Continuous Q-ultraspherical Polynomials Research Articles

Dual addition formulas: the case of continuous q-ultraspherical and q-Hermite polynomials

We settle the dual addition formula for continuous q-ultraspherical polynomials as an expansion in terms of special q-Racah polynomials for which the constant term is given by the linearization formula for the continuous q-ultraspherical polynomials. In a second proof we derive the dual addition formula from the Rahman–Verma addition formula for these polynomials by using the self-duality of the polynomials. We also consider the limit case of continuous q-Hermite polynomials.

Some q-Supercongruences from Transformation Formulas for Basic Hypergeometric Series

Several new q-supercongruences are obtained using transformation formulas for basic hypergeometric series, together with various techniques such as suitably combining terms, and creative microscoping, a method recently developed by the first author in collaboration with Zudilin. More concretely, the results in this paper include q-analogues of supercongruences (referring to p-adic identities remaining valid for some higher power of p) established by Long, by Long and Ramakrishna, and several other q-supercongruences. The six basic hypergeometric transformation formulas which are made use of are Watson’s transformation, a quadratic transformation of Rahman, a cubic transformation of Gasper and Rahman, a quartic transformation of Gasper and Rahman, a double series transformation of Ismail, Rahman and Suslov, and a new transformation formula for a nonterminating very-well-poised {}_{12}phi _{11} series. Also, the nonterminating q-Dixon summation formula is used. A special case of the new {}_{12}phi _{11} transformation formula is further utilized to obtain a generalization of Rogers’ linearization formula for the continuous q-ultraspherical polynomials.

Matrix-valued orthogonal polynomials related to the quantum analogue of (mathrm{SU}(2) times mathrm{SU}(2), mathrm{diag})

Matrix-valued spherical functions related to the quantum symmetric pair for the quantum analogue of (mathrm{SU}(2) times mathrm{SU}(2), mathrm{diag}) are introduced and studied in detail. The quantum symmetric pair is given in terms of a quantised universal enveloping algebra with a coideal subalgebra. The matrix-valued spherical functions give rise to matrix-valued orthogonal polynomials, which are matrix-valued analogues of a subfamily of Askey–Wilson polynomials. For these matrix-valued orthogonal polynomials, a number of properties are derived using this quantum group interpretation: the orthogonality relations from the Schur orthogonality relations, the three-term recurrence relation and the structure of the weight matrix in terms of Chebyshev polynomials from tensor product decompositions, and the matrix-valued Askey–Wilson type q-difference operators from the action of the Casimir elements. A more analytic study of the weight gives an explicit LDU-decomposition in terms of continuous q-ultraspherical polynomials. The LDU-decomposition gives the possibility to find explicit expressions of the matrix entries of the matrix-valued orthogonal polynomials in terms of continuous q-ultraspherical polynomials and q-Racah polynomials.

Explicit matrix inverses for lower triangular matrices with entries involving continuous [formula omitted]-ultraspherical polynomials

For a one-parameter family of lower triangular matrices with entries involving continuous q-ultraspherical polynomials we give an explicit lower triangular inverse matrix, with entries involving again continuous q-ultraspherical functions. The matrices are q-analogues of results given by Cagliero and Koornwinder recently. The proofs are not q-analogues of the Cagliero–Koornwinder case, but are of a different nature involving q-Racah polynomials. Some applications of these new formulas are given. Also the limit β→0 is studied and gives rise to continuous q-Hermite polynomials for 0<q<1 and q>1.

Characterizations of Continuous and Discrete q-Ultraspherical Polynomials

Abstract We characterize the continuous q-ultraspherical polynomials in terms of the special form of the coefficients in the expansion DqPn(x) in the basis {Pn(x)}, Dq being the Askey-Wilson divided difference operator. The polynomials are assumed to be symmetric, and the connection coefficients are multiples of the reciprocal of the square of the L2 norm of the polynomials. A similar characterization is given for the discrete q-ultraspherical polynomials. A new proof of the evaluation of the connection coefficients for big q-Jacobi polynomials is given.

