Basic Hypergeometric Functions Research Articles

Hypergeometric generating functions for values of Dirichlet and other L functions.

Although there is vast literature on the values of L functions at nonpositive integers, the recent appearance of some of these values as the coefficients of specializations of knot invariants comes as a surprise. Using work of G. E. Andrews [(1981) Adv. Math. 41, 173-185; (1986) q-Series: Their Development and Application in Analysis, Combinatories, Physics, and Computer Algebra, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics 66 (Am. Math. Soc, Providence, RI); (1975) Problems and Prospects for Basic Hypergeometric Series: The Theory and Application of Special Functions (Academic, New York); and (1992) Illinois J. Math. 36, 251-274], we revisit this old subject and provide uniform and general results giving such generating functions as specializations of basic hypergeometric functions. For example, we obtain such generating functions for all nontrivial Dirichlet L functions.

Contiguous relations of hypergeometric series

The 15 Gauss contiguous relations for 2F1 hypergeometric series imply that any three 2F1 series whose corresponding parameters differ by integers are linearly related (over the field of rational functions in the parameters). We prove several properties of coefficients of these general contiguous relations, and use the results to propose effective ways to compute contiguous relations. We also discuss contiguous relations of generalized and basic hypergeometric functions, and several applications of them.

Representations for the first associated q-classical orthogonal polynomials

Let {p k(x; q)} be any system of the q-classical orthogonal polynomials, and let ϱ be the corresponding weight function, satisfying the q-difference equation D q ( σϱ)= τϱ, where σ and τ are polynomials of degree at most 2 and exactly 1, respectively. Further, let { p k (1)( x; q)} be associated polynomials of the polynomials {p k(x; q)} . Explicit forms of the coefficients b n, k and c n, k in the expansions p n−1 (1)(x;q)= ∑ k=0 n−1 b n,kϑ k(x), p n−1 (1)(x;q)= ∑ k=0 n−1 c n,kp k(x; q) are given in terms of basic hypergeometric functions. Here ϑ k ( x) equals x k if σ +(0)=0, or ( x; q) k if σ +(1)=0, where σ +( x)≔ σ( x)+( q−1) xτ( x). The most important representatives of those two classes are the families of little q-Jacobi and big q-Jacobi polynomials, respectively. Writing the second-order nonhomogeneous q-difference equation satisfied by p n−1 (1)( x; q) in a special form, recurrence relations (in k) for b n, k and c n, k are obtained in terms of σ and τ.

Q-Integral and Moment Representations for q-Orthogonal Polynomials

AbstractWe develop a method for deriving integral representations of certain orthogonal polynomials as moments. These moment representations are applied to find linear and multilinear generating functions for q-orthogonal polynomials. As a byproduct we establish new transformation formulas for combinations of basic hypergeometric functions, including a new representation of the q-exponential function εq.

The Terwilliger Algebra of a 2-Homogeneous Bipartite Distance-Regular Graph

Let Γ denote a 2-homogeneous bipartite distance-regular graph with diameter D⩾3 and valency k⩾3. Assume that Γ is not isomorphic to a Hamming cube. Fix a vertex x of Γ, and let T=T(x) denote the Terwilliger algebra of T with respect to x. We give three sets of generators for T, two of which satisfy the relations of the quantum universal enveloping algebra of the Lie algebra sl(2). We then describe the simple T-modules. We give a pair of canonical bases for each simple T-module, and we give the overlap function for these bases in terms of a basic hypergeometric function. Finally, we give two generators for the center of T.

