Convex and starllke generalized hypergeometric functions associated with the hardy pace

In the present work the space $L_{p;r} $ which is continuously embedded into $L_{p} $ is introduced. The corresponding Hardy spaces of analytic functions are defined as well. Some properties of the functions from these spaces are studied. The analogs of some results in the classical theory of Hardy spaces are proved for the new spaces. It is shown that the Cauchy singular integral operator is bounded in $L_{p;r} $. The problem of basisness of the system $left{Aleft(tright)e^{{mathop{rm int}} }; Bleft(tright)e^{-{mathop{rm int}} } right}_{nin Z_{+} }, $ is also considered. It is shown that under an additional condition this system forms a basis in $L_{p;r} $ if and only if the Riemann-Hilbert problem has a unique solution in corresponding Hardy class ${ H}_{p;r}^{+} times { H}_{p;r}^{+} $.

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.

Linear combinations of composition operators induced by linear fractional maps on the infinite polydisk

We introduce a method for the fast numerical approximation of linear, second-order parabolic partial differential equations (PDEs for short) with time-independent coefficients based on model order reduction techniques and the Laplace transform. We start by applying this transform to the evolution problem, thus yielding a time-independent boundary value problem solely depending on the complex Laplace variable. In an offline stage, we judiciously sample the Laplace variable and numerically solve the corresponding collection of high-fidelity or full-order problems. Next, we apply a proper orthogonal decomposition (POD) to this collection of solutions in order to obtain a reduced basis in the Laplace domain. We project the linear parabolic problem onto this basis and then, using any suitable time-stepping method, we solve the evolution problem. A key insight to justify the implementation and analysis of the proposed method consists of using Hardy spaces of analytic functions and establishing, through the Paley-Wiener theorem, an isometry between the solution of the time-dependent problem and its Laplace transform. As a result, one may conclude that computing a POD with samples taken in the Laplace domain produces an exponentially accurate reduced basis for the time-dependent problem. Numerical experiments illustrate the performance of the method in terms of accuracy and, in particular, speed-up when compared to the solution obtained by solving the full-order model.

Let E \subset(-1,1) be a compact set, let μ be a positive Borel measure with support \supp μ =E , and let H p (G),1≤ p ≤∈fty, be the Hardy space of analytic functions on the open unit disk G with circumference Γ={z \colon |z|=1} . Let Δ n,p be the error in best approximation of the Markov function \frac{1}{2π i} ∈t_E \frac{d μ(x)}{z-x} in the space L p (Γ) by meromorphic functions that can be represented in the form h=P/Q , where P ∈ H p (G),Q is a polynomial of degree at most n , Q\not \equiv 0 . We investigate the rate of decrease of Δ n,p ,1≤ p ≤∈fty , and its connection with n -widths. The convergence of the best meromorphic approximants and the limiting distribution of poles of the best approximants are described in the case when 1 - ∈fty.

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

Because of the many relations connecting the special functions to each other, and to the elementary functions, it is natural to inquire whether more general functions can be developed so that the special functions and elementary functions are merely specializations of these general functions. General functions of this nature have in fact been developed and are collectively referred to as functions of the hypergeometric type. There are several varieties of these functions, but the most common are the standard hypergeometric function ( which we discuss in this chapter) and the confluent hypergeometric function (Chap. 10).

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.

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.

6–7 - Definite Integrals of Special Functions

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.

In recent years, multiple-input-multiple-output communication has a great demand due to its flexibility and well technical implementation. Though it has been approached by many researchers, in this paper, authors taken an approach with Nakagami-m fading channel as it is a generalized fading channel for the performance of space shift keying (SSK) modulation. Simple closed form expression for the probability of bit error of SSK modulation is presented using probability density function based method for independent and identically distributed (IID) Nakagami-m fading channel. An upper bound expression for IID and correlated Nakagami-m fading channel is also presented. Obtained probability of error are in the form of Gauss hypergeometric function, confluent hypergeometric function and Gamma function which can be numerically evaluated and verified by Monte Carlo simulation.