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In this paper, some new families of orthogonal functions in two variables produced by using Fourier transform of bivariate orthogonal polynomials and their orthogonality relations obtained from Parseval identity are introduced.

Bicomplex convolutional neural networks (BCCNN) are a natural extension of the quaternion convolutional neural networks for the bicomplex case. As it happens with the quaternionic case, BCCNN has the capability of learning and modelling external dependencies that exist between neighbour features of an input vector and internal latent dependencies within the feature. This property arises from the fact that, under certain circumstances, it is possible to deal with the bicomplex number in a component-wise way. In this paper, we present a BCCNN, and we apply it to a classification task involving the colourized version of the well-known dataset MNIST. Besides the novelty of considering bicomplex numbers, our CNN considers an activation function a Bessel-type function. As we see, our results present better results compared with the one where the classical ReLU activation function is considered.

In this work, we study sums of finite products of Pell polynomials and express them in terms of some special orthogonal polynomials. Furthermore, each of the obtained expression is represented as linear combinations of classical polynomials involving hypergeometric functions by means of explicit computations.

Using an experimental mathematics approach, new relations are obtained between Dirichlet-like series for certain periodic coefficients and the moments of certain families of orthogonal polynomials.In addition to the classical hypergeometric orthogonal polynomials, of Racah type and continuous dual Hahn type, a new similar family of orthogonal polynomials intervenes.; En utilisant une approche de mathématiques expérimentales, de nouvelles relations sont obtenues entre les séries de Dirichlet pour certains coefficients périodiques et les moments de certaines familles de polynômes orthogonaux. Outre les polynômes orthogonaux hypergéométriques classiques, de type Racah et dual Hahn continu, une nouvelle famille similaire de polynômes orthogonaux intervient.

The identity 2F1(α,β;2α;z)=(1−z2)−β2F1(β2,β+12;α+12;(z2−z)2) given, either by I.S. Gradshteyn and I.M. Ryzhik in Table of integrals series and products named 9.134 or in the handbook “mathematical functions with formulas, graphs and mathematical tables” done by Abramowitz-Stegun named 15.3.20 or in the book “special functions” done by G. Andrews, R. Askey and R. Roy named 3.1.7 page 127 with a slight modification is true provided that {2α+1,α+32} are not natural numbers and α−β is not an integer (see Gradshteyn, Ryzhik, 9.130). In this manuscript we consider a case where α−β is an integer by taking β=2a, α=−n+1. We give and prove the right identity for any positive integer a and for any any positive integer n.

Several known families of relations among the multiple zeta values proved by using the generating functions are introduced to focus on the connection between the generating functions of relations among multiple zeta values and the hypergeometric functions. The infinite product expansion of the sine function and the gamma function representation of the Aomoto-Drinfel'd generating function are appeared in two of such results. It is also pointed out that the restricted sum formula due to Eie-Liaw-Ong in 2009 is able to obtain as a simple rewrite of a formula which had been proved by the author in 1999., Mathematical structures of integrable systems, its deepening and expansion. September 9-11, 2019. edited by Takao Suzuki. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

Recently the first author proved a congruence proposed in 2006 by Adamchuk: [Formula: see text] for any prime [Formula: see text]. In this paper, we provide more examples (with proofs) of congruences of the same kind [Formula: see text] where [Formula: see text] is a prime such that [Formula: see text], [Formula: see text] is a fraction in [Formula: see text] and [Formula: see text] is a [Formula: see text]-adic integer. The key ingredients are the [Formula: see text]-adic Gamma function [Formula: see text] and a special class of computer-discovered hypergeometric identities.

A new auxiliary result pertaining to twice (q1,q2)-differentiable functions is derived. Using this new auxiliary result, some new versions of Hermite–Hadamard’s inequality involving the class of generalized m-convex functions are obtained. Finally, to demonstrate the significance of the main outcomes, some applications are discussed for hypergeometric functions, Mittag–Leffler functions, and bounded functions.

The Grad–Shafranov plasma equilibrium equation was originally solved analytically in toroidal geometry, which fitted the geometric shape of the first Tokamaks. The poloidal surface of the Tokamak has evolved over the years from a circular to a D-shaped ellipse. The natural geometry that describes such a shape is the prolate elliptical one, i.e., the cap-cyclide coordinate system. When written in this geometry, the Grad–Shafranov equation can be solved in terms of the general Heun function. In this paper, we obtain the complete analytical solution of the Grad–Shafranov equation in terms of the general Heun functions and compare the result with the limiting case of the standard toroidal geometry written in terms of the Fock functions.