Recursion formulae for basic hypergeometric functions

We show that the basic hypergeometric functions F k(ω) ≔ ∏ i=1 r(a i;q) k (q;q) k∏ j=1 s(b j;q) k ω k (−1) kq ( k 2 ) s+1−r r+1φ s+1 a 1q k,…,a rq k,αq k+1 b 1q k,…,b sq k,αβq 2k+2 q;ωq k(s+1−r) , satisfy a recurrence relation of the form ∑ i=0 ϑ A i(k)+ 1 ω B i(k) F k+i(ω)=0, ϑ= max(r+1, s+2), where A i(k), B i(k) are rational functions of q k , and B 0( k)= B ϑ ( k)≡0. When r= s+1 and ω= q, this result can be refined. Namely, we show that the functions F k(q) ≔ ∏ i=1 s+1(a i;q) k (q;q) k∏ j=1 s(b j;q) k q k s+2φ s+1 a 1q k,…,a s+1q k,αq k+1 b 1q k,…,b sq k,αβq 2k+2 q;q , satisfy a recurrence relation of order s+1, ∑ i=0 s+1 C i(k)F k+i(q)=0 with rational coefficients in q k .

G-Continued Fractions for Basic Hypergeometric Functions

G-Continued Fractions for Basic Hypergeometric Functions

On the q-polynomials in the exponential lattice x(s)=c 1 qs +c 3

The main goal of this paper is to continue the sutudy of the q-polynomials on non-uniform lattices by using the approach introduced by Nikiforov and Uvarov in 1983. We consider the q-polynomials on the non-uniform exponential lattice x(s)= c 1 qs +c 3 and study some of their properties (differentiation formulas, structure relations, represntation in terms of hypergeometric and basic hypergeometric functions, etc). Special emphasis is given to q-analogues of the Charlier orthogonal polynomials. For these polynomials (Charlier) we compute the main data, i.e., the coefficients of the three-term recurrence relation, structure relation, the square of the norm, etc, in the exponential lattices , respectively.

Moments and generating functions for the absortion distribution and its negative binomial analogue

The moments of the absorption are difficult to obtain. The generating functions are basic hypergeometric functions. This paper shows how to define two shift operators that allow elementary arguments to be used to develop recursions for the expected values of general functions. The exact moments of the distribution follow. The generating function for the negative binomial analogue gives the moments directly.

Casorati determinant solutions for the q-difference sixth Painlevé equation

The q-difference Painlevé VI equation is investigated on the basis of Hirota forms. It is shown that it admits solutions expressed by Casorati determinants whose entries are given by basic hypergeometric functions.

Overlaps of the biorthogonal (3) coupling coefficients and related basic hypergeometric and otherq-factorial series

The intermediate expansion technique is developed for the overlap coefficients of different biorthogonal coupled states and coupling coefficients of the quantum algebra and group SU(3) with repeating irreducible representations and the role of separate basic and classical hypergeometric functions is demonstrated. Some expansion coefficients such as new prime overlap functions (equivalent to the bilinear combinations of the different boundary isofactors) are expressed in terms of the balanced (Saalschutzian) basic or classical hypergeometric series. These new overlap functions, their analytical inversion, some known triangular expansion matrices, transition matrices related to definite Racah coefficients, the well-poised or series and compositions of q-factorial series resembling the well-poised and series or equivalent to and series provide themselves as generators for mutual expansion of different non-orthonormal systems of and SU(3) isofactors, which may be orthonormalized to the paracanonical or (with a definite caution) canonical version.

The Many Faces of the Chiral Potts Model

In this talk, we give a brief overview of several aspects of the theory of the chiral Potts model, including higher-genus solutions of the star–triangle and tetrahedron equations, cyclic representations of affine quantum groups, basic hypergeometric functions at root of unity, and possible applications.

Contiguous relations, basic hypergeometric functions, and orthogonal polynomials. III. Associated continuous dual q-Hahn polynomials

Explicit solutions for the three-term recurrence satisfied by associated continuous dual q-Hahn polynomials are obtained. A minimal solution is identified and an explicit expression for the related continued fraction is derived. The absolutely continuous component of the spectral measure is obtained. Eleven limit cases are discussed in some detail. These include associated big q-Laguerre, associated Wall, associated Al-Salam-Chihara, associated Al-Salam-Carlitz I, and associated continuous q-Hermite polynomials.

