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The purpose of this note is to characterize all the sequences of orthogonal polynomials $$(P_n)_{n\ge 0}$$ such that $$\begin{aligned} \frac{\triangle }{\mathbf{\triangle } x(s-1/2)}P_{n+1}(x(s-1/2))=c_n(\triangle +2\,\mathrm {I})P_n(x(s-1/2)), \end{aligned}$$ where $$\,\mathrm {I}$$ is the identity operator, x defines a class of lattices with, generally, nonuniform step-size, and $$\triangle f(s)=f(s+1)-f(s)$$ . The proposed method can be applied to similar and to more general problems involving the mentioned operators, in order to obtain new characterization theorems for some specific families of classical orthogonal polynomials on lattices.

We examine the properties of distributions with the density of the form: [see formula in PDF] wherec,a1, …,anare some parameters andAna suitable constant. We find general forms ofAn, ofk-th moment and ofk-th polynomial orthogonal with respect to such measures. We also calculate Cauchy transforms of these measures. We indicate connections of such distributions with distributions and polynomials forming the so called Askey–Wilson scheme. On the way we prove several identities concerning rational symmetric functions. Finally, we consider the case of parametersa1, …,anforming conjugate pairs and give some multivariate interpretations based on the obtained distributions at least for the casesn= 2, 4, 6.

Recently, it was observed that the roots of the Askey-Wilson polynomial are retrieved at the unique global minimum of an associated strictly convex Morse function [J. F. van Diejen and E. Emsiz, Lett. Math. Phys. 109 (2019), pp. 89–112]. The purpose of the present note is to infer that the corresponding gradient flow converges to the roots in question at an exponential rate.

The research of the first author was supported by a Vice-Chancellor’s Postdoctoral Fellowship from the University of Pretoria. The research by the second author was partially supported by the National Research Foundation of South Africa under grant number 108763.

The Askey--Wilson polynomials are the most general classical orthogonal polynomials that are known and the Nassrallah--Rahman integral is a very general extension of Euler's integral representation of the classical $_2F_1$ function. Based on a $q$-series transformation formula and the Nassrallah--Rahman integral we prove a $q$--beta integral which has twelve parameters, with several other results, both classical and new, included as special cases. This $q$-beta integral also allows us to derive a curious double $q$--series transformation formula, which includes one formula of Al--Salam and Ismail as a special case

To derive an eigenvalue problem for the associated Askey–Wilson polynomials, we consider an auxiliary function in two variables which is related to the associated Askey–Wilson polynomials introduced by Ismail and Rahman. The Askey–Wilson operator, applied in each variable separately, maps this function to the ordinary Askey–Wilson polynomials with different sets of parameters. A third Askey–Wilson operator is found with the help of a computer algebra program which links the two, and an eigenvalue problem is stated.

In this lecture we discuss properties of orthogonal polynomials for weights which are semiclassical perturbations of classical orthogonality weights. We use the moments, together with the connection between orthogonal polynomials and Painleve equations to obtain explicit expressions for the recurrence coefficients of polynomials associated with a semiclassical Laguerre and a generalized Freud weight. We analyze the asymptotic behavior of generalized Freud polynomials in two different contexts. We show that unique, positive solutions of the nonlinear difference equation satisfied by the recurrence coefficients exist for all real values of the parameter involved in the semiclassical perturbation but that these solutions are very sensitive to the initial conditions. We prove properties of the zeros of semiclassical Laguerre and generalized Freud polynomials and determine the coefficients an,n+j in the differential-difference equation $$\displaystyle x\frac {d}{dx}P_n(x)=\sum _{k=-1}^{0}a_{n,n+k}P_{n+k}(x), $$ where Pn(x) are the generalized Freud polynomials. Finally, we show that the only monic orthogonal polynomials \(\{P_n\}_{n=0}^{\infty }\) that satisfy $$\displaystyle \pi (x)\mathcal {D}_{q}^2P_{n}(x)=\sum _{j=-2}^{2}a_{n,n+j}P_{n+j}(x),\; x=\cos \theta ,\;~ a_{n,n-2}\neq 0,~ n=2,3,\dots , $$ where π(x) is a polynomial of degree at most 4 and \(\mathcal {D}_{q}\) is the Askey–Wilson operator, are Askey–Wilson polynomials and their special or limiting cases, using this relation to derive bounds for the extreme zeros of Askey–Wilson polynomials.