We introduce two extensions of the canonical Feller–Spitzer distribution from the class of Bessel densities, which comprise two distinct stochastically decreasing one-parameter families of positive absolutely continuous infinitely divisible distributions with monotone densities, whose upper tails exhibit a power decay. The densities of the members of the first class are expressed in terms of the modified Bessel function of the first kind, whereas the members of the second class have the densities of their Lévy measure given by virtue of the same function. The Laplace transforms for both these families possess closed–form representations in terms of specific hypergeometric functions. We obtain the explicit expressions by virtue of the particular parameter value for the moments of the distributions considered and establish the monotonicity of the mean, variance, skewness and excess kurtosis within the families. We derive numerous properties of members of these classes by employing both new and previously known properties of the special functions involved and determine the variance function for the natural exponential family generated by a member of the second class.

As of late, the study of Fractional Calculus (FC) and Special Functions (SFs) has been interestingly prompted in various realms of mathematics, engineering and sciences. This is due to the considerable demonstrated potential of their applications. Among these SFs, Gamma function and Mittag-Leffler functions are the most renowned and distinguished. Numerous authors continue to study this line. The current analysis attempts to introduce and further examine new modifications of Gamma and Kummer function in terms of Mittag-Leffler functions, respectively. Several attributes and formulations of this new Kummer-type function that include integral representations, Beta transform, Laplace transform, derivative formulas, and recurrence relation are investigated. Furthermore, outcomes of Riemann-Liouville fractional integral and fractional derivative in relation to this newly established Kummer function are also investigated.

It is well-known that the Kummer quadratic transformation formula is valid provided that its parameters fulfill some specific conditions (see Gradshteyn, Ryzhik, Tables of Integrals, Series and Products, 9.130, 9.134.1). Very recently, one of us established a new identity when one of these conditions is not fulfilled. In this paper, we aim to discuss another isolated case which completely different from the first. Moreover, in the end, we mention two interesting consequences of these two new results.

The deformed Landau problem under a electromagnetic field is studied, where the Heisenberg algebra is constructed in detail in non-commutative phase space in the presence of a minimal length. We show that, in the presence of a minimal length, the momentum space is more practical to solve any problem of eigenvalues. From the Nikiforov-Uvarov method, the energy eigenvalues are obtained and the corresponding wave functions are expressed in terms of hypergeometric functions. The fortuitous degeneration observed in the spectrum shows that the formulation of the minimal length complements that of the non-commutative phase space., Изучается деформированная задача Ландау в электромагнитном поле, в которой алгебра Гейзенберга подробно строится в некоммутативном фазовом пространстве при наличии минимальной длины. Мы показываем, что при наличии минимальной длины импульсное пространство более практично для решения любой проблемы собственных значений. С помощью метода Никифорова-Уварова получаются собственные значения энергии, а соответствующие волновые функции выражаются через гипергеометрические функции. Случайное вырождение, наблюдаемое в спектре, показывает, что формулировка минимальной длины дополняет формулировку некоммутативного фазового пространства., Вестник КРАУНЦ. Физико-математические науки, Выпуск 4 2020, Pages 188-198

In this paper, we introduce a logarithmic Mittag-Leffler function and discuss some of its properties. The application of these properties become helpful in extension of Pochhammer’s type contour integral representations and Rodrigues formulae of some known hypergeometric functions. On application point of view, some relations are discussed which are useful in interpreting the phenomenon of spread of infectious diseases in terms of Lauricella’s multiple hypergeometric functions. 2010Mathematics Subject Classification: 33E12, 33A17, 34A08. Resumen En este artículo, presentamos una función logarítmica de Mittag-Leffler y discutir algunas de sus propiedades. La aplicación de estas propiedades se vuelven útiles en la extensión de la integral de contorno de tipo de Pochhammer representaciones y fórmulas de Rodrigues de algunos conocidos hipergeométricos funciones. Desde el punto de vista de la aplicación, se discuten algunas relaciones que son útiles para interpretar el fenómeno de la propagación de infecciones enfermedades en términos de las múltiples funciones hipergeométricas de Lauricella.

