Transformations of the Hypergeometric 4F3 with One Unit Shift: A Group Theoretic Study

We study the group of transformations of 4F3 hypergeometric functions evaluated at unity with one unit shift in parameters. We reveal the general form of this family of transformations and its group property. Next, we use explicitly known transformations to generate a subgroup whose structure is then thoroughly studied. Using some known results for 3F2 transformation groups, we show that this subgroup is isomorphic to the direct product of the symmetric group of degree 5 and 5-dimensional integer lattice. We investigate the relation between two-term 4F3 transformations from our group and three-term 3F2 transformations and present a method for computing the coefficients of the contiguous relations for 3F2 functions evaluated at unity. We further furnish a class of summation formulas associated with the elements of our group. In the appendix to this paper, we give a collection of Wolfram Mathematica® routines facilitating the group calculations.

The generalized hypergeometric difference equation

A difference equation analogue of the generalized hypergeometric differential equation is defined, its contiguous relations are developed, and its relation to numerous well-known classical special functions are demonstrated.

On an extension of the generalized Pochhammer symbol and its applications to hypergeometric functions

The main object of this paper is to present a generalization of the Pochhammer symbol. We present some contiguous relations of this generalized Pochhammer symbol and use it to give an extension of the generalized hypergeometric function [Formula: see text]. Finally, we present some properties and generating functions of this extended generalized hypergeometric function.

The hypergeometric systems and the contiguity operators on the Grassmannian manifold and on the associated configuration space

The hypergeometric system [Formula: see text] and the contiguity operators defined on the space of [Formula: see text] matrices [Formula: see text] induce the systems and the contiguity operators on the Grassmannian manifold [Formula: see text] and on the configuration space [Formula: see text] where [Formula: see text] is the group consisting of diagonal matrices of size [Formula: see text]. The first purpose of this paper is to give a rigorous treatment of these systems and contiguity operators, the second is to derive a condition that the system [Formula: see text] is reducible, and the third is to derive the contiguity relations and the contiguous relations satisfied by the solutions of the system on [Formula: see text] or the solutions of the system on [Formula: see text]