The $q$-difference equation, the shift and the contiguity relations of the Askey-Wilson polynomials are cast in the framework of the three and four-dimensional degenerate Sklyanin algebras $\mathfrak{ska}_3$ and $\mathfrak{ska}_4$. It is shown that the $q$-para Racah polynomials corresponding to a non-conventional truncation of the Askey-Wilson polynomials form a basis for a finite-dimensional representation of $\mathfrak{ska}_4$. The first order Heun operators defined by a degree raising condition on polynomials are shown to form a five-dimensional vector space that encompasses $\mathfrak{ska}_4$. The most general quadratic expression in the five basis operators and such that it raises degrees by no more than one is identified with the Heun-Askey-Wilson operator., Comment: 19 pages, 32 references

Given an entire function f ( z ) f(z) that has only negative zeros, we shall prove that all the functions of type f ( m ) ( x ) / f ( n ) ( x ) , m > n f^{(m)}(x)/f^{(n)}(x),\ m>n are completely monotonic. Examples of this type are given for Laguerre polynomials, ultraspherical polynomials, Jacobi polynomials, Stieltjes-Wigert polynomials, q q -Laguerre polynomials, Askey-Wilson polynomials, Ramanujan function, q q -exponential functions, q q -Bessel functions, Euler’s gamma function, Airy function, modified Bessel functions of the first and the second kind, and the confluent basic hypergeometric series.

In this work, we define and introduce a new kind of the Apostol type Frobenius-Euler polynomials based on the (p, q)-calculus and investigate their some properties, recurrence relationships and so on. We give some identities at this polynomial. Moreover, we get (p, q)-extension of Carlitz’s main result in [1].

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 ψ 6 6 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.

In the present work, we establish a general inverse series relations which unify the polynomials of Askey–Wilson, q-Racah, and q-Konhauser. The q-extensions of Riordan’s inverse series relations [Combinatorial Identities, John Wiley and Sons, Inc. 1968] are obtained by means of the main two theorems. Then we emphasize on the special cases, namely the q-Bessel function together with a q-Neumann expansion, and an inverse pair associated with the partition identities.

In his famous theorem (1982), Douglas Leonard characterized the $q$-Racah polynomials and their relatives in the Askey scheme from the duality property of $Q$-polynomial distance-regular graphs. In this paper we consider a nonsymmetric (or Laurent) version of the $q$-Racah polynomials in the above situation. Let $\Gamma$ denote a $Q$-polynomial distance-regular graph that contains a Delsarte clique $C$. Assume that $\Gamma$ has $q$-Racah type. Fix a vertex $x \in C$. We partition the vertex set of $\Gamma$ according to the path-length distance to both $x$ and $C$. The linear span of the characteristic vectors corresponding to the cells in this partition has an irreducible module structure for the universal double affine Hecke algebra $\hat{H}_q$ of type $(C^{\vee}_1, C_1)$. From this module, we naturally obtain a finite sequence of orthogonal Laurent polynomials. We prove the orthogonality relations for these polynomials, using the $\hat{H}_q$-module and the theory of Leonard systems. Changing $\hat{H}_q$ by $\hat{H}_{q^{-1}}$ we show how our Laurent polynomials are related to the nonsymmetric Askey-Wilson polynomials, and therefore how our Laurent polynomials can be viewed as nonsymmetric $q$-Racah polynomials., Comment: 38 pages, 3 figures

We introduce variations of the Robinson–Schensted correspondence parametrized by positive integers p. Each variation gives a bijection between permutations and pairs of standard tableaux of the same shape. In addition to sharing many of the properties of the classical Schensted algorithm, the new algorithms are designed to be compatible with certain permutation statistics introduced by Haglund in the study of Macdonald polynomials. In particular, these algorithms provide an elementary bijective proof converting Haglund's combinatorial formula for Macdonald polynomials to an explicit combinatorial Schur expansion of Macdonald polynomials indexed by partitions μ satisfying μ 1 ≤ 3 and μ 2 ≤ 2 . We challenge the research community to extend this RSK-based approach to more general classes of partitions.

In this paper, we mainly study the uniqueness of specific q-shift difference polynomials and of meromorphic functions, which share a common small function and get the corresponding results. In addition, we also investigate the problem of value distribution on q-shift difference polynomials of entire functions.

The authors wish to make the following corrections to their paper [...]