The abstract is available here: https://uscholar.univie.ac.at/o:1628932

In 1985, Howard introduced degenerate Cauchy polynomials together with degenerate Bernoulli polynomials. His degenerate Bernoulli polynomials have been studied by many authors, but his degenerate Cauchy polynomials have been forgotten. In this paper, we introduce some kinds of hypergeometric degenerate Cauchy numbers and polynomials from the different viewpoints. By studying the properties of the first one, we give their expressions and determine the coefficients. Concerning the second one, called H-degenerate Cauchy polynomials, we show several identities and study zeta functions interpolating these polynomials.

According to K. Takeuchi ([Tku1], [Tku2]), all the arithmetic triangle groups are listed up (1977). They are classified in 19 commensurable classes (Table 2.1 below). It corresponds a quaternion algebra for each class. (a) The first class is of non-compact type, and it induces the usual elliptic modular function. (b) Among 18 remained classes, in 16 cases we have triangle unit groups. (c) For the rest 2 cases (the class II and XII) it appears quadrangle unit groups. Already we reported the result about the case (b) in the RIMS workshop of the previous year. There, we showed how to determine the Shimura canonical model modular function for them. As an application we exposed several defining equations of the Hilbert class fields of CM fields of higher degree. In this survey article we explain how to obtain the exact Shimura canonical model for the class II quadrangle case (that is for the quaternion algebra (-3, 2/Q)) using the modular function coming from the Appell's hypergeometric differential equation. We state only the framework of the argument together with several back grounds. Detailed proofs will be published elesewhere. This Shimura curve is studied by many mathematicians. See [Elk] for it., "Algebraic Number Theory and Related Topics 2016". November 28 - December 2, 2016. edited by Yasuo Ohno, Hiroshi Tsunogai and Toshiro Hiranouchi. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

We propose a new model for regression and dependence analysis when addressing spatial data with possibly heavy tails and an asymmetric marginal distribution. We first propose a stationary process with t marginals obtained through scale mixing of a Gaussian process with an inverse square root process with Gamma marginals. We then generalize this construction by considering a skew‐Gaussian process, thus obtaining a process with skew‐t marginal distributions. For the proposed (skew) t process, we study the second‐order and geometrical properties and in the t case, we provide analytic expressions for the bivariate distribution. In an extensive simulation study, we investigate the use of the weighted pairwise likelihood as a method of estimation for the t process. Moreover we compare the performance of the optimal linear predictor of the t process versus the optimal Gaussian predictor. Finally, the effectiveness of our methodology is illustrated by analyzing a georeferenced dataset on maximum temperatures in Australia.

In this paper, we develop some integral transforms involving a product of Humbert and Bessel functions with a weight e-γx2. These integral transforms will be evaluated in terms of hypergeometric functions. Various transformation formulae are also evaluated in terms of Appell functions to complete this study. Some special cases of the evaluated integrals yield some infinite series of generalized hypergeometric and Appell functions. As application, one of our main results is investigated to give an expression of the Generalized Humbert-Gaussian beams (GHGBs) propagating through a paraxial ABCD optical system.

We present some solutions of the three-dimensional Laplace equation in terms of linear combinations of generalized hyperogeometric functions in prolate elliptic geometry, which simulates the current tokamak shapes. Such solutions are valid for particular parameter values. The derived solutions are compared with the solutions obtained in the standard toroidal geometry.

The major goal of the current article is to create new formulas and connections between several well-known polynomials and the Euler polynomials. These formulas are developed using some of these polynomials’ well-known fundamental characteristics as well as those of the Euler polynomials. In terms of the Euler polynomials, new formulas for the derivatives of various symmetric and non-symmetric polynomials, including the well-known classical orthogonal polynomials, are given. This leads to the deduction of several new connection formulas between various polynomials and the Euler polynomials. As an important application, new closed forms for the definite integrals for the product of various symmetric and non-symmetric polynomials with the Euler polynomials are established based on the newly derived connection formulas.

This paper presents new results of Bernoulli polynomials. New derivative expressions of some celebrated orthogonal polynomials and other polynomials are given in terms of Bernoulli polynomials. Hence, some new connection formulas between these polynomials and Bernoulli polynomials are also deduced. The linking coefficients involve hypergeometric functions of different arguments that can be summed in some cases. Formulas that express some celebrated numbers in terms of Bernoulli numbers are displayed. Based on the new connection formulas between different polynomials and Bernoulli polynomials, along with some well-known integrals involving these polynomials, new closed forms for some definite integrals are given.

