The Abel Lemma and the -Gosper Algorithm

Chu has recently shown that the Abel lemma on summation by parts reveals the telescoping nature of Bailey's bilateral summation formula. We present a systematic approach to compute Abel pairs for bilateral and unilateral basic hypergeometric summation formulas by using the -Gosper algorithm. It is demonstrated that Abel pairs can be derived from Gosper pairs. This approach applies to many classical summation formulas.

     

Extensions of the well-poised and elliptic well-poised Bailey lemma

We establish a number of extensions of the well-poised Bailey lemma and elliptic well-poised Bailey lemma. As application we prove some new transformation formulae for basic and elliptic hyper-geometric series, and embed some recent identities of Andrews, Berkovich and Spiridonov in a well-poised Bailey tree.     

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.     

Abel's lemma on summation by parts and basic hypergeometric series

Basic hypergeometric series identities are revisited systematically by means of Abel's lemma on summation by parts. Several new formulae and transformations are also established. The author is convinced that Abel's lemma on summation by parts is a natural choice in dealing with basic hypergeometric series.     

On some generalizations of Ramanujan’s continued fraction identities

In this note we establish continued fraction developments for the ratios of the basic hypergeometric function₂ϕ₁(a,b;c;x) with several of its contiguous functions. We thus generalize and give a unified approach to establishing several continued fraction identities including those of Srinivasa Ramanujan.     

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.     

q-Difference equation and the Cauchy operator identities

In this paper, we verify the Cauchy operator identities by a new method. And by using the Cauchy operator identities, we obtain a generating function for Rogers-Szegö polynomials. Applying the technique of parameter augmentation to two multiple generalizations of q-Chu-Vandermonde summation theorem given by Milne, we also obtain two multiple generalizations of the Kalnins-Miller transformation.     

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.     

A q-Analogue of Weber-Schafheitlin Integral of Bessel Functions

In an attempt to find a q-analogue of Weber and Schafheitlin's integral ₀ x ^– J (ax) J (bx) dx which is discontinuous on the diagonal a = b the integral ₀ x ^– J ⁽²⁾ (a(1 – q)x; q)J ⁽¹⁾ (b(1 – q)x; q) dx is evaluated where J ⁽¹⁾ (x; q) and J ⁽²⁾ (x; q) are two of Jackson's three q-Bessel functions. It is found that the question of discontinuity becomes irrelevant in this case. Evaluations of this integral are also made in some interesting special cases. A biorthogonality formula is found as well as a Neumann series expansion for x in terms of J ⁽²⁾ _+1+2n ((1 – q)x; q). Finally, a q-Lommel function is introduced.     

Identities of the Rogers-Ramanujan-Bailey type

A multiparameter generalization of the Bailey pair is defined in such a way as to include as special cases all Bailey pairs considered by W.N. Bailey in his paper [Identities of the Rogers-Ramanujan type, Proc. London Math. Soc. (2) 50 (1949) 421-435]. This leads to the derivation of a number of elegant new Rogers-Ramanujan type identities.     

Some curious q-series expansions and beta integral evaluations

We deduce several curious q-series expansions by applying inverse relations to certain identities for basic hypergeometric series. After rewriting some of these expansions in terms of q-integrals, we obtain, in the limit q→ 1, some curious beta-type integral evaluations which appear to be new. Dedicated to Dick Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—15A09, 33D15, 33E20; Secondary—05A30 M. Schlosser was fully supported by an APART fellowship of the Austrian Academy of Sciences.     

Method of separation of the variables for basic analogs of equations of mathematical physics

We apply method of separation of the variables to q-analogs of several equations of mathematical physics. Dedicated to Dick Askey on his 70th birthday. 2000 Mathematics Subject Classification Primary—33D45, 42C10; Secondary—33D15.     

Some Orthogonal Very-Well-Poised 8ϕ7-Functions that Generalize Askey–Wilson Polynomials

In a recent paper Ismail et al. (Algebraic Methods and q-Special Functions (J.F. van Diejen and L. Vinet, eds.) CRM Proceding and Lecture Notes, Vol. 22, American Mathematical Society, 1999, pp. 183–200) have established a continuous orthogonality relation and some other properties of a ₂₁-Bessel function on a q-quadratic grid. Dick Askey (private communication) suggested that the Bessel-type orthogonality found in Ismail et al. (1999) at the ₂₁-level has really a general character and can be extended up to the ₈₇-level. Very-well-poised ₈₇-functions are known as a nonterminating version of the classical Askey–Wilson polynomials (SIAM J. Math. Anal. 10 (1979), 1008–1016; Memoirs Amer. Math. Soc. Number 319 (1985)). Askey's conjecture has been proved by the author in J. Phys. A: Math. Gen. 30 (1997), 5877–5885. In the present paper which is an extended version of Suslov (1997) we discuss in detail properties of the orthogonal ₈₇-functions. Another type of the orthogonality relation for a very-well-poised ₈₇-function was recently found by Askey et al. J. Comp. Appl. Math. 68 (1996), 25–55.     

The nearest neighbor random walk on subspaces of a vector space and rate of convergence

LetX be the collection ofk-dimensional subspaces of ann-dimensional vector spaceV _n overGF(q). A metric may be defined onX by letting

Two new families of <Emphasis Type="Italic">q</Emphasis>-positive integers

Let n,p and k be three non negative integers. We prove that the apparently rational fractions of q:

The Bailey Lemma and Kostka Polynomials

Using the theory of Kostka polynomials, we prove an A _n–1 version of Bailey's lemma at integral level. Exploiting a new, conjectural expansion for Kostka numbers, this is then generalized to fractional levels, leading to a new expression for admissible characters of A⁽¹⁾ _n–1 and to identities for A-type branching functions.     

Some Expansions in Basic Fourier Series and Related Topics

We consider explicit expansions of some elementary and q-functions in basic Fourier series introduced recently by Bustoz and Suslov. Natural q-extensions of the Bernoulli and Euler polynomials, numbers, and the Riemann zeta function are discussed as a by-product.     

Carlitz and the general 3φ2

In a letter dated March 3, 1971, L. Carlitz defined a sequence of polynomials, Φ _n (a,b; x, y; z), generalizing the Al-Salam & Carlitz polynomials, but closely related thereto. He concluded the letter by stating: “It would be of interest to find properties of Φ _n (a, b; x, y; z) when all the parameters are free.” In this paper, we reproduce the Carlitz letter and show how a study of Carlitz’s polynomials leads to a clearer understanding of the general ₃Φ₂ (a, b, c; d; e; q, z). Dedicated to my friend, Richard Askey. 2000 Mathematics Subject Classification Primary—33D20. G. E. Andrews: Partially supported by National Science Foundation Grant DMS 0200047.     

Some systems of multivariable orthogonal q-Racah polynomials

In 1991 Tratnik derived two systems of multivariable orthogonal Racah polynomials and considered their limit cases. q-Extensions of these systems are derived, yielding systems of multivariable orthogonal q-Racah polynomials, from which systems of multivariable orthogonal q-Hahn, dual q-Hahn, q-Krawtchouk, q-Meixner, and q-Charlier polynomials follow as special or limit cases. Dedicated to Richard Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—33D50; Secondary—33C50 Supported in part by NSERC grant #A6197.