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Axioms, Volume 1, Issue 3 (December 2012), Pages 238-403

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Research

Open AccessArticle A Class of Extended Fractional Derivative Operators and Associated Generating Relations Involving Hypergeometric Functions
Axioms 2012, 1(3), 238-258; doi:10.3390/axioms1030238
Received: 29 June 2012 / Revised: 8 September 2012 / Accepted: 12 September 2012 / Published: 5 October 2012
Cited by 13 | PDF Full-text (238 KB) | HTML Full-text | XML Full-text
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Axioms: Feature Papers)
Open AccessArticle The Sum of a Finite Group of Weights of a Hopf Algebra
Axioms 2012, 1(3), 259-290; doi:10.3390/axioms1030259
Received: 2 July 2012 / Revised: 17 September 2012 / Accepted: 17 September 2012 / Published: 5 October 2012
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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. [...] Read more.
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). Full article
(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations)
Open AccessCommunication The Hecke Bicategory
Axioms 2012, 1(3), 291-323; doi:10.3390/axioms1030291
Received: 19 July 2012 / Revised: 4 September 2012 / Accepted: 5 September 2012 / Published: 9 October 2012
Cited by 2 | PDF Full-text (285 KB) | HTML Full-text | XML Full-text
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations)
Open AccessArticle Frobenius–Schur Indicator for Categories with Duality
Axioms 2012, 1(3), 324-364; doi:10.3390/axioms1030324
Received: 2 July 2012 / Revised: 23 August 2012 / Accepted: 29 September 2012 / Published: 23 October 2012
PDF Full-text (443 KB)
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations)
Open AccessArticle New Curious Bilateral q-Series Identities
Axioms 2012, 1(3), 365-371; doi:10.3390/axioms1030365
Received: 17 September 2012 / Revised: 9 October 2012 / Accepted: 17 October 2012 / Published: 31 October 2012
Cited by 1 | PDF Full-text (144 KB) | HTML Full-text | XML Full-text
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 [...] Read more.
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. Full article
Open AccessArticle The Cranks for 5-Core Partitions
Axioms 2012, 1(3), 372-383; doi:10.3390/axioms1030372
Received: 7 August 2012 / Revised: 21 September 2012 / Accepted: 26 November 2012 / Published: 3 December 2012
Cited by 1 | PDF Full-text (288 KB) | HTML Full-text | XML Full-text
Abstract
It is well known that the number of 5-core partitions of 5kn + 5k − 1 is a multiple of 5k. In [1] a statistic called a crank was developed to sort the 5-core partitions of 5 [...] Read more.
It is well known that the number of 5-core partitions of 5kn + 5k − 1 is a multiple of 5k. 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 5kn + 5k − 1 into 5k classes of equal size. Full article
Open AccessArticle On the Equilibria of Generalized Dynamical Systems
Axioms 2012, 1(3), 384-394; doi:10.3390/axioms1030384
Received: 24 September 2012 / Revised: 26 November 2012 / Accepted: 26 November 2012 / Published: 6 December 2012
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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 [...] Read more.
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. Full article
Open AccessArticle Generating Functions for q-Apostol Type Frobenius–Euler Numbers and Polynomials
Axioms 2012, 1(3), 395-403; doi:10.3390/axioms1030395
Received: 30 October 2012 / Revised: 26 November 2012 / Accepted: 28 November 2012 / Published: 7 December 2012
Cited by 7 | PDF Full-text (158 KB) | HTML Full-text | XML Full-text
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 [...] Read more.
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. Full article

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