En este artículo, presentamos una función logarítmica de Mittag-Leffler y analizamos algunas de sus propiedades. La aplicación de estas propiedades se vuelve útil en la extensión de las representaciones integrales de contorno tipo de Pochhammer y las fórmulas de Rodrigues de algunas funciones hipergeométricas conocidas. Desde el punto de vista de la aplicación, se discuten algunas relaciones útiles para interpretar el fenómeno de propagación de enfermedades infecciosas en términos de las múltiples funciones hipergeométricas de Lauricella. In this paper, we introduce a logarithmic Mittag-Leffler function and discuss some of its properties. The application of these properties become helpful in extension of Pochhammer’s type contour integral representations and Rodrigues formulae of some known hypergeometric functions. On application point of view, some relations are discussed which are useful in interpreting the phenomenon of spread of infectious diseases in terms of Lauricella’s multiple hypergeometric functions.

In this article, we derived conditions on the coefficient functions a(z) and b(z) of the differential equations y″(z)+a(z)y′(z)+b(z)y(z)=0 and z2y″(z)+a(z)zy′(z)+b(z)y(z)=0, such their solution f(z) with normalization f(0)=0=f′(0)−1 is starlike in the lemniscate domain, equivalently zf′(z)/f(z)≺1+z. We provide several examples with graphical presentations for a clear view of the obtained results.

Orientador: Edmundo Capelas de Oliveira Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica Resumo: Efetuamos um levantamento histórico concernente ao cálculo integral e diferencial de ordem arbitrária, também conhecido como cálculo de ordem fracionária ou ainda cálculo fracionário, com o intuito de justificar sua importância, nos dias de hoje, a partir de uma audaciosa e profética frase proferida por Leibniz. A partir das várias definições para derivada de ordem arbitrária, em particular, as definições de Riemann, Liouville, Riemann-Liouville, Grünwald-Letnikov, Weyl e Caputo, elucidamos e justificamos a importância de cada uma delas, nas aplicações, quando associadas ao estudo de uma equação diferencial parcial de ordem arbitrária. Justificamos que, para problemas modelados pelas assim chamadas equações diferenciais de ordem arbitrária, o enfoque conforme proposto por Caputo parece ser o mais conveniente Abstract: We propose a hystorical review associated with the integral and differential calculus of arbitrary order, known as calculus of fractional order or also fractional calculus with the objective to justify its importance nowadays as of an audacious and profetic phrasis said by Leibniz. By means of several definitions associated with the derivative of fractional order, specifically, the definitions of Riemann, Liouville, Riemann-Liouville, Grünwald-Letnikov,Weyl and Caputo, we discuss and justify the importance of each one, in the applications, when associated with the study to the so-called differential equations of arbitrary order. We also justify that the derivative as proposed by Caputo is the most convenient in problems modelled by a fractional differential equation Mestrado Mestre em Matemática

We investigate the gravitational effect on Landau levels. We show that the familiar infinite Landau degeneracy of the energy levels of a quantum particle moving inside a uniform and constant magnetic field is removed by the interaction of the particle with a gravitational field. Two independent approaches are used to solve the relevant Schr\"odinger equation within the Newtonian approximation. It is found that both approaches yield qualitatively similar results within their respective approximations. With the goal of clarifying some results found in the literature concerning the use of a third independent approach for extracting the quantization condition based on a similar differential equation, we show that such an approach cannot yield a general and yet consistent result. We point out to the more accurate, but impractical, way to use such an approach; a way which does in principle yield a consistent quantization condition. We discuss how our results could be used to contribute in a novel way to the existing methods for testing gravity at the tabletop experiments level as well as at the astrophysical observational level by deriving the corrections brought by Yukawa-like and power-law deviations from the inverse-square law. The full relativistic regime is also examined in detail., Comment: 37 pages, no figures, matches the published version

The popular concept of slash distribution is generalized by considering the quotient Z = X/Y of independent random variables X and Y, where X is any continuous random variable and Y has a general beta distribution. The density of Z can usually be expressed by means of generalized hypergeometric functions. We study the distribution of Z for various parent distributions of X and indicate a possible application in finance.

