The Hypergeometric Function

This paper deals with extended M-series, which is extension of the generalized M-series [12]. Mittag-Leffler function, $\omega$ - hypergeometric function, generalized $\omega-$ Gauss hypergeometric function, $\omega-$ confluent hypergeometric function, Bessel-Maitland function can be deduced as special cases of our finding. Moreover, we obtain some theorem for extended M-series by using fractional calculus operators and many results associated with Riemann-Liouville, Weyl and ErdelyiKober operators. We begin our study from the following definitions.

The object of this paper is to present generalizations of some recent results ot S.S. Miller and P.T. Mocanu on the univalence and starlikeness of Kummer' confluent hypergeometric function. Some applications involving generalized hypergeometric functions associated with the Hardy space of analytic functions are also considered.

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.

Log-convexity and log-concavity of hypergeometric-like functions

6–7 - Definite Integrals of Special Functions

A universal practical formula is given for calculating an integral which includes two confluent hypergeometric functions, power and exponential functions; then by means of this formula, the expressions of the radial matrix elements for a relativistic harmonic oscillator are given.

In this thesis, we find an exact formula for the weighted average of the symmetric square L-values at the center. The average is taken over a Hecke eigen basis of cusp forms of SL2(Z) with a fixed weight 2k. The weights are the n-th Fourier Coefficients of these functions. The terms in the formula involve quadratic Dirichlet L-values at the center, Confluent Hypergeometric functions, and some arithmetic functions. The main ingredient, and the starting point, is a formula due Shimura, which relates the symmetric square L-function of a Hecke eigen form f to the inner product of f with the product of the theta function, θ; and a real analytic Eisenstein series of half integral weight, E. We apply Michel-Ramakrishnan's averaging technique on Shimura's formula to write the weighted average of symmetric square L-values in terms of the Fourier coefficients of the Eisenstein series. There are two complications. First, the levels of θ x E and f are different. Second, E is not holomorphic. That is why we first take trace of θ x E, and then we take the holomorphic projection. Computing the Fourier coefficients of the resulting function gives us the exact formula desired. Finally, we deduce the asymptotic behavior of these formulas as k → ∞.

Based on the Hadamard product of power series, polynomial series expansions for confluent hypergeometric functions M ( a , c ; ⋅ ) M(a,c;\cdot ) and for Gaussian hypergeometric functions F ( a , b ; c ; ⋅ ) F(a,b;c;\cdot ) are introduced and studied. It turns out that the partial sums provide an interesting alternative for the numerical evaluation of the functions M ( a , c ; ⋅ ) M(a,c;\cdot ) and F ( a , b ; c ; ⋅ ) F(a,b;c;\cdot ) , in particular, if the parameters are also viewed as variables.

I. M. Gel’fand and his collaborators created the theory of general hypergeometric functions (multivariate hypergeometric functions on lattices and hypergeometric functions on Grassmannians). These functions are multivariate generalizations of hypergeometric functions of one variable. They are solutions of certain systems of differential equations (general hypergeometric systems) and are related to the Radon transform. At the present time, a deep connection of these hypergeometric functions with representations of groups is absent. But it is clear that this connection exists. Remark that the theory of q-analogues of Grassmannians and of related generalizations of Gel’fand hypergeometric functions is under elaboration (see, for example, [432]).

Some new re-lines with special functions for their solution

Changes of coordinates represent one of the most effective ways of deriving solvable potentials from ordinary differential equations for separate special functions. Here we relax the standard Hermiticity requirement and find an innovative construction which leads to unusual, complex potentials. Their energy spectrum is shown to stay real after a weakening of the Hermiticity of the Schr?dinger equation to its mere invariance under the combined (parity) and (time-reversal) symmetry. This ultimately results in richer bound-state spectra. Some of our new exactly solvable potentials generalize the current textbook models. Details are given for constructions based on the hypergeometric and confluent hypergeometric special functions.

The main objective of this research paper is to explore further generalization of confluent hypergeometric and Whittaker functions by introducing a new parameter $k>0$, 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 hypergeometric functions are one of the most important and special functions in mathematics. They are the generalization of the exponential functions. Particularly the ordinary hypergeometric function 2F1(a, b; c; z) is represented by hypergeometric series and is a solution to a second order differential equation. Similarly, Laplace transform is a form of integral transform that converts linear differential equations to algebraic equations. This paper aims to study the convergence of hypergeometric function and Laplace transform of some hypergeometric functions. Moreover, some relationships between Laplace transformation and hypergeometric functions is established in the concluding section of this paper.

Exact solutions of the Klein–Gordon–Fock (KGF) general relativistic equation that describe the dynamics of a massive, electrically charged scalar particle in the curved spacetime geometry of an electrically charged, rotating Kerr–Newman–(anti) de Sitter black hole are investigated. In the general case of a rotating, charged, cosmological black hole the solution of the KGF equation with the method of separation of variables results in Fuchsian differential equations for the radial and angular parts which for most of the parameter space contain more than three finite singularities and thereby generalise the Heun differential equations. For particular values of the physical parameters (i.e. mass of the scalar particle) these Fuchsian equations reduce to the case of the Heun equation and the closed form analytic solutions we derive are expressed in terms of Heun functions. For other values of the parameters some of the extra singular points are false singular points. We derive the conditions on the coefficients of the generalised Fuchsian equation such that a singular point is a false point. In such a case the exact solution of the Fuchsian equation can in principle be simplified and expressed in terms of Heun functions. This is the generalisation of the case of a Heun equation with a false singular point in which the exact solution of Heun’s differential equation is expressed in terms of Gauß hypergeometric function. We also derive the exact solutions of the radial and angular equations for a charged massive scalar particle in the Kerr–Newman spacetime. The analytic solutions are expressed in terms of confluent Heun functions. Moreover, we derived the constraints on the parameters of the theory such that the solution simplifies and expressed in terms of confluent Kummer hypergeometric functions. We also investigate the radial solutions in the KN case in the regions near the event horizon and far from the black hole. Finally, we construct several expansions of the solutions of the Heun equation in terms of generalised hypergeometric functions of Lauricella–Appell.

A theory of confluent hypergeometric functions is developed, based upon the methods described in the first three papers (I, II and III) of this series for replacing the divergent parts of asymptotic expansions by easily calculable series involving one or other of the four ‘ basic converging factors ’ which were investigated and tabulated in I. This theory is then illustrated by application to the special cases of exponential-type integrals, parabolic cylinder functions, modified Bessel functions, and ordinary Bessel functions.