provided to demonstrate computational techniques for obtaining the fractional Laplacian using special functions, Cauchy’s residue theorem and integral identities. An application to defining the Riesz derivative in the classical sense at odd numbers is also considered at the end.

A note on the axisymmetric diffusion equation

We consider the explicit solution to the axisymmetric diffusion equation. We recast the solution in the form of a Mellin inversion formula, and outline a method to compute a formula for \(u(r,t)\) as a series using the Cauchy residue theorem. As a consequence, we are able to represent the solution to the axisymmetric diffusion equation as a rapidly converging series. doi:10.1017/S1446181121000110

On generalized Bessel–Maitland function

An approach to the generalized Bessel–Maitland function is proposed in the present paper. It is denoted by $\mathcal{J}_{\nu , \lambda }^{\mu }$ J ν , λ μ , where $\mu >0$ μ > 0 and $\lambda ,\nu \in \mathbb{C\ }$ λ , ν ∈ C get increasing interest from both theoretical mathematicians and applied scientists. The main objective is to establish the integral representation of $\mathcal{J}_{\nu ,\lambda }^{\mu }$ J ν , λ μ by applying Gauss’s multiplication theorem and the representation for the beta function as well as Mellin–Barnes representation using the residue theorem. Moreover, the mth derivative of $\mathcal{J}_{\nu ,\lambda }^{\mu }$ J ν , λ μ is considered, and it turns out that it is expressed as the Fox–Wright function. In addition, the recurrence formulae and other identities involving the derivatives are derived. Finally, the monotonicity of the ratio between two modified Bessel–Maitland functions $\mathcal{I}_{\nu ,\lambda }^{\mu }$ I ν , λ μ defined by $\mathcal{I}_{\nu ,\lambda }^{\mu }(z)=i^{-2\lambda -\nu }\mathcal{J}_{ \nu ,\lambda }^{\mu }(iz)$ I ν , λ μ ( z ) = i − 2 λ − ν J ν , λ μ ( i z ) of a different order, the ratio between modified Bessel–Maitland and hyperbolic functions, and some monotonicity results for $\mathcal{I}_{\nu ,\lambda }^{\mu }(z)$ I ν , λ μ ( z ) are obtained where the main idea of the proofs comes from the monotonicity of the quotient of two Maclaurin series. As an application, some inequalities (like Turán-type inequalities and their reverse) are proved. Further investigations on this function are underway and will be reported in a forthcoming paper.

A NOTE ON THE AXISYMMETRIC DIFFUSION EQUATION

We consider the explicit solution to the axisymmetric diffusion equation. We recast the solution in the form of a Mellin inversion formula, and outline a method to compute a formula for $u(r,t)$ as a series using the Cauchy residue theorem. As a consequence, we are able to represent the solution to the axisymmetric diffusion equation as a rapidly converging series.

Two families of pro-𝑝 groups that are not absolute Galois groups

Let 𝑝 be a prime. We produce two new families of pro-𝑝 groups which are not realizable as absolute Galois groups of fields. To prove this, we use the 1-smoothness property of absolute Galois pro-𝑝 groups. Moreover, we show in these families, one has several pro-𝑝 groups which may not be ruled out as absolute Galois groups employing the quadraticity of Galois cohomology (a consequence of the norm residue theorem), or the vanishing of Massey products in Galois cohomology.

A novel proof of two partial fraction decompositions

In this paper, by constructing contour integral and using Cauchy’s residue theorem, we provide a novel proof of Chu’s two partial fraction decompositions.

Gupta Transform Approach to the Series RL and RC Networks with Steady Excitation Sources

The analysis of electric networks circuits is an essential course in engineering. The response of such networks is usually obtained by mathematical approaches such as Laplace Transform, Calculus Approach, Convolution Theorem Approach, Residue Theorem Approach. This paper presents a new integral transform called Gupta Transform for obtaining the complete response of the series RL and RC networks circuits with a steady voltage source. The response obtained will provide electric current or charge flowing through series RL and RC networks circuits with a steady voltage source. In this paper, the response of the series RL and RC networks circuits with steady excitation source is provided as a demonstration of the application of the new integral transform called Gupta Transform.