To study an arithmetic nature of the values of hyper-geometric functions (and their derivatives including those with respect to parameter), it is common practice to use one either Siegel's method or the method based on the effective construction of the linear approximating form. The main distinction between these methods consists in the mode of construction of the first approximating form. Applying Siegel's method allows us to construct such a form by means of a pigeonhole principle that makes it possible to establish, in certain cases, the algebraic independence of the values of corresponding functions. The Siegel's method can be usually applied just while considering hyper-geometric functions with rational parameters. The effective method has a certain advantage here, for in some cases this method enables us to carry out corresponding investigation also for the functions with irrational parameters. Another advantage of the effective method is that it provides obtaining of high- accuracy quantitative results. By quantitative results one usually implies the estimates of the moduli of the linear forms in the values of corresponding functions. The effective method has made it possible to obtain, in some cases, estimates being precise regarding the height of such forms with calculation of the corresponding constants. A drawback of the effective method is the narrow domain of its applications. The effective construction of the linear approximating form, which initiates reasoning, is always a challenge. So far, an effective construction of the approximating form for the product of powers of hyper-geometric functions (with the rare exceptions) failed.In both aforementioned methods one proves previously linear independence (in Siegel's method, as a rule, algebraic independence,) of the functions under consideration. Such a proof is often considered as an independent result.In this paper, by means of the method especially elaborated for this purpose we establish linear independence of some hyper-geometric functions and their derivatives (including those with respect to parameter) over the field of rational fractions. Subsequently, it will be possible to apply this result to investigate arithmetic properties of the values of such functions. Herewith we mean the application of the effective method to achieve the sufficiently accurate quantitative result.

Given two polynomial sets $\{ P_n(x) \}_{n\geq 0},$ and $\{ Q_n(x) \}_{n\geq 0}$ such that $\deg ( P_n(x) ) = \deg ( Q_n(x) )=n.$ The so-called connection problem between them asks to find coefficients $\alpha_{n,k}$ in the expression $\displaystyle Q_n(x) =\sum_{k=0}^{n} \alpha_{n,k} P_k(x).$ The connection problem for different types of polynomials has a long history, and it is still of interest. The connection coefficients play an important role in many problems in pure and applied mathematics, especially in combinatorics, mathematical physics and quantum chemical applications. For the particular case $Q_n(x)=x^n$ the connection problem is called the inversion problem associated to $\{P_n(x)\}_{n\geq 0}.$ The particular case $Q_n(x)=P'_{n+1}(x)$ is called the derivative connecting problem for polynomial family $\{ P_n(x) \}_{n\geq 0}.$ In this paper, we give a closed-form expression of the inversion and the derivative coefficients for hypergeometric polynomials of the form $${}_2 F_1 \left[ \left. \begin{array}{c} -n, a \\ b \end{array} \right | z \right], {}_2 F_1 \left[ \left. \begin{array}{c} -n, n+a \\ b \end{array} \right | z \right], {}_2 F_1 \left[ \left. \begin{array}{c} -n, a \\ \pm n +b \end{array} \right | z \right],$$ where $\displaystyle {}_2 F_1 \left[ \left. \begin{array}{c} a, b \\ c \end{array} \right | z \right] =\sum_{k=0}^{\infty} \frac{(a)_k (b)_k}{(c)_k} \frac{z^k}{k!},$ is the Gauss hypergeometric function and $(x)_n$ denotes the Pochhammer symbol defined by $$\displaystyle (x)_n=\begin{cases}1, n=0, \\x(x+1)(x+2)\cdots (x+n-1) , n>0.\end{cases}$$ All polynomials are considered over the field of real numbers.

We obtain stochastic duality functions for specific Markov processes using representation theory of Lie algebras. The duality functions come from the kernel of a unitary intertwiner between $*$-representations, which provides (generalized) orthogonality relations for the duality functions. In particular, we consider representations of the Heisenberg algebra and $\mathfrak{su}(1,1)$. Both cases lead to orthogonal (self-)duality functions in terms of hypergeometric functions for specific interacting particle processes and interacting diffusion processes., 23 pages

In this paper, we first obtain the Fourier transforms of some finite bivariate orthogonal polynomials and then by using the Parseval identity, we introduce some new families of bivariate orthogonal functions.

The purpose of this paper is to construct a unified generating function involving the families of the higher-order hypergeometric Bernoulli polynomials and Lagrange–Hermite polynomials. Using the generating function and their functional equations, we investigate some properties of these polynomials. Moreover, we derive several connected formulas and relations including the Miller–Lee polynomials, the Laguerre polynomials, and the Lagrange Hermite–Miller–Lee polynomials.

