Continuous q-Hermite Polynomials Research Articles

Q-fractional integral operators with two parameters

We use the Poisson kernel of the continuous q-Hermite polynomials to introduce families of integral operators. One of them is semigroups of linear operators. We describe the eigenvalues and eigenfunctions of one family of operators. The action of the semigroups of operators on the Askey–Wilson polynomials is shown to only change the parameters but preserves the degrees, hence we produce transmutation relations for the Askey–Wilson polynomials. The transmutation relations are then used to derive bilinear generating functions involving the Askey–Wilson polynomials.

A unified construction of all the hypergeometric and basic hypergeometric families of orthogonal polynomial sequences

We construct a set H of orthogonal polynomial sequences that contains all the families in the Askey scheme and the q-Askey scheme. The polynomial sequences in H are solutions of a generalized first-order difference equation which is determined by three linearly recurrent sequences of numbers. Two of these sequences are solutions of the difference equation sk+3=z(sk+2−sk+1)+sk, where z is a complex parameter, and the other sequence satisfies a related difference equation of order five.We obtain explicit expressions for the coefficients of the orthogonal polynomials and for the generalized moments with respect to a basis of Newton type of the space of polynomials. We also obtain explicit formulas for the coefficients of the three-term recurrence relation satisfied by the polynomial sequences in H.The set H contains all the 15 families in the Askey scheme of hypergeometric orthogonal polynomials [8, p. 183] and all the 29 families of basic hypergeometric orthogonal polynomial sequences in the q-Askey scheme [8, p. 413]. Each of these families is obtained by direct substitution of appropriate values for the parameters in our general formulas. The only cases that require some limits are the Hermite and continuous q-Hermite polynomials. We present the values of the parameters for some of the families.

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.

Bivariate continuous q-Hermite polynomials and deformed quantum Serre relations

To Nicolás Andruskiewitsch on his 60th birthday, with admiration We introduce bivariate versions of the continuous [Formula: see text]-Hermite polynomials. We obtain algebraic properties for them (generating function, explicit expressions in terms of the univariate ones, backward difference equations and recurrence relations) and analytic properties (determining the orthogonality measure). We find a direct link between bivariate continuous [Formula: see text]-Hermite polynomials and the star product method of [S. Kolb and M. Yakimov, Symmetric pairs for Nichols algebras of diagonal type via star products, Adv. Math. 365 (2020), Article ID: 107042, 69 pp.] for quantum symmetric pairs to establish deformed quantum Serre relations for quasi-split quantum symmetric pairs of Kac–Moody type. We prove that these defining relations are obtained from the usual quantum Serre relations by replacing all monomials by multivariate orthogonal polynomials.

Connection relations for q-Taylor polynomial bases

We derive the connection relations between three q-Taylor polynomial bases. We also derive the connection relations between two of these bases and the continuous q-Hermite polynomials, as well as several new generating functions for the continuous q-Hermite polynomials. Our results also lead to some new integral evaluations. Using one of these connection relations, we give a new proof of a general addition theorem of Suslov for the q-exponential function Eq of Ismail and Zhang.

Approach of the continuous q-Hermite polynomials to x-representation of q-oscillator algebra and its coherent states

We use the recursion relations of the continuous [Formula: see text]-Hermite polynomials and obtain the [Formula: see text]-difference realizations of the ladder operators of a [Formula: see text]-oscillator algebra in terms of the Askey–Wilson operator. For [Formula: see text]-deformed coherent states associated with a disc in the radius [Formula: see text], we obtain a compact form in [Formula: see text]-representation by using the generating function of the continuous [Formula: see text]-Hermite polynomials, too. In this way, we obtain a [Formula: see text]-difference realization for the [Formula: see text]-oscillator algebra in the finite interval [Formula: see text] as a [Formula: see text]-generalization of known differential formalism with respect to [Formula: see text] in the interval [Formula: see text] of the simple harmonic oscillator.

