New series expansions of the Gauss hypergeometric function

Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Pöschl–Teller–Ginocchio potential wave functions

This paper deals with presenting a survey-cum-expository account of some developments concerning the approximation properties of well known Phillips operators. These operators are sometimes called as genuine Szasz Durrmeyer operators, because of their property of reproducing constants as well as linear functions. We give the alternate form to present these operators in terms of Hypergeometric functions, which are related to the modified Bessel’s function of first kind of index 1. Also, we observe that the r-th moment can be represented in terms of confluent hypergeometric functions, and further it can be written in terms of generalized Laguerre polynomials. In addition, we will present some known results on such operators, which include simultaneous approximation, linear and iterative combinations, global direct and inverse results, rate of convergence for functions of bounded variation, and q-analogues of these operators.

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).

For primes p ≡ 1 (mod 12), we present an explicit relation between the traces of Frobenius on a family of elliptic curves with j-invariant 1728 / t and values of a particular 2 F 1 -hypergeometric function over F p . We also give a formula for traces of Hecke operators on spaces of cusp forms of weight k and level 1 in terms of the same traces of Frobenius. This leads to formulas for Ramanujan's T -function in terms of hypergeometric functions.

In the spirit of Hasanov, Srivastava, and Turaev (2006), we introduce new inverse operators together with a more general operator and find a summation formula for the last one. Based on these operators and the earlier known q-analogues of the Burchnall-Chaundy operators, we find 15 symbolic operator formulas. Then, 10 expansions for the q-analogues of Srivastava’s three triple hypergeometric functions in terms of ϕ34q-hypergeometric and q-Kampé de Fériet functions are derived. These expansions readily reduce to 10 new expansions for the three triple Srivastava hypergeometric functions in terms of F34 hypergeometric and Kampé de Fériet functions.

Expansions in series of Coulomb and hypergeometric functions for the solutions of the generalized spheroidal wave equations (GSWEs) are analysed and written together in pairs. Each pair consists of a solution in series of hypergeometric functions and another in series of Coulomb wavefunctions and has the same recurrence relations for the series coefficients, but the solutions inside it present different radii of convergence. Expansions without a phase parameter are derived by truncating the series with a phase parameter. For the Whittaker-Hill equation, solutions are found by treating that equation as a particular case of GSWE while, for the confluent GSWE, solutions, with and without a phase parameter, are given as pairs of series of Coulomb wavefunctions. Amongst the applications there are equations for the time dependence of Dirac test fields in some nonflat Friedmann-Robertson-Walker spacetimes, the radial Schrödinger equation for an electron in the field of two Coulombian centres and the Schrödinger equation for the Razavy-type quasi-exactly solvable potentials. For these problems it is possible to find wavefunctions in terms of infinite series, regular and convergent over the entire range of the independent variable, by matching expansions belonging to one or more of the above pairs. The infinite-series solutions for the Razavy-type potentials, in addition to the polynomial ones, suggest that the whole energy spectra may be determined without appealing to perturbation theory or semi-classical methods of approximation.

New solutions for the elliptic Darboux equation are obtained as particular cases of solutions constructed for Heun's general equation. We consider two groups of power series expansions and two new groups of expansions in series of Gauss hypergeometric functions. The convergence of one group in power series is determined by means of ratio tests for infinite series, while the other groups are designed to solve problems which admit finite‐series solutions. Actually, we envisage periodic quasi‐exactly solvable potentials for which the stationary one‐dimensional Schrödinger equation is reduced to the Darboux equation. In general, finite‐ and infinite‐series solutions are obtained from power series expansions for Heun's equation. However, we show that the Schrödinger equation admits additional finite‐series expansions in terms of hypergeometric functions for a family of associated Lamé potentials used in band theory of solids.For each finite‐series solution, we find as well four infinite‐series expansions which are bounded and convergent for all values of the independent variable.

Abstract. In this paper, the expansion of a basic confluent hypergeometric function in terms of an incomplete q-gamma function is investigated and shown to be an asymptotic expansion for a basic confluent hypergeometric function and a q-analogue of the Hadamard expansion. The general expansion associated with a basic hypergeometric function is introduced. In special cases, this provides alternative derivations of the q-Hadamard expansion. Expansions and asymptotics for modified q-Bessel functions are derived as special cases from the general expansion.

: 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.

Inspired by recent developments in stereology, rotational versions of the Crofton formula are derived. The first version involves rotation averages of Minkowski functionals. It is shown that for the special case where the Minkowski functional is surface area, the rotation average can be expressed in terms of hypergeometric functions. The second rotational version of the Crofton formula solves the ‘opposite’ problem of finding functions with rotation averages equal to the Minkowski functionals. For the case of surface area, hypergeometric functions appear again. The second type of rotational Crofton formula has applications in local stereology. As a by-product, a formula involving mixed volumes is found.

We define a function in terms of quotients of the p-adic gamma function which generalizes earlier work of the author on extending hypergeometric functions over finite fields to the p-adic setting. We prove, for primes p >3, that the trace of Frobenius of any elliptic curve over Fp, whose j -invariant does not equal 0 or 1728, is just a special value of this function. This generalizes results of Fuselier and Lennon which evaluate the trace of Frobenius in terms of hypergeometric functions over Fp when p 1 .mod 12/. 1. Introduction and statement of results Let Fp denote the finite field with p, a prime, elements. Consider E=Q an elliptic curve with an integral model of discriminantA.E/. We denote Ep the reduction of E modulo p. We note that Ep is nonsingular, and hence an elliptic curve over Fp, if and only if p› A.E/, in which case we say p is a prime of good reduction. Regardless, we define

Letx 0 (t),x 0 ɛℝ4 be a homothetic solution of the planar three-body problem with total energyh, described in relative coordinates with respect to one body. It is shown that the variational equation of the problem atx 0 (t) can be solved explicitly in terms of hypergeometric functions. This is done by using the scaled true anomaly of the one-dimensional Kepler motion as the independent variable. The classical theorems about hypergeometric functions allow a simple calculation of all the values needed in applications. By means of this theory the past of a homothetic triple close encounter may be described in a linearized approximation.

Let $X_0^6(1)/W_6$ be the Atkin-Lehner quotient of the Shimura curve $X_0^6(1)$ associated to a maximal order in an indefinite quaternion algebra of discriminant $6$ over $\mathbb Q$. By realizing modular forms on $X_0^6(1)/W_6$ in two ways, one in terms of hypergeometric functions and the other in terms of Borcherds forms, and using Schofer's formula for values of Borcherds forms at CM-points, we obtain special values of certain hypergeometric functions in terms of periods of elliptic curves over $\overline\mathbb Q$ with complex multiplication.

ABSTRACTGeneralized Abel type integral equations with Gauss, Kummer's and Humbert's confluent hypergeometric functions in the kernel and generalized Abel type integral equations with localized fractional integrals are considered. The left-hand sides of these equations are inversed by using generalized fractional derivatives. Explicit solutions of the equations in the class of locally summable functions are obtained. They are represented in terms of hypergeometric functions. Asymptotic power exponential type expansions of the generalized and localized fractional integrals are obtained. The base solutions of the generalized Abel type integral equation are given in the form of asymptotic series.

The radius of convergence of dynamical mean-field theory