Advances in Generalized Hypergeometric Functions, Integral Transforms and Number Theory

In the paper, we prove a limit theorem in the sense of the weak convergence of probability measures for the modified Mellin transform $\mathcal{Z}\left(s\right)$, $s=\sigma +it$, with fixed $1/2<\sigma <1$, of the square ${\left|\zeta (1/2+it)\right|}^2$ of the Riemann zeta-function. We consider probability measures defined by means of $\mathcal{Z}(\sigma +i\phi (t\left)\right)$, where $\phi \left(t\right)$, $t⩾t_0>0$, is an increasing to $+\infty$ differentiable function with monotonically decreasing derivative ${\phi }^{\prime }\left(t\right)$ satisfying a certain normalizing estimate related to the mean square of the function $\mathcal{Z}(\sigma +i\phi (t\left)\right)$. This allows us to extend the distribution laws for $\mathcal{Z}\left(s\right)$. Full article

This paper shows that certain${ }_3{F}_4$ hypergeometric functions can be expanded in sums of pair products of${ }_2{F}_3$ functions, which reduce in special cases to${ }_2{F}_3$ functions expanded in sums of pair products of${ }_1{F}_2$ functions. This expands the class of hypergeometric functions having summation theorems beyond those expressible as pair-products of generalized Whittaker functions,${ }_2{F}_1$ functions, and${ }_3{F}_2$ functions into the realm of${ }_p{F}_q$ functions where $p<q$ for both the summand and terms in the series. In addition to its intrinsic value, this result has a specific application in calculating the response of the atoms to laser stimulation in the Strong Field Approximation. Full article

Let $\mathcal{P}$ be the set of generalized prime numbers, and ${\zeta }_{\mathcal{P}}\left(s\right)$, $s=\sigma +it$, denote the Beurling zeta-function associated with $\mathcal{P}$. In the paper, we consider the approximation of analytic functions by using shifts ${\zeta }_{\mathcal{P}}(s+i\tau )$, $\tau \in \mathbb{R}$. We assume the classical axioms for the number of generalized integers and the mean of the generalized von Mangoldt function, the linear independence of the set $\{logp:p\in \mathcal{P}\}$, and the existence of a bounded mean square for ${\zeta }_{\mathcal{P}}\left(s\right)$. Under the above hypotheses, we obtain the universality of the function ${\zeta }_{\mathcal{P}}\left(s\right)$. This means that the set of shifts ${\zeta }_{\mathcal{P}}(s+i\tau )$ approximating a given analytic function defined on a certain strip $\widehat{\sigma }<\sigma <1$ has a positive lower density. This result opens a new chapter in the theory of Beurling zeta functions. Moreover, it supports the Linnik–Ibragimov conjecture on the universality of Dirichlet series. For the proof, a probabilistic approach is applied. Full article

The Bessel function of the first kind $J_N\left(kx\right)$ is expanded in a Fourier–Legendre series, as is the modified Bessel function of the first kind $I_N\left(kx\right)$. The purpose of these expansions in Legendre polynomials was not an attempt to rival established numerical methods for calculating Bessel functions but to provide a form for $J_N\left(kx\right)$ useful for analytical work in the area of strong laser fields, where analytical integration over scattering angles is essential. Despite their primary purpose, one can easily truncate the series at 21 terms to provide 33-digit accuracy that matches the IEEE extended precision in some compilers. The analytical theme is furthered by showing that infinite series of like-powered contributors (involving ${ }_1{F}_2$ hypergeometric functions) extracted from the Fourier–Legendre series may be summed, having values that are inverse powers of the eight primes $1/\left(2^i{3}^j{5}^k{7}^l{11}^m{13}^n{17}^o{19}^p\right)$ multiplying powers of the coefficient k. Full article

