Integration of Hypergeometric Superhyperbolic Functions Using Generalized Hypergeometric Function and Its Applications

In the present paper, we study the arithmetic properties of power expansions related to generalized hypergeometric differential equations and series. Defining the series f(z),g(z) in powers of z so that f(z) and \(f\left( z \right)\log z + g\left( z \right)\) satisfy a hypergeometric equation under a special choice of parameters, we prove that the series \(q\left( z \right) = ze^{{{g\left( {Cz} \right)} \mathord{\left/ {\vphantom {{g\left( {Cz} \right)} {f\left( {Cz} \right)}}} \right. \kern-\nulldelimiterspace} {f\left( {Cz} \right)}}}\) in powers of z and its inversion z(q) in powers of q have integer coefficients (here the constant C depends on the parameters of the hypergeometric equation). The existence of an integral expansion z(q) for differential equations of second and third order is a classical result; for orders higher than 3 some partial results were recently established by Lian and Yau. In our proof we generalize the scheme of their arguments by using Dwork's p-adic technique.

While investigating the Lauricella's list of 14 complete second-order hypergeometric series in three variables, Srivastava noticed the existence of three additional complete triple hypergeometric series of the second order, which were denoted by <TEX>$H_A$</TEX>, <TEX>$H_B$</TEX> and <TEX>$H_C$</TEX>. Each of these three triple hypergeometric functions <TEX>$H_A$</TEX>, <TEX>$H_B$</TEX> and <TEX>$H_C$</TEX> has been investigated extensively in many different ways including, for example, in the problem of finding their integral representations of one kind or the other. Here, in this paper, we aim at presenting further integral representations for the Srivatava's triple hypergeometric function <TEX>$H_B$</TEX>.

We present the fermionic representation for the q-deformed hypergeometric functions related to Schur polynomials. We show that these multivariate hypergeometric functions are tau-functions of the KP hierarchy, and at the same time they are the ratios of Toda lattice tau-functions, considered by Takasaki, evaluated at certain values of higher Toda lattice times. The variables of the hypergeometric functions are related to the higher times of those hierarchies via a Miwa change of variables. The discrete Toda lattice variable shifts parameters of hypergeometric functions. Hypergeometric functions of type pΦs can also be viewed as a group 2-cocycle for the ΨDO on the circle (the group times are higher times of TL hierarchy and the arguments of a hypergeometric function). We obtain the determinant representation and the integral representation of a special type of KP tau-functions, these results generalize some of the results of Milne concerning multivariate hypergeometric functions. We write down a system of partial differential equations for these tau-functions (string equations).

Coxeter group actions on Saalschützian [formula omitted] series and very-well-poised [formula omitted] series

: The usual power series representations for hypergeometric functions in two variables have a limited range of validity. In particular, they are of little use when the magnitude of one of the variables becomes very large. Using a Barnes-type integral representation and the concept of analytic continuation, the region of utility is extended to the desired domain. The poles that occur in the Barnes-type integral are assumed to be simple. Thus, explicit asymptotic expansions are obtained for each of the fourteen hypergeometric functions that belong to this class. Hypergeometric functions, Functions with a large argument, Functions of two variables, Contour integration, Power series expansions.

The main aim of this research paper is to introduce a new extension of the Gauss hypergeometric function and confluent hypergeometric function by using an extended beta function. Some functional relations, summation relations, integral representations, linear transformation formulas, and derivative formulas for these extended functions are derived. We also introduce the logarithmic convexity and some important inequalities for extended beta function.

In this present paper, first, we investigate a new generalized Pochhammer's symbol and its various properties in terms of a new symbol $(s; k)$, where $s; k &gt; 0$. Then, we define a new generalization of gamma and beta functions and their various associated properties in the form of $(s; k)$. Also, we define a new generalization of hypergeometric functions and develop differential equations for generalized hypergeometric functions in the form of $(s; k)$. We present that generalized hypergeometric functions are the solution of the said differential equation. Furthermore, some useful results and properties and integral representation related to these generalized Pochhammer's symbol, gamma function, beta function, and hypergeometric functions are presented.

The l=0 partial wave projected Coulomb off-shell T matrix Tc,l=0 in momentum representation is obtained in closed form. Problems existing in the literature concerning the half- and on-shell behavior of Tc and Tc,l are discussed and clarified by means of explicit formulas. The remaining derivations in this paper are based on Tc,l=0. We consider the class of N-term separable potentials where the form factors are rational functions of p2 (in momentum representation). We prove that the l=0 T matrix corresponding to the Coulomb potential plus any such so-called rational separable potential has a very simple form, namely, it can be written in terms of rational functions and the (simple) hypergeometric function with parameters (1, iγ; 1+iγ), where γ is the well-known Coulomb parameter. Explicit analytic formulas are derived for a number of simple members of the class, the Yamaguchi potential being one of them. In this particular case the expressions of Zachary and of Bajzer are reproduced who used a method based on the O4 symmetry.