Hypergeometric structures in single and multiscale Feynman integrals emerge in a wide class of topologies. Using integration-by-parts relations, associated master or scalar integrals have to be calculated. For this purpose it appears useful to devise an automated method which recognizes the respective (partial) differential equations related to the corresponding higher transcendental functions. We solve these equations through associated recursions of the expansion coefficient of the multivalued formal Taylor series. The expansion coefficients can be determined using either the package Sigma in the case of linear difference equations or by applying heuristic methods in the case of partial linear difference equations. In the present context a new type of sums occurs, the Hurwitz harmonic sums, and generalized versions of them. The code HypSeries transforming classes of differential equations into analytic series expansions is described. Also partial difference equations having rational solutions and rational function solutions of Pochhammer symbols are considered, for which the code solvePartialLDE is designed. Generalized hypergeometric functions, Appell-, Kampé de Fériet-, Horn-, Lauricella-Saran-, Srivasta-, and Exton–type functions are considered. We illustrate the algorithms by examples.

The main motive of this study is to present a new class of generalized integral formulae which involve a generating function of two variables G(u, x). By this approach we deduce a set of new outcomes, which are integrals associated with generalized hypergeometric function, Laguerre, Hermite and Bessel polynomials, Kampe de Feriet hypergeometric series of two variables, Lauricella function and several special cases of our main results.

▪ International Conference on Mathematical Analysis and Computing, ICMAC 2019, Kalavakkam, 23 December 2019through 24 December 2019. ▪ Volume Editors; Mohapatra R.N., Yugesh S., Kalpana G., Kalaivani C. A brief tour of more than one hundred years of the historical development of some of the popular integral and related operators in Geometric Function Theory (GFT) is given in this article. The strengths and discovery of the methods used in these operators lie in their ability to unify a large number of diverse operators and results. We also address some of the q- analogues of the integral operators in GFT. Since there are several surveys and books in GFT, we present here only a selection of the results related to our precise objectives. © 2021, The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

In this paper, we show that a closed-form formula for the generalized Clebsch–Gordan integral and the Fourier–Legendre expansion theory allow to evaluate hypergeometric series involving powers of the normalized central binomial coefficient $${\frac{1}{4^{n}}\genfrac(){0.0pt}0{2n}{n}}$$ .

In this paper, we show how to construct exceptional orthogonal polynomials (XOP) using isospectral deformations of classical orthogonal polynomials. The construction is based on confluent Darboux transformations, where repeated factorizations at the same eigenvalue are allowed. These factorizations allow us to construct Sturm-Liouville problems with polynomial eigenfunctions that have an arbitrary number of realvalued parameters. We illustrate this new construction by exhibiting the class of deformed Gegenbauer polynomials, which are XOP families that are isospectral deformations of classical Gegenbauer polynomials.

This paper is devoted to the analytical treatment of trinomial equations of the form \(y^n+y=x,\) where \(y\) is the unknown and \(x\in\mathbb{C}\) is a free parameter. It is well-known that, for degree \(n\geq 5,\) algebraic equations cannot be solved by radicals; nevertheless, roots are described in terms of univariate hypergeometric or elliptic functions. This classical piece of research was founded by Hermite, Kronecker, Birkeland, Mellin and Brioschi, and continued by many other Authors. The approach mostly adopted in recent and less recent papers on this subject (see [1,2] for example) requires the use of power series, following the seminal work of Lagrange [3]. Our intent is to revisit the trinomial equation solvers proposed by the Italian mathematician Davide Besso in the late nineteenth century, in consideration of the fact that, by exploiting computer algebra, these methods take on an applicative and not purely theoretical relevance.

The main purpose of the current article is to develop new specific and general linearization formulas of some classes of Jacobi polynomials. The basic idea behind the derivation of these formulas is based on reducing the linearization coefficients which are represented in terms of the Kamp&eacute, de F&eacute, riet function for some particular choices of the involved parameters. In some cases, the required reduction is performed with the aid of some standard reduction formulas for certain hypergeometric functions of unit argument, while, in other cases, the reduction cannot be done via standard formulas, so we resort to certain symbolic algebraic computation, and specifically the algorithms of Zeilberger, Petkovsek, and van Hoeij. Some new linearization formulas of ultraspherical polynomials and third-and fourth-kinds Chebyshev polynomials are established.

