Little and big q-Jacobi polynomials and the Askey–Wilson algebra

The Ramanujan Journal - The little and big q-Jacobi polynomials are shown to arise as basis vectors for representations of the Askey–Wilson algebra. The operators that these polynomials...     

A q-series expansion formula and the Askey–Wilson polynomials

Previously, we proved a q-series expansion formula which allows us to recover many important classical results for q-series. Based on this formula, we derive a new q-formula in this paper, which clearly includes infinitely many q-identities. This new formula is used to give a new proof of the orthogonality relation for the Askey–Wilson polynomials. A curious q-transformation formula is proved, and many applications of this transformation to Hecke type series are given. Some Lambert series identities are also derived.     

The structure relation for Askey–Wilson polynomials

An explicit structure relation for Askey–Wilson polynomials is given. This involves a divided q-difference operator which is skew symmetric with respect to the Askey–Wilson inner product and which sends polynomials of degree n   to polynomials of degree n+1$n+1$. By specialization of parameters and by taking limits, similar structure relations, as well as lowering and raising relations, can be obtained for other families in the q-Askey scheme and the Askey scheme. This is explicitly discussed for Jacobi polynomials, continuous q-Jacobi polynomials, continuous q-ultraspherical polynomials, and for big q-Jacobi polynomials. An already known structure relation for this last family can be obtained from the new structure relation by using the three-term recurrence relation and the second order q-difference formula. The results are also put in the framework of a more general theory. Their relationship with earlier work by Zhedanov and Bangerezako is discussed. There is also a connection with the string equation in discrete matrix models and with the Sklyanin algebra.     

The factorization method for the Askey–Wilson polynomials

A special Infeld–Hull factorization is given for the Askey–Wilson second order q-difference operator. It is then shown how to deduce a generalization of the corresponding Askey–Wilson polynomials.     

Multivariable Askey–Wilson function and bispectrality

For every positive integer d, we define a meromorphic function F _d (n;z), where n,z∈ℂ ^d , which is a natural extension of the multivariable Askey–Wilson polynomials of Gasper and Rahman (Theory and Applications of Special Functions, Dev. Math., vol. 13, pp. 209–219, Springer, New York, 2005). It is defined as a product of very-well-poised ₈ φ ₇ series and we show that it is a common eigenfunction of two commutative algebras A_z{mathcal{A}}_{z} and A_n{mathcal{A}}_{n} of difference operators acting on z and n, with eigenvalues depending on n and z, respectively. In particular, this leads to certain identities connecting products of very-well-poised ₈ φ ₇ series.     

Integral representations and holonomic q-difference structure of Askey–Wilson type functions

We derive in a unified way the difference equations for Askey–Wilson polynomials and their Stieltjes transforms, by using basic properties of the de Rham cohomology associated with q-integral representations (Jackson integrals of BC ₁ type) of these functions.     

Meromorphic Functions of Finite φ-order and Linear Askey–Wilson Divided Difference Equations

The growth of meromorphic solutions of linear difference equations containing Askey–Wilson divided difference operators is estimated. The φ-order is used as a general growth indicator,which covers the growth spectrum between the logarithmic order ρlog(f) and the classical order ρ(f)of a meromorphic function f.     

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.     

Burchnall–Chaundy polynomials and Dunkl–Darboux operators

Mathematical Notes - Certain properties of Burchnall–Chaundy polynomials are studied. The first two nonzero coefficients following the leading coefficient are calculated in explicit form. The...     

Daubechies–Matzinger wavelets and Lorentz polynomials

We construct new compactly supported wavelets and investigate their asymptotic regularity; they appear to be more regular than the Daubechies ones. These new wavelets are associated to Bernstein–Lorentz polynomials (Daubechies–Volkmer’s wavelets) and Hermite–Féjer polynomials (Lemarié–Matzinger’s wavelets) and this property enables us to derive some improved regularity ratio bounds. This revised version was published online in June 2006 with corrections to the Cover Date.     

Narayana polynomials and Hall–Littlewood symmetric functions

We show that Narayana polynomials are a specialization of row Hall–Littlewood symmetric functions. Using λ-ring calculus, we generalize to Narayana polynomials the formulas of Koshy and Jonah for Catalan numbers.