We extend previous research to derive three additional M-1-dimensional integral representations over the interval $[0,1]$. The prior version covered the interval $[0,\infty ]$. This extension applies to products of M Slater orbitals, since they (and wave functions derived from them) appear in quantum transition amplitudes. It enables the magnitudes of coordinate vector differences (square roots of polynomials) $|{\mathbf{x}}_1-{\mathbf{x}}_2|=\sqrt{x_1^2-2x_1{x}_{2}cos\theta +x_2^2}$ to be shifted from disjoint products of functions into a single quadratic form, allowing for the completion of its square. The M-1-dimensional integral representations of M Slater orbitals that both this extension and the prior version introduce provide alternatives to Fourier transforms and are much more compact. The latter introduce a 3M-dimensional momentum integral for M products of Slater orbitals (in M separate denominators), followed in many cases by another set of M-1-dimensional integral representations to combine those denominators into one denominator having a single (momentum) quadratic form. The current and prior methods are also slightly more compact than Gaussian transforms that introduce an M-dimensional integral for products of M Slater orbitals while simultaneously moving them into a single (spatial) quadratic form in a common exponential. One may also use addition theorems for extracting the angular variables or even direct integration at times. Each method has its strengths and weaknesses. We found that these M-1-dimensional integral representations over the interval $[0,1]$ are numerically stable, as was the prior version, having integrals running over the interval $[0,\infty ]$, and one does not need to test for a sufficiently large upper integration limit as one does for the latter approach. For analytical reductions of integrals arising from any of the three, however, there is the possible drawback for large M of there being fewer tabled integrals over $[0,1]$ than over $[0,\infty ]$. In particular, the results of both prior and current representations have integration variables residing within square roots asarguments of Macdonald functions. In a number of cases, these can be converted to Meijer G-functions whose arguments have the form $(a{x}^2+bx+c)/x$, for which a single tabled integral exists for the integrals from running over the interval $[0,\infty ]$ of the prior paper, and from which other forms can be found using the techniques given therein. This is not so for integral representations over the interval $[0,1]$. Finally, we introduce a fourth integral representation that is not easily generalizable to large M but may well provide a bridge for finding the requisite integrals for such Meijer G-functions over $[0,1]$. Full article

The essential goal of this work is to suggest applying the multi-dimensional Sumdu generalized Laplace transform decomposition for solving pseudo-parabolic equations. This method is a combination of the multi-dimensional Sumudu transform, the generalized Laplace transform, and the decomposition method. We provided some examples to show the effectiveness and the ability of this approach to solve linear and nonlinear problems. The results show that the proposed method is reliable and easy for obtaining approximate solutions of FPDEs and is more precise if we compare it with existing methods. Full article

Many different number systems have been the topic of research. One of the recently studied number systems is that of hybrid numbers, which are generalizations of other number systems. In this work, we introduce and study the hybrid hyper k-Pell, hybrid hyper k-Pell–Lucas, and hybrid hyper Modified k-Pell numbers. In order to study these new sequences, we established new properties, generating functions, and the Binet formula of the hyper k-Pell, hyper k-Pell–Lucas, and hyper Modified k-Pell sequences. Thus, we present some algebraic properties, recurrence relations, generating functions, the Binet formulas, and some identities for the hybrid hyper k-Pell, hybrid hyper k-Pell–Lucas, and hybrid hyper Modified k-Pell numbers. Full article

This paper provides both analytical and numerical solutions of (PDEs) involving time-fractional derivatives. We implemented three powerful techniques, including the modified variational iteration technique, the modified Adomian decomposition technique, and the modified homotopy analysis technique, to obtain an approximate solution for the bounded space variable $\nu$. The Laplace transformation is used in the time-fractional derivative operator to enhance the proposed numerical methods’ performance and accuracy and find an approximate solution to time-fractional Fornberg–Whitham equations. To confirm the accuracy of the proposed methods, we evaluate homogeneous time-fractional Fornberg–Whitham equations in terms of non-integer order and variable coefficients. The obtained results of the modified methods are shown through tables and graphs. Full article