Under a certain condition, we find the explicit formulas for the trace functions of certain intertwining operators among gl(n)-modules, introduced by Etingof in connection with the solutions of the Calogero-Sutherland model. If n = 2, the master function of the trace function is exactly the classical Gauss hypergeometric function. When n > 2, the master functions of the trace functions are a ne...

In this paper we provide an extensive classification of one and two dimensional diffusion processes which admit an exact solution to the Kolmogorov (and hence Black-Scholes) equation (in terms of hypergeometric functions). By identifying the one-dimensional solvable processes with the class of integrable superpotentials introduced recently in supersymmetric quantum mechanics, we obtain new anal...

The Abramov-Petkovšek reduction computes an additive decomposition of a hypergeometric term, which extends the functionality of the Gosper algorithm for indefinite hypergeometric summation. We modify the AbramovPetkovšek reduction so as to decompose a hypergeometric term as the sum of a summable term and a non-summable one. The outputs of the Abramov-Petkovšek reduction and our modified version...

In this course we will study multivariate hypergeometric functions in the sense of Gel’fand, Kapranov, and Zelevinsky (GKZ systems). These functions generalize the classical hypergeometric functions of Gauss, Horn, Appell, and Lauricella. We will emphasize the algebraic methods of Saito, Sturmfels, and Takayama to construct hypergeometric series and the connection with deformation techniques in...

The reductions of the Heun equation to the hypergeometric equation by rational transformations of its independent variable are enumerated and classified. Heun-tohypergeometric reductions are similar to classical hypergeometric identities, but the conditions for the existence of a reduction involve features of the Heun equation that the hypergeometric equation does not possess; namely, its cross...

A Gaming Application of the Negative Hypergeometric Distribution by Steven Jones Dr. Rohan Dalpatadu, Advisory Committee Chair Associate Professor of Mathematical Sciences University of Nevada, Las Vegas The Negative Hypergeometric distribution represents waiting times when drawing from a finite sample without replacement. It is analogous to the negative binomial, which models the distribution ...

Starting from the second-order diierence hypergeometric equation satissed by the set of discrete orthogonal polynomials fp n g, we nd the analytical expressions of the expansion coeecients of any polynomial r m (x) and of the product r m (x)q j (x) in series of the set fp n g. These coeecients are given in terms of the polynomial coeecients of the second-order diierence equations satissed by th...

Carlson and Shaffer [B.C. Carlson, D.B. Shaffer, Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal. 15 (1984) 737–745] have introduced a linear operator associated with the Gaussian hypergeometric function which has been generalized by Dziok and Srivastava [J. Dziok, H.M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function, Appl....

We consider the applicability (or terminating condition) of the well-known Zeilberger’s algorithm and give the complete solution to this problem for the case where the original hypergeometric term F(n; k) is a rational function. We specify a class of identities ∑n k=0 F(n; k) = 0; F(n; k)∈C(n; k), that cannot be proven by Zeilberger’s algorithm. Additionally, we give examples showing that the s...

q-series, which is also called basic hypergeometric series, plays a very important role in many fields, such as affine root systems, Lie algebras and groups, number theory, orthogonal polynomials, physics, and so on. Inequality technique is one of the useful tools in the study of special functions. There are many papers about it 1–6 . In 1 , the authors gave some inequalities for hypergeometric...