A number of generalized Hurwitz&ndash, Lerch zeta functions have been presented and investigated. In this study, by choosing a known extended Hurwitz&ndash, Lerch zeta function of two variables, which has been very recently presented, in a systematic way, we propose to establish certain formulas and representations for this extended Hurwitz&ndash, Lerch zeta function such as integral representations, generating functions, derivative formulas and recurrence relations. We also point out that the results presented here can be reduced to yield corresponding results for several less generalized Hurwitz&ndash, Lerch zeta functions than the extended Hurwitz&ndash, Lerch zeta function considered here. For further investigation, among possibly various more generalized Hurwitz&ndash, Lerch zeta functions than the one considered here, two more generalized settings are provided.

We study an energy-dependent potential related to the Rosen&ndash, Morse potential. We give in closed-form the expression of a system of eigenfunctions of the Schr&ouml, dinger operator in terms of a class of functions associated to a family of hypergeometric para-orthogonal polynomials on the unit circle. We also present modified relations of orthogonality and an asymptotic formula. Consequently, bound state solutions can be obtained for some values of the parameters that define the model. As a particular case, we obtain the symmetric trigonometric Rosen&ndash, Morse potential for which there exists an orthogonal basis of eigenstates in a Hilbert space. By comparing the existent solutions for the symmetric trigonometric Rosen&ndash, Morse potential, an identity involving Gegenbauer polynomials is obtained.

In this article, we extend the integration by parts formulae (IbPF) for the laws of Bessel bridges obtained in a recent work by Elad Altman and Zambotti to linear functionals. Our proof relies on properties of hypergeometric functions, thus providing a new interpretation of these formulae., 13 pages

We consider one-dependent random walks on $\mathbb{Z}^d$ in random hypergeometric environment for $d\ge 3$. These are memory-one walks in a large class of environments parameterized by positive weights on directed edges and on pairs of directed edges which includes the class of Dirichlet environments as a special case. We show that the walk is a.s. transient for any choice of the parameters, and moreover that the return time has some finite positive moment. We then give a characterization for the existence of an invariant measure for the process from the point of view of the walker which is absolutely continuous with respect to the initial distribution on the environment in terms of a function $\kappa$ of the initial weights. These results generalize [Sab11] and [Sab13] on random walks in Dirichlet environment. It turns out that $\kappa$ coincides with the corresponding parameter in the Dirichlet case, and so in particular the existence of such invariant measures is independent of the weights on pairs of directed edges, and determined solely by the weights on directed edges., Comment: 23 pages

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2020, Director: Jordi Marzo Sánchez, [en] In this work we will prove a theorem that Bernstein proved in 1937. This theorem states that there are no quadrature formulas with equal weights (of Chebyshev) in the interval $[-1,1]$ $$ \int_{-1}^{1} f(x) d x \approx \frac{2}{n} \sum_{k=1}^{n} f\left(x_{k}\right) $$ that are true for polynomials $f$ of degree $\leq n$, with nodes $x_{k} \in[-1,1]$, if $n \geq 10$. We will also see some results related to the distribution of these nodes when $n$ is large.

In this paper, we define a new extension of Srivastava's triple hypergeometric functions by using a new extension of Pochhammer's symbol that was recently proposed by Srivastava, Rahman and Nisar [H. M. Srivastava, G. Rahman and K. S. Nisar, Some extensions of the Pochhammer symbol and the associated hypergeometric functions, Iran. J. Sci. Technol. Trans. A Sci. 43 2019, 5, 2601-2606]. We present their certain basic properties such as integral representations, derivative formulas, and recurrence relations. Also, certain new special cases have been identified and some known results are recovered from main results. © 2020 Walter de Gruyter GmbH, Berlin/Boston 2020.

