Expansions for some confluent hypergeometric 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.

The basic parabolic cylinder (bpc) distributions of the firstBasic parabolic cylinder (bpc) distributionof the first kind and second kindsBasic parabolic cylinder (bpc) distributionof the second kind were derived by OgasawaraOgasawara (Commun Stat Theor Methods [17]) to obtain the absolute momentsMomentsabsolute of the normal variable underStripe truncation stripe (zebraicTruncationzebraic or tigerishTruncationtigerish) truncationTruncationstripe. The bpc distribution of the third kindBasic parabolic cylinder (bpc) distributionof the third kind or simply bpc is introduced as a general bpc, whose special cases are the bpc’s of the first and second kinds. TheWeighted or incomplete parabolic cylinder function weighted or incomplete parabolic cylinder functionParabolic cylinder functionincomplete is obtained using theWeighted or incomplete Kummer confluent hypergeometric function weighted or incomplete Kummer confluent hypergeometric functionKummer confluent hypergeometric function given by OgasawaraOgasawara [J Multivar Anal [18]) to have the absolute momentsMomentsabsolute. The multivariate bpc distributionMultivariate basic parabolic cylinder (bpc) distribution is also derived to obtain the absolute momentsMomentsabsolute of the normal vectorNormalvector underTruncationsectional sectional truncationSectional truncation, which is the multivariate version ofTruncationstripe stripe truncationStripe truncation.

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.

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).

6–7 - Definite Integrals of Special Functions

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.

Option prices can be used to extract the implied risk-neutral density functions of the future underlying asset prices and returns. These market expectations provide valuable information that can be helpful to policy makers and investors. We tested the accuracy and stability of four nonstructural models in estimating the “true” risk-neutral density functions from option prices: the density functional based on the confluent hypergeometric function, the mixture of lognormal distributions, the smoothed implied volatility smile and Edgeworth expansions. The “true” risk-neutral density is unknown, so it was generated using the Carr–Geman–Madan–Yor (CGMY) Gamma-Ornstein–Uhlenbeck (Gamma-OU) model, a structural model able to generate flexible “true” risk-neutral densities. We observed that the density functional based on the confluent hypergeometric function and mixture of lognormal distributions outperformed the smoothed implied volatility smile and the Edgeworth expansion models in capturing the “true” risk-neutral density. The smoothed implied volatility smile had the best performance in terms of stability. This work aims to be more exhaustive than previous studies, by testing a wider variety of nonstructural models. In addition, we tested the effectiveness of these models in more demanding conditions that are closer to reality. In fact, the use of the CGMY Gamma-OU model to build the “true” risk-neutral density allows densities with a higher probability of extreme events and more leptokurtosis, and also generates sudden, discontinuous moves in prices, which has advantages over the diffusion models used in previous studies.

The function whose properties are discussed in this note, is a special form of Whittaker's Confluent Hypergeometric Function, Wkm(z). It is the general solution of the Differential Equationand can be obtained in the form of a series, terminating only when k is half a positive odd integer, viz.,

Introducing independent parameters and to represent the strength of the attractive and repulsive components, respectively, we write the Kratzer molecular potential as This parameterisation is not only natural, but also convenient for the construction of ladder operators for the system. Adopting the straightforward method of deriving recurrence relations among confluent hypergeometric functions, we construct seven pairs of ladder operators for the Kratzer potential system. Detailed analysis of the laddering actions of these operators is given to show that they connect eigenstates of equal energy but belong to a hierarchy of Kratzer potential systems corresponding to different values of the parameters and Significantly, it is pointed out that it may not be possible to construct, in the position representation, a ladder operator which would connect different eigenstates belonging to the same potential . Transition to the hydrogen atom case is discussed. A number (14 altogether) of functional relations among the confluent hypergeometric functions have been derived and reported separately in an appendix.

Extended Hurwitz results for hypergeometric functions arising in spectral theory

A class of identities relating Whittaker and Bessel functions

The object of this paper is to evaluate an infinite integral involving the product of Meijer's G-function (5) and Kampé de Fériet function (1) in terms of Kampé de Fériet function. A number of papers of Bailey (3,4), Ragab (7,8), Slater (9), and Srivastava (10) have appeared, evaluating an integral in terms of a hypergeometric function of two variables or in terms of an E-function. Their results are obviously the particular cases of my result. Since Meijer's G-function is the most general function of one variable which can be expressed in terms of special functions (5) and Kampé de Fériet's function being the most general hypergeometric function of two variables, the integral given by me is the most general integral ever obtained and generalizes most of the results obtained so far for the integral of Mellin type in terms of generalized hypergeometric series. This is because the Kampé de Fériet function reduces to the product of two generalized hypergeometric functions by choosing parameters suitably.

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.

An exact analytical solution for the universal probability distribution of the order parameter fluctuations as well as for the universal statistical and thermodynamic functions of an ideal gas in the whole critical region of Bose–Einstein condensation (BEC) is obtained. A universal constraint nonlinearity is found that is responsible for all nontrivial critical phenomena of the BEC phase transition. Simple analytical approximations, which describe the universal structure of the critical region in terms of confluent hypergeometric or parabolic cylinder functions, as well as asymptotics of the exact solution are derived. The results for the order parameter, all higher order moments of BEC fluctuations and thermodynamic quantities, including specific heat, perfectly match the known asymptotics outside critical region as well as the phenomenological renormalization-group ansatz with known critical exponents in the close vicinity of the critical point. Thus, a full analytical solution to a long-standing problem of finding a universal structure of the λ point for BEC in an ideal gas is found.

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.