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 . 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.

On discrete q-ultraspherical polynomials and their duals

A special case of the big q-Jacobi polynomials P n ( x ; a , b , c ; q ) , which corresponds to a = b = − c , is shown to satisfy a discrete orthogonality relation for imaginary values of the parameter a (outside of its commonly known domain 0 < a < q −1 ). Since P n ( x ; q α , q α , − q α ; q ) tend to Gegenbauer (or ultraspherical) polynomials in the limit as q → 1 , this family represents another q-extension of these classical polynomials, different from the continuous q-ultraspherical polynomials of Rogers. For a dual family with respect to the polynomials P n ( x ; a , a , − a ; q ) (i.e., for dual discrete q-ultraspherical polynomials) we also find new orthogonality relations with extremal measures.

Classical Orthogonal Polynomials as Moments

AbstractWe show that the Meixner, Pollaczek, Meixner-Pollaczek, the continuous q-ultraspherical polynomials and Al-Salam-Chihara polynomials, in certain normalization, are moments of probability measures.We use this fact to derive bilinear and multilinear generating functions for some of these polynomials. We also comment on the corresponding formulas for the Charlier, Hermite and Laguerre polynomials.

Identities for q-Ultraspherical Polynomials and Jacobi Functions

A q-analogue of a result by Badertscher and Koornwinder [Canad. J. Math. 44 (1992), 750-773] relating the action of a Hahn polynomial of differential operator argument on ultraspherical polynomials to an ultraspherical polynomial of shifted order and degree is derived. The q-analogue involves q-Hahn polynomials, continuous q-ultraspherical polynomials, and a shift operator. Another limit as q tends to 1 yields an identity for Jacobi functions. Combination with another result of Badertscher and Koornwinder gives a curious formula for Jacobi functions.

A model for the continuous q-ultraspherical polynomials

An algebraic interpretation for two classes of continuous q-polynomials is provided. Rogers’ continuous q-Hermite polynomials and continuous q-ultraspherical polynomials are shown to realize, respectively, bases for representation spaces of the q-Heisenberg algebra and a q-deformation of the Euclidean algebra in these dimensions. A generating function for the continuous q-Hermite polynomials and a q-analog of the Fourier–Gegenbauer expansion are naturally obtained from these models.

Identities for 𝑞-ultraspherical polynomials and Jacobi functions

A q-analogue of a result by Badertscher and Koornwinder [Canad. J. Math. 44 (1992), 750-773] relating the action of a Hahn polynomial of differential operator argument on ultraspherical polynomials to an ultraspherical polynomial of shifted order and degree is derived. The q-analogue involves q-Hahn polynomials, continuous q-ultraspherical polynomials, and a shift operator. Another limit as q tends to 1 yields an identity for Jacobi functions. Combination with another result of Badertscher and Koornwinder gives a curious formula for Jacobi functions.

A Nonterminating q-Clausen Formula and Some Related Product Formulas

This paper uses Gasper–s proof of Rogers– linearization formula for the continuous q-ultraspherical polynomials and a quadratic transformation formula for well-poised basic hypergeometric ${}_2 \phi _1 $ series to derive a nonterminating q-analogue of Clausen–s formula for the square of a certain hypergeometric series. This formula is extended to a q-analogue of the Ramanujan and Bailey extension of Clausen–s formula by employing the Gasper and Rahman nonterminating q-extension of the Sears–Carlitz quadratic transformation formula. Additional product formulas are derived.

Asymptotics of the Askey–Wilson and q-Jacobi Polynomials

We derive explicit representations and complete asymptotic expansions for the Askey–Wilson ${}_4 \phi _3 $ polynomials and the little and big q-Jacobi polynomials. We also give an alternate proof of a Dirichlet–Mehler type formula for the continuous q-ultraspherical polynomials. We also determine the asymptotic behavior of the q-Racah polynomials.