On position and momentum operators in the q-oscillator algebra

The aim of this paper is to study the position and momentum operators in q-deformed oscillator algebras. The natural form of the position operator is Xp=qpN(a++a)qpN, where p is a real number. This operator is an operator representable by a Jacobi matrix. Using the theory of Jacobi matrices, the theory of classical moment problem and the theory of basic hypergeometric functions, it is shown that, depending on values of q and p, Xp can be unbounded symmetric operator [which has the deficiency indices (1,1) and, hence, is not self-adjoint, but has self-adjoint extensions], bounded self-adjoint operator with continuous simple spectrum or self-adjoint operator of trace class (therefore, with discrete spectrum with zero as the point of accumulation of eigenvalues). The connection of the q-deformed Heisenberg relation PX−qXP=1 for the position and momentum operators with a q-deformation of the quantum harmonic oscillator is also considered.

Contiguous relations, continued fractions and orthogonality: An 8Ø 7 model

A contiguous relation for very well poised 8ø7 basic hypergeometric functions is used to derive an explict expression for the associated continued fraction. When the continued fraction terminates, it has a singularity structure of simple poles which can be associated with a system of biorthogonal rational functions having a discrete weight function. However, in the nonterminating case, the continued fraction does not have explicit singularities and thus we are unable to exhibit the associated orthogonality. This is the first example of this type that we have encountered.

Transformations of Certain Generalized q-Hypergeometric Functions of 2 Variables

The object of the present paper is to derive a number of transformations involving certain families of basic (or q-) hypergeometric functions of two variables. These results would provide basic (or q-) extensions of numerous (known or new) transformations associated with the ordinary Kampé de Fériet functions with different sets of arguments. Some further consequences of our main transformation formulas are also considered.

Ramanujan-type continuous measures for classicalq-polynomials

It is shown that Ramanujan-type measures for a hierarchy of classical q-orthogonal polynomials can be systematically built from simple cases of continuous q-Hermite and q−1-Hermite polynomials using the Berg-Ismail procedure of attaching generating functions to measures. Applications of this technique also leads to the evaluation of Ramanujan-type integrals for Al-Salam-Chihara polynomials where 0 1, as well as for the product of four particular nonterminating basic hypergeometric functions 2φ1.

Some Families of Multilinear q-Generating Functions and Combinatorial q-Series Identities

A number of linear, bilinear, and multilinear generating functions are obtained here for various families of q-orthogonal polynomials. In particular, the bilinear and multilinear generating functions derived here for the q-laguerre polynomials would provide q-extensions of numerous results for the classical Laguerre polynomials, including (for example) the celebrated Hille-Hardy formula. Many combinatorial q-series identities are also proven by applying the theory of basic (or q-) hypergeometric functions.

Linearization of the product of symmetric orthogonal polynomials

A new constructive approach is given to the linearization formulas of symmetric orthogonal polynomials. We use the monic three-term recurrence relation of an orthogonal polynomial system to set up a partial difference equation problem for the product of two polynomials and solve it in terms of the initial data. To this end, an auxiliary function of four integer variables is introduced, which may be seen as a discrete analogue of Riemann's function. As an application, we derive the linearization formulas for the associated Hermite polynomials and for their continuousq-analogues. The linearization coefficients are represented here in terms of3F2 and3Φ2 (basic) hypergeometric functions, respectively. We also give some partial results in the case of the associated continuousq-ultraspherical polynomials.

Quantum algebra embeddings: deforming functionals and algebraic approach

The study of subalgebras of Lie algebras arising in physical models has been important for many applications. In the present paper we examine the q-deformation of such embeddings; the Lie algebras are then replaced by quantum algebras. Two methods are presented: one based upon deforming functionals, and a direct algebraic approach. A number of examples are given, e.g., [Formula: see text] and [Formula: see text]. For the last example, we give the q-boson construction, and the relevant overlap coefficients are related to a generalized basic hypergeometric function [Formula: see text].