Beaton, Owczarek and Xu (2019) studied generating functions of Kreweras walks and of reverse Kreweras walks in the quarter plane, with interacting boundaries. They proved that for the reverse Kreweras step set, the generating function is always algebraic, and for the Kreweras step set, the generating function is always D-finite. However, apart from the particular case where the interactions are symmetric in $x$ and~$y$, they left open the question of whether the latter one is algebraic. Using computer algebra tools, we confirm their intuition that the generating function of Kreweras walks is not algebraic, apart from the particular case already identified., Comment: 13 pages, accepted at FPSAC'21

In this thesis, the theory of classical orthogonal polynomials which are Hermite, Laguerre and Jacobi polynomials will be studied. To begin with, we will supply an outline regarding the special functions. Followed by examples of properties for orthogonal polynomials in Chapter 2. In the third chapter, we begin classical orthogonal polynomials. To start with, we collate the orthogonal relation, Rodrigues formulas followed by the norm of the classical orthogonal polynomials. In the same chapter, the division of the collected examples of classical orthogonal polynomials into three chapters and assign them the weight function, intermission of the orthogonality, followed by differential equations, hypergeometric representation. To finalise we explain limit relations between polynomials. Keywords: Classical orthogonal polynomials, hypergeometric functions, second order differential equations, Rodrigues formula. ÖZ: Bu tezde Hermite, Laguerre ve Jacobi olan klasik ortogonal polinomlar açıklanmıştır. Öncelikli olarak özel fonksiyonlar hakkında bilgi verilmiştir. İlerleyen bölümlerinde ise ortogonal polinomların özelikleri anlatılmıştır. Daha sonraki bölümde de klasik ortogonal polinomlar tanımlanarak ortogonallik ilişkisi anlatılmıştır. Rodrigues formülü ile klasik ortogonal polinomlar için norm hesabı yapılmıştır. Daha sonra ise klasik ortogonal polinom örneklerinin üç bölüme ayrıldığını görürüz. Bunların her biri için ayrı ayrı ağırlık fonksiyonları, ortogonallik aralığı, ikinci dereceden diferensiyel denklemi ve hipergeometrik gösterimi verilerek anlatılmıştır. Tezin son bölümünde de polinomlar arasındaki limit ilişkileri açıklanmıştır. Anahtar Kelimeler: Klasik ortogonal polinomlar, hipergeometrik fonksiyon, ikinci dereceden diferansiyel denklem, Rodrigues formülü. Master of Science in Mathematics. Thesis (M.S.)--Eastern Mediterranean University, Faculty of Arts and Sciences, Dept. of Mathematics, 2020. Supervisor: Prof. Dr. Sonuç Zorlu Oğurlu.

It is well known that Feynman integrals in dimensional regularization often evaluate to functions of hypergeometric type. Inspired by a recent proposal for a coaction on one-loop Feynman integrals in dimensional regularization, we use intersection numbers and twisted homology theory to define a coaction on certain hypergeometric functions. The functions we consider admit an integral representation where both the integrand and the contour of integration are associated with positive geometries. As in dimensionally- regularized Feynman integrals, endpoint singularities are regularized by means of exponents controlled by a small parameter ϵ. We show that the coaction defined on this class of integral is consistent, upon expansion in ϵ, with the well-known coaction on multiple polylogarithms. We illustrate the validity of our construction by explicitly determining the coaction on various types of hypergeometric $_{p+1}$F$_{p}$ and Appell functions. It is well known that Feynman integrals in dimensional regularization often evaluate to functions of hypergeometric type. Inspired by a recent proposal for a coaction on one-loop Feynman integrals in dimensional regularization, we use intersection numbers and twisted homology theory to define a coaction on certain hypergeometric functions. The functions we consider admit an integral representation where both the integrand and the contour of integration are associated with positive geometries. As in dimensionally-regularized Feynman integrals, endpoint singularities are regularized by means of exponents controlled by a small parameter $\epsilon$. We show that the coaction defined on this class of integral is consistent, upon expansion in $\epsilon$, with the well-known coaction on multiple polylogarithms. We illustrate the validity of our construction by explicitly determining the coaction on various types of hypergeometric ${}_{p+1}F_p$ and Appell functions.

В статье представлен выборочный обзор начально-граничной задачи на собственные значения для дифференциального уравнения колебаний балки с изменяющимся сечением. Рассмотрен случай линейно изменяющегося сечения и приведены решения уравнения колебаний балки для различных случаев с использованием функций Бесселя, Ханкеля, Макдональда и гипергеометрических функций.

We prove two integral transformations that relate different constructions of rational approximations to [math] . The first one relates a double integral over the unit square and a Barnes-type integral. The second one relates two Barnes-type integrals and was discovered and proved by W. Zudilin using an automated proof method. Here we offer a proof based on more classical means such as contiguous relations, the second Barnes lemma and the duplication formula for the gamma function.