On a generalization of the Rogers generating function

We derive a generalization of the Rogers generating function for the continuous q-ultraspherical/Rogers polynomials whose coefficient is a ϕ12. From that expansion, we derive corresponding specialization and limit transition expansions for the continuous q-Hermite, continuous q-Legendre, Laguerre, and Chebyshev polynomials of the first kind. Using a generalized expansion of the Rogers generating function in terms of the Askey–Wilson polynomials by Ismail & Simeonov whose coefficient is a ϕ78, we derive corresponding generalized expansions for the Wilson, continuous q-Jacobi, and Jacobi polynomials. By comparing the coefficients of the Askey–Wilson expansion to our continuous q-ultraspherical/Rogers expansion, we derive a new quadratic transformation for basic hypergeometric functions which relates an ϕ78 to a ϕ12. We also obtain several definite integral representations which correspond to the above mentioned expansions through the use of orthogonality.

The symmetric q-oscillator algebra: q-coherent states, q-Bargmann–Fock realization and continuous q-Hermite polynomials with 0 < q < 1

The symmetric [Formula: see text]-analysis is used to construct a type of minimum-uncertainty [Formula: see text]-coherent states in the Fock representation space of the symmetric [Formula: see text]-oscillator ∗-algebra with [Formula: see text]. Then, its corresponding [Formula: see text]-Hermite polynomials are derived by using the [Formula: see text]-Bargmann–Fock realization of the symmetric [Formula: see text]-oscillator algebra.

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.

Non-Symmetric Basic Hypergeometric Polynomials and Representation Theory for Confluent Cherednik Algebras

In this paper we introduce a basic representation for the confluent Cherednik algebras $\mathcal H_{\rm V}$, $\mathcal H_{\rm III}$, $\mathcal H_{\rm III}^{D_7}$ and $\mathcal H_{\rm III}^{D_8}$ defined in arXiv:1307.6140. To prove faithfulness of this basic representation, we introduce the non-symmetric versions of the continuous dual $q$-Hahn, Al-Salam-Chihara, continuous big $q$-Hermite and continuous $q$-Hermite polynomials.

Askey-Wilson integral and its generalizations

We expand the Askey-Wilson (AW) density in a series of products of continuous q-Hermite polynomials times the density that makes these polynomials orthogonal. As a by-product we obtain the value of the AW integral as well as the values of integrals of q-Hermite polynomial times the AW density (q-Hermite moments of AW density). Our approach uses nice, old formulae of Carlitz and is general enough to venture a generalization. We prove that it is possible and pave the way how to do it. As a result, we obtain a system of recurrences that if solved successfully gives a sequence of generalized AW densities with more and more parameters. MSC: 33D45, 05A30, 05E05.

Some q-generating functions of the Carlitz and Srivastava–Agarwal types associated with the generalized Hahn polynomials and the generalized Rogers–Szegö polynomials

Motivated essentially by the fact that generating functions play an important rôle in the investigation of many potentially useful properties of the sequences which they generate, various families of q-generating functions of the Carlitz and Srivastava–Agarwal types have been investigated in the literature for the continuous q-Hermite polynomials, the Hahn polynomials, the Rogers–Szegö polynomials, and indeed also for many other q-polynomial systems. The main object of this paper is to apply a certain q-exponential decomposition technique with a view to deriving several bilinear and trilinear q-generating functions associated with the generalized Hahn polynomials. As a simple consequence of one of the main results derived in this paper, we are led naturally to the corrected version of a bilinear q-generating function which appeared in one of Carlitz’s papers.

ON SUMMABLE, POSITIVE POISSON–MEHLER KERNELS BUILT OF AL-SALAM–CHIHARA AND RELATED POLYNOMIALS

Using special technique of expanding ratio of densities in an infinite series of polynomials orthogonal with respect to one of the densities, we obtain simple, closed forms of certain kernels built of the so-called Al-Salam–Chihara (ASC) polynomials. We consider also kernels built of some other families of polynomials such as the so-called big continuous q-Hermite polynomials that are related to the ASC polynomials. The constructed kernels are symmetric and asymmetric. Being the ratios of the densities they are automatically positive. We expand also reciprocals of some of the kernels, getting nice identities built of the ASC polynomials involving six variables like e.g., formula (3.6). These expansions lead to asymmetric, positive and summable kernels. The particular cases (referring to q = 1 and q = 0) lead to the kernels build of certain linear combinations of the ordinary Hermite and Chebyshev polynomials.

