Given the hypergeometric function F (a, b; c; z) = ∞ P n=0 (a)n(b)n (c)n(1)n zn, we place conditions on a, b and c to guarante that zpF (a, b; c; z) will be in various subclasses of p-valent starlike and p-valent convex functions. Operators related to the hypergeometric function are also examined.

We present the τ -functions for the hypergeometric solutions to the q-Painlevé system of type E (1) 7 in a determinant formula whose entries are given by the basic hypergeometric function 8W7. By using the W (D5) symmetry of the function 8W7, we construct a set of twelve solutions and describe the action of W̃ (D (1) 6 ) on the set.

The object of the present paper is to investigate several general families of bilinear and bilateral generating functions with different argument for the Gauss’ hypergeometric polynomials. Mathematics Subject Classification(2010): Primary 42C05, Secondary 33C45. Keywords—Appell’s functions, Gauss hypergeometric functions, Heat polynomials, Kampe’ de Fe’riet function, Laguerre polynomials, Lauri...

The mathematical properties of the generalized hypergeometric function are now well established, due largely to the efforts of Bailey, Watson, Slater, and many others. There are available a number of books on the subject ['], [6] which contain extensive bibliographies together with an outline of the history of the investigations. A large portion of the published work concerning these functions ...

The main objective of this work is to develop a pair of Lommel matrix functions suggested by the hypergeometric matrix functions and some of their properties are studied. Some properties of the hypergeometric and Bessel matrix functions are obtained.

Using Krattenthaler’s operator method, we give a new proof of Warnaar’s recent elliptic extension of Krattenthaler’s matrix inversion. Further, using a theta function identity closely related to Warnaar’s inversion, we derive summation and transformation formulas for elliptic hypergeometric series of Karlsson–Minton-type. A special case yields a particular summation that was used by Warnaar to ...

A recurrence relation between equal mass two-loop sunrise diagrams differing in dimensionality by 2 is derived and it's solution in terms of Gauss' 2 F 1 and Appell's F 2 hypergeometric functions is presented. For arbitrary space-time dimension d the imaginary part of the diagram on the cut is found to be the 2 F 1 hypergeometric function with argument proportional to the maximum of the Kibble ...

The applicability or terminating condition for the ordinary case of Zeilberger’s algorithm was recently obtained by Abramov. For the qanalogue, the question of whether a bivariate q-hypergeometric term has a qZ-pair remains open. Le has found a solution to this problem when the given bivariate q-hypergeometric term is a rational function in certain powers of q. We solve the problem for the gene...

The 2f1-f2 distortion product otoacoustic emission (DPOAE) is thought to arise primarily from the complex interaction of components that come from two different cochlear locations. Such distortion has its origin in the nonlinear interaction on the basilar membrane of the excitation patterns resulting from the two stimulus tones, f1 and f2. Here we examine the spatial extent of initial generatio...

In recent years, several interesting families of generating functions for various classes of hypergeometric and generalized hypergeometric functions in one, two and more variables were investigated systematically. Here, in this sequel, we aim at establishing several (presumably) new generating relations for the generalized Gauss type hypergeometric functions which are introduced by means of som...