Al-Salam Carlitz Research Articles

A Review of q-Difference Equations for Al-Salam–Carlitz Polynomials and Applications to U(n + 1) Type Generating Functions and Ramanujan’s Integrals

In this review paper, our aim is to study the current research progress of q-difference equations for generalized Al-Salam–Carlitz polynomials related to theta functions and to give an extension of q-difference equations for q-exponential operators and q-difference equations for Rogers–Szegö polynomials. Then, we continue to generalize certain generating functions for Al-Salam–Carlitz polynomials via q-difference equations. We provide a proof of Rogers formula for general Al-Salam–Carlitz polynomials and obtain transformational identities using q-difference equations. In addition, we gain U(n+1)-type generating functions and Ramanujan’s integrals involving general Al-Salam–Carlitz polynomials via q-difference equations. Finally, we derive two extensions of the Andrews–Askey integral via q-difference equations.

Exactly solvable discrete time birth and death processes

We present 15 explicit examples of discrete time birth and death processes which are exactly solvable. They are related to hypergeometric orthogonal polynomials of the Askey scheme having discrete orthogonality measures. Namely, they are the Krawtchouk, three different kinds of q-Krawtchouk, (dual, q)-Hahn, (q)-Racah, Al-Salam–Carlitz II, q-Meixner, q-Charlier, dual big q-Jacobi, and dual big q-Laguerre polynomials. The birth and death rates are determined by using the difference equations governing the polynomials. The stationary distributions are the normalized orthogonality measures of the polynomials. The transition probabilities are neatly expressed by the normalized polynomials and the corresponding eigenvalues. This paper is simply the discrete time versions of the known solutions of the continuous time birth and death processes.

Densely generated 2D q-Appell polynomials of Bessel type and q-addition formulas

The article aims to introduce a densely generated class of $2D$ $q$-Appell polynomials of Bessel type via generating equation and to establish its $q$-determinant form. It is advantageous to consider the $2D$ $q$-Bernoulli, $2D$ $q$-Roger Szeg\"{o} and $2D$ $q$-Al-Salam Carlitz polynomials of Bessel type as their special members. The $q$-determinant forms and certain $q$-addition formulas are derived for these polynomials. The article concludes with a brief view on discrete $q$-Bessel convolution of the $2D$ $q$-Appell polynomials.

Stieltjes’ theorem for classical discrete orthogonal polynomials

The purpose of this note is to establish, from the hypergeometric-type difference equation introduced by Nikiforov and Uvarov, new tractable sufficient conditions for the monotonicity with respect to a real parameter of zeros of classical discrete orthogonal polynomials. This result allows one to carry out a systematic study of the monotonicity of zeros of classical orthogonal polynomials on linear, quadratic, q-linear, and q-quadratic grids. In particular, we analyze in a simple and unified way the monotonicity of the zeros of Hahn, Charlier, Krawtchouk, Meixner, Racah, dual Hahn, q-Meixner, quantum q-Krawtchouk, q-Krawtchouk, affine q-Krawtchouk, q-Charlier, Al-Salam–Carlitz, q-Hahn, little q-Jacobi, little q-Laguerre/Wall, q-Bessel, q-Racah, and dual q-Hahn polynomials.

On Second Order q-Difference Equations Satisfied by Al-Salam–Carlitz I-Sobolev Type Polynomials of Higher Order

This contribution deals with the sequence {Un(a)(x;q,j)}n≥0 of monic polynomials in x, orthogonal with respect to a Sobolev-type inner product related to the Al-Salam–Carlitz I orthogonal polynomials, and involving an arbitrary number j of q-derivatives on the two boundaries of the corresponding orthogonality interval, for some fixed real number q∈(0,1). We provide several versions of the corresponding connection formulas, ladder operators, and several versions of the second order q-difference equations satisfied by polynomials in this sequence. As a novel contribution to the literature, we provide certain three term recurrence formula with rational coefficients satisfied by Un(a)(x;q,j), which paves the way to establish an appealing generalization of the so-called J-fractions to the framework of Sobolev-type orthogonality.