Lifting q-difference operators for Askey-Wilson polynomials and their weight function

We determine an explicit form of a q-difference operator that transforms the continuous q-Hermite polynomials Hn(x|q) of Rogers into the Askey-Wilson polynomials pn(x; a, b, c, d|q) on the top level in the Askey q-scheme. This operator represents a special convolution-type product of four one-parameter q-difference operators of the form ɛq(cqDq) (where cq are some constants), defined as Exton’s q-exponential function ɛq(z) in terms of the Askey-Wilson divided q-difference operator Dq. We also determine another q-difference operator that lifts the orthogonality weight function for the continuous q-Hermite polynomialsHn(x|q) up to the weight function, associated with the Askey-Wilson polynomials pn(x; a, b, c, d|q).

On lifting q-difference operators in the Askey scheme of basic hypergeometric polynomials

We construct a q-difference operator that lifts the continuous q-Hermite polynomials Hn(x| q) of Rogers up to the continuous big q-Hermite polynomials Hn(x; a| q) on the next level in the Askey scheme of basic hypergeometric polynomials. This operator is defined as Exton's q-exponential function εq(aqDq) in terms of the Askey–Wilson divided q-difference operator Dq and it represents a particular q-extension of the standard shift operator . We next show that one can move two steps more upwards in order first to reach the Al-Salam–Chihara family of polynomials Qn(x; a, b | q), and then the continuous dual q-Hahn polynomials pn(x; a, b, c| q). In both these cases, lifting operators, respectively, turn out to be convolution-type products of two and three one-parameter q-difference operators of the same type εq(aqDq) at the initial step. At each step, we also determine q-difference operators that lift the weight function for the continuous q-Hermite polynomials Hn(x| q) successively up to the weight functions for Hn(x; a| q), Qn(x; a, b | q) and pn(x; a, b, c| q).

Q-Extension of Mehta's eigenvectors of the finite Fourier transform for q, a root of unity

It is shown that the continuous q-Hermite polynomials for q, a root of unity, have simple transformation properties with respect to the classical Fourier transform. This result is then used to construct q-extended eigenvectors of the finite Fourier transform in terms of these polynomials.

On integral and finite Fourier transforms of continuous q-Hermite polynomials

We give an overview of the remarkably simple transformation properties of the continuous q-Hermite polynomials Hn(x|q) of Rogers with respect to the classical Fourier integral transform. The behavior of the q-Hermite polynomials under the finite Fourier transform and an explicit form of the q-extended eigenfunctions of the finite Fourier transform, defined in terms of these polynomials, are also discussed.

On continuous q-Hermite polynomials and the classical Fourier transform

We prove that the classical Fourier-transform operator intertwines two q-difference equations for the continuous q-Hermite polynomials Hn(x|q) of Rogers, which are associated with the two distinct sets of values for the parameter q: 0 < q < 1 and 1 < q < ∞.

Factorization of the q-difference equation for continuous q-Jacobi polynomials

The formalism of factorization recently used by Atakishiyev et al (2007 J. Phys. A: Math. Theor. 40 9311–7) to study the q-difference equation for continuous q-Hermite polynomials is extended to the continuous q-Jacobi polynomials. Particular cases of continuous q-Laguerre, q-Hermite, q-Legendre and q-ultraspherical polynomials are easily recovered.

On q-extended eigenvectors of the integral and finite Fourier transforms

Mehta has shown that eigenvectors of the N × N finite Fourier transform can be written in terms of the standard Hermite eigenfunctions of the quantum harmonic oscillator (1987 J. Math. Phys. 28 781). Here, we construct a one-parameter family of q-extensions of these eigenvectors, based on the continuous q-Hermite polynomials of Rogers. In the limit when q → 1 these q-extensions coincide with Mehta's eigenvectors, and in the continuum limit as N → ∞ they give rise to q-extensions of eigenfunctions of the Fourier integral transform.