On the Sears–Slater basic hypergeometric transformations

We introduce the concept of a Jackson integral of type BC ₁ which is a q-series permitting Weyl group symmetry. Using this, we give a simple proof of transformation formulas for a very-well-poised-balanced _2r ψ _2r hypergeometric series discovered by Sears and Slater. This work was supported in part by Grant-in-Aid for Scientific Research (C) No. 17540034 from the Ministry of Education, Culture, Sports, Science and Technology, Japan.     

A bibasic hypergeometric transformation associated with combinatorial identities of the Rogers-Ramanujan type

During the last five decades, a number of combinatorial generalizations and interpretations have occurred for the identities of the Rogers-Ramanujan type. The object of this paper is to give a most general known analytic auxiliary functional generalization which can be used to give combinatorial interpretations of generalizedq-identities of the Rogers-Ramanujan type. The derivation realise the theory of basic hypergeometric series with two unconnected bases.     

Multivariate basic hypergeometric series associated to root systems of type Am

In previous paper [Ann. Comb. 4 (2000) 347-373], we established and proved two new q-beta integrals and two multivariate basic hypergeometric series associated with the root system of A_m. In this paper we give a detailed proof of the multivariate basic hypergeometric series obtained in [Ann. Comb. 4 (2000) 347-373].     

Reduction and transformation formulae for bivariate basic hypergeometric series

The main object of this paper is to establish several bivariate basic hypergeometric series identities by means of elementary series manipulation. Some of them can be applied to yield transformation and reduction formulae for q-Kampé de Fériet functions.     

The Cauchy operator for basic hypergeometric series

We introduce the Cauchy augmentation operator for basic hypergeometric series. Heine's transformation formula and Sears' transformation formula can be easily obtained by the symmetric property of some parameters in operator identities. The Cauchy operator involves two parameters, and it can be considered as a generalization of the operator T(bD_q). Using this operator, we obtain extensions of the Askey–Wilson integral, the Askey–Roy integral, Sears' two-term summation formula, as well as the q-analogs of Barnes' lemmas. Finally, we find that the Cauchy operator is also suitable for the study of the bivariate Rogers–Szegö polynomials, or the continuous big q-Hermite polynomials.     

Classes of series identities and associated hypergeometric reduction formulas

The main object of the present paper is to investigate some classes of series identities and their applications and consequences leading naturally to several (known or new) hypergeometric reduction formulas. We also indicate how some of these series identities and reduction formulas would yield several series identities which emerged recently in the context of fractional calculus (that is, calculus of integrals and derivatives of any arbitrary real or complex order).     

Twisted Bimodules and Hochschild Cohomology for Self-injective Algebras of Class A n , II

Up to derived equivalence, the representation-finite self-injective algebras of class A _n are divided into the wreath-like algebras (containing all Brauer tree algebras) and the Möbius algebras. In Part I (Forum Math. 11 (1999), 177–201), the ring structure of Hochschild cohomology of wreath-like algebras was determined, the key observation being that kernels in a minimal bimodule resolution of the algebras are twisted bimodules. In this paper we prove that also for Möbius algebras certain kernels in a minimal bimodule resolution carry the structure of a twisted bimodule. As an application we obtain detailed information on subrings of the Hochschild cohomology rings of Möbius algebras.     

An Simple Elementary Proof for the Inequality dn〈3^n

In this paper, by using a variation of the Chebyshev＇s method, we present a very simple, elementary proof of an inequality which has applications in number theory.     

Continuity theorems for the M/M/1/n queueing system

In this paper continuity theorems are established for the number of losses during a busy period of the M/M/1/n queue. We consider an M/GI/1/n queueing system where the service time probability distribution, slightly different in a certain sense from the exponential distribution, is approximated by that exponential distribution. Continuity theorems are obtained in the form of one or two-sided stochastic inequalities. The paper shows how the bounds of these inequalities are changed if further assumptions, associated with specific properties of the service time distribution (precisely described in the paper), are made. Specifically, some parametric families of service time distributions are discussed, and the paper establishes uniform estimates (given for all possible values of the parameter) and local estimates (where the parameter is fixed and takes only the given value). The analysis of the paper is based on the level crossing approach and some characterization properties of the exponential distribution. Dedicated to Vladimir Mikhailovich Zolotarev, Victor Makarovich Kruglov, and to the memory of Vladimir Vyacheslavovich Kalashnikov.     