Quasi-orthogonality and zeros of some [formula omitted] and [formula omitted] polynomials

We state and prove the q-extension of a result due to Johnston and Jordaan (cf. Johnston and Jordaan (2015)) and make use of this result, the orthogonality of q-Laguerre, little q-Jacobi, q-Meixner and Al-Salam–Carlitz I polynomials as well as contiguous relations satisfied by the polynomials, to establish the quasi-orthogonality of certain 2ϕ2 and 3ϕ2 polynomials. The location and interlacing properties of the real zeros of these quasi-orthogonal polynomials are studied. Interlacing properties of the zeros of q-Laguerre quasi-orthogonal polynomials Ln(δ)(z;q) when −2<δ<−1 with those of Ln−1(δ+1)(z;q) and Ln(δ+1)(z;q) are also considered.

On the generalized q-poly-Euler polynomials of the second kind

In this work, we define the generalized q-poly-Euler numbers of the second kind of order ? and the generalized q-poly-Euler polynomials of the second kind of order ?. We investigate some basic properties for these polynomials and numbers. In addition, we obtain many identities, relations including the Roger-Sz?go polynomials, the Al-Salam Carlitz polynomials, q-analogue Stirling numbers of the second kind and two variable Bernoulli polynomials.

On infinite Jacobi matrices with a trace class resolvent

Let {Pn(x)} be an orthonormal polynomial sequence and denote by {wn(x)} the respective sequence of functions of the second kind. Suppose the Hamburger moment problem for {Pn(x)} is determinate and denote by J the corresponding Jacobi matrix operator on ℓ2. We show that if J is positive definite and J−1 belongs to the trace class then the series on the right-hand side of the defining equation F(z)≔1−z∑n=0∞wn(0)Pn(z)converges locally uniformly on ℂ and it holds true that F(z)=∏n=1∞(1−z∕λn) where {λn;n=1,2,3,…}=specJ. Furthermore, the Al-Salam–Carlitz II polynomials are treated as an example of orthogonal polynomials to which this theorem can be applied.

Subnormality of unbounded composition operators over one-circuit directed graphs: Exotic examples

A recent example of a non-hyponormal injective composition operator in an L2-space generating Stieltjes moment sequences, invented by three of the present authors, was built over a non-locally finite directed tree. The main goal of this paper is to solve the problem of whether there exists such an operator over a locally finite directed graph and, in the affirmative case, to find the simplest possible graph with these properties (simplicity refers to local valency). The problem is solved affirmatively for the locally finite directed graph G2,0, which consists of two branches and one loop. The only simpler directed graph for which the problem remains unsolved consists of one branch and one loop. The consistency condition, the only efficient tool for verifying subnormality of unbounded composition operators, is intensively studied in the context of G2,0, which leads to a constructive method of solving the problem. The method itself is partly based on transforming the Krein and the Friedrichs measures coming either from shifted Al-Salam–Carlitz q-polynomials or from a quartic birth and death process.

Q-difference equations for Askey–Wilson type integrals via q-polynomials

In this paper, we show how to deduce Askey–Wilson type integrals by the method of q-difference equation. In addition, we construct the relation between bilinear generating functions and solutions of q-difference equation. More over, we generalize Bailey's ψ66 summation from the perspective of q-integral by the method of q-difference equation. At last, we deduce U(n+1) type generating functions for Al-Salam–Carlitz polynomials by the method of q-difference equation.

Orthogonality relations for Al-Salam–Carlitz polynomials of type II

By use of a special case of Slater’s q-beta integral evaluation formula, we determine orthogonality relations for the Al-Salam–Carlitz polynomials of type II with respect to a family of measures supported on a discrete subset of R. From spectral analysis of the corresponding second-order q-difference operator we obtain an infinite set of functions that complement the Al-Salam–Carlitz II polynomials to an orthogonal basis of the associated L2-space.

A note on generalized q-difference equations for q-beta and Andrews–Askey integral

Two q-difference equations with solutions expressed by q-exponential operator identities are investigated. As applications, two extensions of Ramanujanʼs formulas for q-beta integral are given, two generalizations of Andrews–Askey integral are obtained. In addition, generating functions for generalized Al-Salam–Carlitz polynomials are deduced. At last, a generalized transformation identity is gained.

