Table of Contents

Abstract: Recently, an extended operator of fractional derivative related to a generalized Beta function was used in order to obtain some generating relations involving the extended hypergeometric functions [1]. The main object of this paper is to present a further generalization of the extended fractional derivative operator and apply the generalized extended fractional derivative operator to derive linear and bilinear generating relations for the generalized extended Gauss, Appell and Lauricella hypergeometric functions in one, two and more variables. Some other properties and relationships involving the Mellin transforms and the generalized extended fractional derivative operator are also given.

Abstract: Motivated by the orthogonality relations for irreducible characters of a finite group, we evaluate the sum of a finite group of linear characters of a Hopf algebra, at all grouplike and skew-primitive elements. We then discuss results for products of skew-primitive elements. Examples include groups, (quantum groups over) Lie algebras, the small quantum groups of Lusztig, and their variations (by Andruskiewitsch and Schneider).

Abstract: We present an application of the program of groupoidification leading up to a sketch of a categorification of the Hecke algebroid—the category of permutation representations of a finite group. As an immediate consequence, we obtain a categorification of the Hecke algebra. We suggest an explicit connection to new higher isomorphisms arising from incidence geometries, which are solutions of the Zamolodchikov tetrahedron equation. This paper is expository in style and is meant as a companion to Higher Dimensional Algebra VII: Groupoidification and an exploration of structures arising in the work in progress, Higher Dimensional Algebra VIII: The Hecke Bicategory, which introduces the Hecke bicategory in detail.

Abstract: We introduce the Frobenius–Schur indicator for categories with duality to give a category-theoretical understanding of various generalizations of the Frobenius–Schur theorem including that for semisimple quasi-Hopf algebras, weak Hopf C*-algebras and association schemes. Our framework also clarifies a mechanism of how the “twisted” theory arises from the ordinary case. As a demonstration, we establish twisted versions of the Frobenius–Schur theorem for various algebraic objects. We also give several applications to the quantum SL₂.

Abstract: By applying a classical method, already employed by Cauchy, to a terminating curious summation by one of the authors, a new curious bilateral q-series identity is derived. We also apply the same method to a quadratic summation by Gessel and Stanton, and to a cubic summation by Gasper, respectively, to derive a bilateral quadratic and a bilateral cubic summation formula.

Abstract: It is well known that the number of 5-core partitions of 5^kn + 5^k − 1 is a multiple of 5^k. In [1] a statistic called a crank was developed to sort the 5-core partitions of 5n + 4 and 25n + 24 into 5 and 25 classes of equal size, respectively. In this paper we will develop the cranks that can be used to sort the 5-core partitions of 5^kn + 5^k − 1 into 5^k classes of equal size.

Abstract: This research work presents original properties of the equilibrium critical (ideal) points sets for an important class of generalized dynamical systems. The existence and significant results regarding such points are specified. Strong connections with the Vector Optimization by the Efficiency and the Potential Theory together with its applications following Choquet’s boundaries are provided.

Abstract: The aim of this paper is to construct generating functions, related to nonnegative real parameters, for q-Eulerian type polynomials and numbers (or q-Apostol type Frobenius–Euler polynomials and numbers). We derive some identities for these polynomials and numbers based on the generating functions and functional equations. We also give multiplication formula for the generalized Apostol type Frobenius–Euler polynomials.