Bivariate Goncarov polynomials are a basis of the solutions of the bivariate Goncarov Interpolation Problem in numerical analysis. A sequence of bivariate Goncarov polynomials is determined by a set of nodes Z = { ( x i , j , y i , j ) ź R 2 } and is an affine sequence if Z is an affine transformation of the lattice grid N 2 , i.e.,ź ( x i , j , y i , j ) T = A ( i , j ) T + ( c 1 , c 2 ) T for some 2×2 matrix A and constants c 1 , c 2 . In this paper we prove that a sequence of bivariate Goncarov polynomials is of binomial type if and only if it is an affine sequence with c 1 = c 2 = 0 . Such polynomials form a higher-dimensional analog of the Abel polynomial A n ( x ; a ) = x ( x - a n ) n - 1 . We present explicit formulas for a general sequence of bivariate affine Goncarov polynomials and its exponential generating function, and use the algebraic properties of Goncarov polynomials to give some new two-dimensional generalizations of Abel identities.

We derive two new versions of Cooper's formula for the iterated Askey–Wilson operator. Using the second version of Cooper's formula and the Leibniz rule for the iterated Askey–Wilson operator, we derive several formulas involving this operator. We also give new proofs of Rogers' summation formula for ϕ 5 6 series, Watson's transformation, and we establish a Rodriguez type operational formula for the Askey–Wilson polynomials. In addition we establish two integration by parts formulas for integrals involving the iterated Askey–Wilson operator. Using the first of these integration by parts formulas, we derive a two parameter generating function for the Askey–Wilson polynomials. A generalization of the Leibniz rule for the iterated Askey–Wilson operator is also given and used to derive a multi-sum identity.

Within the Askey scheme of hypergeometric orthogonal polynomials, Racah polynomials stay on the top of the hierarchy and they generalize all of the discrete hypergeometric orthogonal polynomials. In this paper, we investigate asymptotic behaviors of Racah polynomials with varying parameters when the polynomial degree tends to infinity. Using the difference equation technique developed in our earlier papers, we obtain an asymptotic formula in the outer region via ratio asymptotics and then derive asymptotic formulas in the oscillatory region via a matching method. Our asymptotic formulas are explicitly given in terms of the polynomial degree, variable and parameters, using elementary functions such as logarithmic, exponential and rational functions. By taking limits, our results also yield asymptotic formulas for orthogonal polynomials in the lower hierarchy of the Askey scheme such as Hahn polynomials and Krawtchouk polynomials.

We obtain an explicit formula for a certain weighted enumeration of lozenge tilings of a hexagon with an arbitrary triangular hole. The complexity of our expression depends on the distance from the hole to the center of the hexagon. This proves and generalizes conjectures of Ciucu et al., who considered the case of plain enumeration when the triangle is located at or very near the center. Our proof uses Askey-Wilson polynomials as a tool to relate discrete and continuous Selberg-type integrals.

We present an explicit branching formula for the six-parameter Macdonald-Koornwinder polynomials with hyperoctahedral symmetry., 8 pages, LaTeX

We prove an equivalence between the existence of the first structure relation satisfied by a sequence of monic orthogonal polynomials $\{P_n\}_{n=0}^{\infty}$, the orthogonality of the second derivatives $\{\mathbb{D}_{x}^2P_n\}_{n= 2}^{\infty}$ and a generalized Sturm-Liouville type equation. Our treatment of the generalized Bochner theorem leads to explicit solutions of the difference equation [Vinet L., Zhedanov A., J. Comput. Appl. Math. 211 (2008), 45-56], which proves that the only monic orthogonal polynomials that satisfy the first structure relation are Wilson polynomials, continuous dual Hahn polynomials, Askey-Wilson polynomials and their special or limiting cases as one or more parameters tend to $\infty$. This work extends our previous result [arXiv:1711.03349] concerning a conjecture due to Ismail. We also derive a second structure relation for polynomials satisfying the first structure relation.