Q-Oscillator Algebra And d-Orthogonal Polynomials

In this paper we express the matrix coefficients of the Fock representation of a q-oscillator algebra in terms of the d-orthogonal Al-Salam Carlitz polynomials. Also, we derive a generating functions, recurrence relations and q-difference equations for these d-orthogonal polynomials.

Notes on Askey–Roy integral and certain generating functions for [formula omitted]-polynomials

In this paper, generalizations of Askey–Roy integral are given by the technique of transformation. In addition, a recurring formula for Askey–Roy integral and an integral involving Cauchy polynomial are deduced by Askey–Roy integral. Besides, a new transformation identity is gained by Askey–Roy integral. Moreover, a certain generating function for generalized Al-Salam–Carlitz polynomials is obtained by Askey–Roy integral. At last, two generalizations of U(n+1) type generating function for generalized Al-Salam–Carlitz polynomials are derived by Askey–Roy integral.

An operator approach to the Al-Salam–Carlitz polynomials

We present an operator approach to Rogers-type formulas and Mehler’s formula for the Al-Salam–Carlitz polynomials Un(x,y,a;q). By using the q-exponential operator, we obtain a Rogers-type formula, which leads to a linearization formula. With the aid of a bivariate augmentation operator, we get a simple derivation of Mehler’s formula due to Al-Salam and Carlitz [“Some orthogonal q-polynomials,” Math. Nachr. 30, 47 (1965)]. By means of the Cauchy companion augmentation operator, we obtain an equivalent form of Mehler’s formula. We also give several identities on the generating functions for products of the Al-Salam–Carlitz polynomials, which are extensions of the formulas for the Rogers–Szegö polynomials.

A recurring [formula omitted]-integral formula

In this paper, we use the q -Chu–Vandermonde formula to derive a recurring q -integral formula. Some applications of the recurring q -integral formula are also given, which include deriving the 3 ϕ 2 representation of the Al-Salam–Carlitz polynomials and the extension of the terminating Sears’ 3 ϕ 2 transformation formula.

Formula omitted]-integral representation of the Al-Salam–Carlitz polynomials

We use the Andrews–Askey integral and the Leibniz rule for the q -difference operator to give the q -integral representation of the Al-Salam–Carlitz polynomials, which includes the q -integral representation of the Rogers–Szegö polynomials and the q -integral representation of the q -Hermite polynomials as special cases.

A characterization of some q-orthogonal polynomials

We show that the only orthogonal polynomials satisfying a q-difference equation of the form π(x)D q P n (x) = (α n x + β n )P n (x) + γ n P n−1(x) where π(x) is a polynomial of degree 2, are the Al-Salam Carlitz 1, little and big q-Laguerre, the little and big q-Jacobi, and the q-Bessel polynomials. This is a q-analog of the work carried out in [1].

Efficient algorithms for construction of recurrence relations for the expansion and connection coefficients in series of Al-Salam–Carlitz I polynomials

Two formulae expressing explicitly the derivatives and moments of Al- Salam-Carlitz I polynomials of any degree and for any order in terms of Al-Salam-Carlitz I themselves are proved. Two other formulae for the expansion coefficients of general-order derivatives D p f( x), and for the moments xD p f( x),of an arbitrary function f( x)in terms of its original expansion coefficients are also obtained. Application of these formulae for solving q-difference equations with varying coefficients, by reducing them to recurrence relations in the expansion coefficients of the solution, is explained. An algebraic symbolic approach (using Mathematica) in order to build and solve recursively for the connection coefficients between Al-Salam-Carlitz I polynomials and any system of basic hypergeometric orthogonal polynomials, belonging to the q-Hahn class, is described.

Mellin transforms for some families of q-polynomials

By using Ramanujan's q-extension of the Euler integral representation for the gamma function, we derive the Mellin integral transforms for the families of the discrete q-Hermite II, the Al-Salam–Carlitz II, the big q-Laguerre, the big q-Legendre, the big q-Jacobi and the q-Hahn polynomials.