Discrete Fourier–Neumann series

Let J_μ denote the Bessel function of order μ. The systemwith n=0,1,…,α>−1, and where p_s denotes the sth positive zero of J_α(ax), is orthonormal in . In this paper, we study the mean convergence of the Fourier series with respect to this system. Also, we describe the space in which the span of the system is dense.     

Symmetry Classification for Jackson Integrals Associated with the Root System BC n

The Jackson integrals associated with the non-reduced root system are defined as multiple sums which are generalization of the Bailey's very-well-poised ₆₆ sum. They are classified by the number of their parameters when they can be expressed as a product of the Jacobi elliptic theta functions. The sums which appear in the classification list coincide with those investigated individually by Gustafson and van Diejen.     

Some limit relations in the A1 tableau of Dunkl-Cherednik operators

In this paper we study some limit relations involving some q-special functions related with the A₁ (root system) tableau of Dunkl-Cherednik operators. Concretely we consider the limits involving the nonsymmetric q-ultraspherical polynomials (q-Rogers polynomials), ultraspherical polynomials (Gegenbauer polynomials), q-Hermite and Hermite polynomials.     

Construction of the extremal function for a functional on the classH ω (n)

For a functional on the classH _ω ⁽ⁿ⁾ ,n≥3, we construct the extremal function on which the upper bound obtained by A. I. Stepanets is attained. Translated fromMatematicheskie Zametki, Vol. 61, No. 4, pp. 519–529, April, 1997. Translated by N. K. Kulman     

High energy positive solutions of Neumann problem for an elliptic system of equations with critical nonlinearities

In this paper, we consider the Neumann problem for an elliptic system of two equations involving the critical Sobolev exponents. By means of variational method, we establish the existence of positive solutions with high energy. Supported by Special Funds For Major States Basic Research Projects of China(G1999075107) and Knowledge Innovation Funds of CAS in China.     

On Solutions of the q-Hypergeometric Equation with q N = 1

We consider the q-hypergeometric equation with q ^N = 1 and , , . We solve this equation on the space of functions given by a power series multiplied by a power of the logarithmic function. We prove that the subspace of solutions is two-dimensional over the field of quasi-constants. We get a basis for this space explicitly. In terms of this basis, we represent the q-hypergeometric function of the Barnes type constructed by Nishizawa and Ueno. Then we see that this function has logarithmic singularity at the origin. This is a difference between the q-hypergeometric functions with 0 < |q| < 1 and at |q| = 1.     

On the γ n -complete hypergroups and K H hypergroups

The class of γn-complete hypergroups is introduced. Several properties and examples are found both of γn-complete hypergroups and of KH hypergroups.     

Asymptotic Results and a Markovian Approximation for the M(n)/M(n)/s+GI System

In this paper for the M(n)/M(n)/s+GI system, i.e. for a s-server queueing system where the calls in the queue may leave the system due to impatience, we present new asymptotic results for the intensities of calls leaving the system due to impatience and a Markovian system approximation where these results are applied. Furthermore, we present a new proof for the formulae of the conditional density of the virtual waiting time distributions, recently given by Movaghar for the less general M(n)/M/s+GI system. Also we obtain new explicit expressions for refined virtual waiting time characteristics as a byproduct.     

Schur Q-polynomials, multiple hypergeometric series and enumeration of marked shifted tableaux

We study Schur Q-polynomials evaluated on a geometric progression, or equivalently q-enumeration of marked shifted tableaux, seeking explicit formulas that remain regular at q=1. We obtain several such expressions as multiple basic hypergeometric series, and as determinants and pfaffians of continuous q-ultraspherical or continuous q-Jacobi polynomials. As special cases, we obtain simple closed formulas for staircase-type partitions.