On hypergeometric series reductions from integral representations, the Kampe de Feriet function, and elsewhere

Single variable hypergeometric functions pFq arise in connection with the power series solution of the Schrodinger equation or in the summation of perturbation expansions in quantum mechanics. For these applications, it is of interest to obtain analytic expressions, and we present the reduction of a number of cases of pFp and p+1F_p, mainly for p=2 and p=3. These and related series have additional applications in quantum and statistical physics and chemistry.Comment: 17 pages, no figure

On computing some special values of hypergeometric functions

The theoretical computing of special values assumed by the hypergeometric functions has a high interest not only on its own, but also in sight of the remarkable implications to both pure Mathematics and Mathematical Physics. Accordingly, in this paper we continue the path of research started in two our previous papers appeared on [30] and [31] providing some contribution to such functions computability inside and outside their disk of convergence. This is accomplished through two different approaches. The first is to provide some new results in the spirit of theorem 3.1 of 31] obtaining formulae for multivariable hypergeometric functions by generalizing a well known Kummer's identity to the hypergeometric functions of two or more variable like those of Appell and Lauricella.Comment: 21 pages. Sixth version. To appear in Journal of Mathematical Analysis and Application

Geometry of q-Hypergeometric Functions as a Bridge between Yangians and Quantum Affine Algebras

The rational quantized Knizhnik-Zamolodchikov equation (qKZ equation) associated with the Lie algebra $sl_2$ is a system of linear difference equations with values in a tensor product of $sl_2$ Verma modules. We solve the equation in terms of multidimensional $q$-hypergeometric functions and define a natural isomorphism between the space of solutions and the tensor product of the corresponding quantum group $U_q(sl_2)$ Verma modules, where the parameter $q$ is related to the step $p$ of the qKZ equation via $q=e^{pi i/p}$. We construct asymptotic solutions associated with suitable asymptotic zones and compute the transition functions between the asymptotic solutions in terms of the trigonometric $R$-matrices. This description of the transition functions gives a new connection between representation theories of Yangians and quantum loop algebras and is analogous to the Kohno-Drinfeld theorem on the monodromy group of the differential Knizhnik-Zamolodchikov equation. In order to establish these results we construct a discrete Gauss-Manin connection, in particular, a suitable discrete local system, discrete homology and cohomology groups with coefficients in this local system, and identify an associated difference equation with the qKZ equation.Comment: 66 pages, amstex.tex (ver. 2.1) and amssym.tex are required; misprints are correcte

Generalized Heine Identity for Complex Fourier Series of Binomials

In this paper we generalize an identity first given by Heinrich Eduard Heine in his treatise, {\it Handbuch der Kugelfunctionen, Theorie und Anwendungen (1881), which gives a Fourier series for $1/[z-\cos\psi]^{1/2}$, for $z,\psi\in\R$, and $z>1$, in terms of associated Legendre functions of the second kind with odd-half-integer degree and vanishing order. In this paper we give a generalization of this identity as a Fourier series of $1/[z-\cos\psi]^\mu$, where z,\mu\in\C, $|z|>1$, and the coefficients of the expansion are given in terms of the same functions with order given by $\frac12-\mu$. We are also able to compute certain closed-form expressions for associated Legendre functions of the second kind.Comment: 12 page