We study matrix elements of a change of base between two different bases of representations of the quantum algebra $U_q(su(1,1))$. The two bases, which are multivariate versions of Al-Salam--Chihara polynomials, are eigenfunctions of iterated coproducts of twisted primitive elements. The matrix elements are identified with Gasper and Rahman's multivariate Askey-Wilson polynomials, and from this interpretation we derive their orthogonality relations. Furthermore, the matrix elements are shown to be eigenfunctions of the twisted primitive elements after a change of representation, which gives a quantum algebraic derivation of the fact that the multivariate Askey-Wilson polynomials are solutions of a multivariate bispectral $q$-difference problem., Comment: 26 pages

In this paper, we define a sequence of polynomials Pn(α)(x|A,H) depending only on the choice of two analytic functions A and H in a neighborhood of zero. For a pair of compositional inverses A and B, we will show the identity Pn(α)(x|B,H∘B)=Pn(n+1−α)(1−x|A,A′H), which generalize the Carlitz's identity on Bernoulli polynomials.

By studying various properties of some divided difference operators, we prove that Wilson polynomials are solutions of a second-order difference equation of hypergeometric type. Next, some new structure relations are deduced, the inversion and the connection problems are solved using an algorithmic method.

Infinitely many Casoratian identities are derived for the Wilson and Askey-Wilson polynomials in parallel to the Wronskian identities for the Hermite, Laguerre and Jacobi polynomials, which were reported recently by the present authors. These identities form the basis of the equivalence between eigenstate adding and deleting Darboux transformations for solvable (discrete) quantum mechanical systems. Similar identities hold for various reduced form polynomials of the Wilson and Askey-Wilson polynomials, e.g. the continuous q-Jacobi, continuous (dual) (q-)Hahn, Meixner-Pollaczek, Al-Salam-Chihara, continuous (big) q-Hermite, etc., 31 pages, 2 figures. Comments and references added. To appear in Journal of Approximation Theory

In this work, using the inversion coefficients and some connection coefficients between some polynomial sets, we give explicit representations of the moments of all the orthogonal polynomials belonging to the Askey–Wilson scheme. Generating functions for some of these moments are also provided.

We give a general expansion formula of functions in the Askey–Wilson polynomials and using Askey–Wilson orthogonality we evaluate several integrals. Moreover we give a general expansion formula of functions in polynomials of Askey–Wilson type, which are not necessarily orthogonal. Limiting cases give expansions in little and big q -Jacobi type polynomials. We also give a new generating function for Askey–Wilson polynomials and a new evaluation for specialized Askey–Wilson polynomials.

In this paper, we prove the quasi-orthogonality of a family of F 2 2 polynomials and several classes of F 2 3 polynomials that do not appear in the Askey scheme for hypergeometric orthogonal polynomials. Our results include, as a special case, two F 2 3 polynomials considered by Dickinson in 1961. We also discuss the location and interlacing of the real zeros of our polynomials.

We study a family of integrals parameterised by N = 2; 3;::: generalising the Askey-Wilson integral N = 2 which has arisen in the theory of q-analogs of monodromy preserving deformations of linear dierential systems and in theory of the Baxter Q operator for the XXZ open quantum spin chain. These integrals are particular examples of moments dened by weights generalising the Askey-Wilson weight and we show the integrals are characterised by various (N 1)-th order linearq-dierence equations which we construct. In addition we demonstrate that these integrals can be evaluated as a nite sum of (N 1) BC1-type Jackson integrals or 2N+2’2N+1 basic hypergeometric functions.

We present a fourfold series expansion representing the Askey-Wilson polynomials. To obtain the result, a sequential use is made of several summation and transformation formulas for the basic hypergeometric series, including the Verma's q-extension of the Field and Wimp expansion, Andrews' terminating q-analogue of Watson's 3F2 sum, Singh's quadratic transformation. As an application, we present an explicit formula for the Koornwinder polynomial of type BCn (n in Z_>0) with one row diagram. When the parameters are specialized, we recover Lassalle's formula for Macdonald polynomials of type Bn, Cn and Dn with one row diagram, thereby proving his conjectures.

The mixed moments for the Askey–Wilson polynomials are found using a bootstrapping method and connection coefficients. A similar bootstrapping idea on generating functions gives a new Askey–Wilson generating function. Modified generating functions of orthogonal polynomials are shown to generate polynomials satisfying recurrences of known degree greater than three. An important special case of this hierarchy is a polynomial which satisfies a four term recurrence, and its combinatorics is studied.