The special matrix functions have received significant attention in many fields, such as theoretical physics, number theory, probability theory, and the theory of group representations. In 2017, Dwivedi and Sahai introduced the generalized hypergeometric matrix function using matrix parameters and proved the convergence on | z | = 1 {|z|=1} . Recently, hypergeometric matrix functions and their potential applications have played a major role in mathematical physics and engineering. Motivated by aforesaid works and in order to enrich this flourishing field, we investigate the Euler-type integral representations for the generalized hypergeometric matrix function and determine various transformations in terms of hypergeometric matrix functions. Furthermore, unit and half arguments have been provided for several particular cases.

An efficient and accurate contour integration method is presented for large-scale electronic structure calculations based on the Green function. By introducing a continued fraction representation of the Fermi-Dirac function derived from a hypergeometric function, the Matsubara summation is generalized with respect to distribution of poles so that the integration of the Green function can converge rapidly. Numerical illustrations, evaluation of the density matrix for a simple model Green function and a total energy calculation for aluminum bulk within density functional theory, clearly show that the method provides remarkable convergence with a small number of poles, indicating that the method can be applied to not only the electronic structure calculations, but also a wide variety of problems.

The present paper is a further development, after the Great Proletarian Cultural Revolu-tion, of our statistical point of view that the element of homogeneous isotropic turbulenceis a small axially symmetrical vortex from the solution of the Navier-Stokes equations ofmotion. After long years of observation and analysis in practice, we recognize that thestructure of vortices, as elements of turbulence, does not obey the simple law of similarityor self-preservation. The structure undergoes a kind of stretching phenomena in the processof the decay of vortices. Therefore, we first bring out a vortex scale—vortex Reynoldsnumber relation, and then introduce a pseudo-similarity or psoudo-self-preservation conceptinto the theory. We solve the case of small vortex Reynolds number flows. The zerothorder approximation of the solution can be expressed in terms of confluent hypergeometricfunctions. The law of the decay of turbulent energy and the expression for the microscaleof turbulence from the initial to the final period of decay together with the double velocitycorrelation thus calculated all agree very well with the experiments of Batchelor and Town-send. We have also computed the energy spectrum transfer function corresponding to thetriple velocity correlation. But in this range of small Reynolds number flows there arestill no experimental measurements available for its verification. In the Appendix by using the method of multi-Fourier transform, we have derived theFourier transforms of the triple and n-tiple velocity correlations and the simpler form ofthe energy spectrum transfer function corresponding to the triple velocity correlation.

On the enumeration of [formula omitted]-omino towers

The main objective of this research paper is to explore further generalization of confluent hypergeometric and Whittaker functions by introducing a new parameter $k&gt;0$, in generalized extended confluent hypergeometric and Whittaker functions defined by Khan \emph{et al.} \cite{m7}. We also investigate some properties such as integral representations, Mallin transforms, Hankel, Laplace transformations and derivative of these new generalized extended confluent hypergeometric and Whittaker $k$-functions. We also obtain Riemann-Liouville fractional integral and Riemann-Liouville $k$-fractional integral of these new generalized extended Whittaker $k$-function.

The Heckman-Opdam hypergeometric functions of type BC extend classical Jacobi functions in one variable and include the spherical functions of non-compact Grassmann manifolds over the real, complex or quaternionic numbers. There are various limit transitions known for such hypergeometric functions, see e.g. [dJ], [RKV]. In the present paper, we use an explicit form of the Harish-Chandra integral representation as well as an interpolated variant, in order to obtain limit results for three continuous classes of hypergeometric functions of type BC which are distinguished by explicit, sharp and uniform error bounds. The rst limit realizes the approximation of the spherical functions of innite dimensional Grassmannians of xed rank; here hypergeometric functions of type A appear as limits. The second limit is a contraction limit towards Bessel functions of Dunkl type.

The main aim of this paper is to give the definitions of Caputo fractional derivative operators and show their use in the special function theory. For this purpose, we introduce new types of incomplete hypergeometric functions and obtain their integral representations. Furthermore, we define incomplete Caputo fractional derivative operators and show that the images of some elementary functions under the action of incomplete Caputo fractional operators give a new type of incomplete hypergeometric functions. This definition helps us to obtain linear and bilinear generating relations for the new type incomplete Gauss hypergeometric functions.