Product and Addition Formulas for the Continuous q-Ultraspherical Polynomials

Using Askey and Wilson’s orthogonality relation and Rahman’s product formula for Askey–Wilson polynomials a Gegenbauer-type product formula is obtained for continuous q-ultraspherical polynomials. Summation and transformation formulas for balanced hypergeometric series are then employed to derive a q-analogue of Gegenbauer’s addition formula.

An Integral of Products of Ultraspherical Functions and a q -Extension

Let Pn(x) and Qn{x) denote the Legendre polynomial of degree n and the usual second solution to the differential equation, respectively. Din showed that ∫ − 1 1 Q n ( x ) P m ( x ) P l ( x ) d x vanishes when |l−m| < n < l+m, and Askey evaluated the integral for arbitrary integral values ofl, m and n. We extend this to the evaluation of ∫ − 1 1 D n λ ( x ) C m λ ( x ) C l λ ( x ) ( 1 − x 2 ) 2 λ − 1 d x , where C n λ ( x ) is the ultraspherical polynomial and C n λ ( x ) is the appropriate second solution to the ultraspherical differential equation. A q-extension is found using the continuous q-ultraspherical polynomials of Rogers. Again the integral vanishes when |l−m| < n < l+m. It is shown that this vanishing phenomenon holds for quite general orthogonal polynomials. A related integral of the product of three Bessel functions is also evaluated.

Rogers’ Linearization Formula for the Continuous q-Ultraspherical Polynomials and Quadratic Transformation Formulas

A simple proof is given for the Rogers’ linearization formula for the continuous q-ultraspherical polynomials. This formula is then used to derive several quadratic transformation formulas.

A Characterization of the Continuous q-Ultraspherical Polynomials

AbstractIn his Ph.D. thesis Allaway found all polynomials that can be represented asand a0bn≠0. We solve the essentially equivalent problem of finding all symmetric polynomials with when are orthogonal with respect to dα(x). The polynomials are the continuous q-ultraspherical polynomials and some of their limiting cases.

Turán Inequalities for Ultraspherical and Continuous q-Ultraspherical Polynomials

Paul Turán discovered that Legendre polynomials satisfy the inequality \[ P_n^2 - P_{n + 1} P_{n - 1} > 0\quad {\text{for }} - 1 < x < 1. \] It was then discovered that such an inequality holds for Jacobi polynomials, generalized Laguerre polynomials, and Bessel functions. In this paper we prove that the continuous q-ultraspherical polynomials satisfy this inequality also. Writing $F_n^\alpha (x) = {{P_n^\alpha (x)} /{P_n^\alpha}} (1)$ where $P_n^\alpha (x)$ is the nth ultraspherical polynomial we prove an inequality similar to the above, \[ (n + 1)F_n^\alpha F_{n - 1}^\beta - (n - 1)F_{n + 1}^\alpha F_{n - 2}^\beta > 0,\quad 0 < x < 1,\quad \frac{1}{2} < a \leq \beta \leq \alpha + 1.\]

Positivity of the Poisson Kernel for the Continuous q-Ultraspherical Polynomials

Rogers’ [Proc. London Math. Soc., 24 (1893), pp. 117–179] bilinear generating function for the continuous q-Hermite polynomials \[ \frac{{\left( {t^2 ;q} \right)_\infty }} {{\left| {\left( {te^{i(\theta + \psi )} ;q} \right)_\infty \left( {te^{i(\theta - \psi )} ;q} \right)_\infty } \right|^2 }} = \sum\limits_{n = 0}^\infty {\frac{{H_n ( {\cos \theta |q} )H_n ( {\cos \psi |q} )}} {{(q;q)_n }}} t^n , \] where $(q;q)_n = (1 - q)(1 - q^2 ) \cdots (1 - q^n )$ and $(a;q)_\infty = (1 - a)(1 - aq)(1 - aq^2 ) \cdots $, is extended to the continuous q-ultraspherical polynomials $C_n (x;\beta |q)$ and used to give conditions for the positivity of the Poisson kernel for these polynomials. Related bilinear generating functions are also considered.