In this paper we are engaged with classical orthogonal polynomials. We deduce relations for the reproducing kernels for these polynomials without derivations. By the using this form of reproducing kernels of classical orthogonal polynomials we deduce some inequalities for polynomials with special weight function. This enabled us to derive some inequalities for orthonormal polynomial with special weight function. It allows us to deduce general statement for every orthonormal polynomial with a nonnegative weight function on a finite interval of orthogonality. Mathematics Subject Classification: 33C45, 33C47

The following lectures are based on the works (Tamm, Communication complexity of sum-type functions, 1991, [115]; Appl Math Lett 7:39–44, 1994, [116]; Inf Comput 116(2):162–173, 1995, [117]; Tamm, Discret Appl Math 61:271–283, 1995, [118]; Proceedings of 1998 IEEE Symposium on Information Theory, MIT, Cambridge, 1998, [120]; Tamm, IEEE Trans Inf Theory 44(5):2003–2009, 1998, [121]; Numbers, Information and Complexity, Special Volume in Honour of Rudolf Ahlswede, 2000, [122]; Proceedings of the Workshop Codes and Association Schemes, 2001, [123]; Electron J Comb 8(A1):31, 2001, [124]; J Stat Plan Interf 2(2):433–448, 2002, [125]) of Tamm.

Let k, q( ≥ 2) be two positive integers, b ≠ 0 be a complex number, and let H( f, f ′ , …, f(k)) be a differential polynomial with \(\frac{\Gamma } {\gamma } \vert _{H}

The Szegő–Askey polynomials are orthogonal polynomials on the unit circle. In this paper, we study their asymptotic behavior by knowing only their weight function. Using the Riemann–Hilbert method, we obtain global asymptotic formulas in terms of Bessel functions and elementary functions for z in two overlapping regions, which together cover the whole complex plane. Our results agree with those obtained earlier by Temme [Uniform asymptotic expansion for a class of polynomials biorthogonal on the unit circle, Constr. Approx. 2 (1986) 369–376]. Temme's approach started from an explicit expression of the Szegő–Askey polynomials in terms of an2F1-function, and followed by integral methods.

In this article, we first introduce and study a new family of the multi-index and multi-variable Gould-Hopper and Dattoli type polynomials {Hn(cm, cm-1,…, c3, c2)(a1, a2, …, am} defined by (2.1), which are an extension of different types of Her-mite polynomials defined in section 1. We next consider multi-variable linear, bilinear and bilateral generating relations of the newly defined hypergeometric polynomials, using series iteration techniques. Further, we generalize these generating relations in the forms of multiple series identities involving bounded multiple sequences, Fox-Wright hypergeometric function and Srivastava-Daoust multi-variable hypergeometric function.

New formulas for then th moment n(a;b;c;d;q) of the Askey-Wilson polynomials are given. These are derived using analytic techniques, and by considering three combinatorial models for the moments: Motzkin paths, matchings, and staircase tableaux. A related positivity theorem is given and another one is conjectured. R´ esum´ e. Nous pr´ esentons de nouvelles formules pour lesn-moments n(a;b;c;d;q) des polynˆ omes Askey-Wilson. Ils sont calculavec des techniques analytiques, et en considtrois modcombinatoires pour les moments: des chemins de Motzkin, des couplages, et des tableaux escalier. Un th´ eorde positivitliest donn´ e et un autre est conjectur´

We determine the q-orthogonal polynomial solutions to the difference equation , where is the Askey–Wilson divided-difference operator, using an approach that does not appear in the literature. To accomplish this, we construct a polynomial expansion via a Chebyshev basis, which ultimately allows explicit formulas to be derived for the recurrence coefficients of above. From there, we obtain our solutions and discuss some future research.

We give new derivations of properties of the functions of the second kind of the Jacobi, little and big q-Jacobi polynomials, and the symmetric Al-Salam-Chihara polynomials for q>1. We also study the Askey-Wilson functions and the Wilson functions of second kind. An integration by parts formula is derived for the Wilson operator in appropriate Hilbert space. In case of undetermined moment problem, we prove that the function of second kind depends on the measure.

In this paper the ladder operator approach has been applied to deduce the Mehler–Heine type formulas for orthogonal polynomials with respect to the modified Jacobi weight \[ ω α , β , h ( x ) = h ( x ) ( 1 − x ) α ( 1 + x ) β , x ∈ [ − 1 , 1 ] , \omega _{\alpha ,\beta ,h}(x)=h(x) (1-x)^\alpha (1+x)^\beta , \hspace {0.5cm} x\in [-1,1], \] where α , β > 0 , \alpha ,\beta >0, and with h h real analytic and strictly positive on [ − 1 , 1 ] . [-1, 1].