A generalised integration by parts formula for sequences of absolutely continuous functions that satisfy the w−Appell condition and different estimates for the remainder are provided. Applications for particular instances of such sequences are pointed out as well.

In this paper, using similar symbolical method of Burchnall and Chaundy formulas of expansion for the generalized hypergeometric function were constructed. By means of the found formulas of expansion the formulas of an analytic continuation for hypergeometric function of Clausen is defined. The obtained formulas of an analytic continuation express known hypergeometric Appell function F2 (a; b1,...

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...

We describe an involution on a set of sequences associated with lattice paths with north or east steps constrained to lie between two arbitrary boundaries. This involution yields recursions (from which determinantal formulas can be derived) for the number and area enumerator of such paths. An analogous involution can be defined for parking functions with arbitrary lower and upper bounds. From t...

Within the algebraic framework of Hopf algebras, random walks and associated diffusion equations (master equations) are constructed and studied for two basic operator algebras of Quantum Mechanics i.e the Heisenberg-Weyl algebra (hw) and its q-deformed version hwq. This is done by means of functionals determined by the associated coherent state density operators. The ensuing master equations ad...

Abstract Some examples of Appell–Dunkl sequences are shown using determined operators. Specifically, whose generating functions the form $$E_{\alpha }(xt)/(1\pm t^m)$$ E α ( x t ) /</...

The HYPERDIRE is a project devoted to creation of a set of Mathematica based programs for differential reduction of hypergeometric functions. The current version includes two parts: one, pfq, is relevant to manipulation with hypergeometric functions p+1Fp, and second, AppellF1F4, to manipulations with Appell’s hypergeometric functions of two variables F1, F2, F3, F4. As application, a few multi...