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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.

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 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 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

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.

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.

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 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.

If $P(z)$ is a polynomial of degree $n$ having all its zeros in $|z|\le k, k\le 1,$ Rather, Gulzar and Ahangar [8] proved that for every $\alpha \in \mathbb{C}$ with $|\alpha|\ge k$ and $\gamma >0,$ \[ n(|\alpha|-k)\Bigg\{\int_0^{2\pi}\Big|\frac{P(e^{i\theta})}{D_{\alpha}P(e^{i\theta})}\Big|^\gamma d\theta\Bigg\}^\frac{1}{\gamma}&\leq \Bigg\{\int_0^{2\pi}\Big|1+ke^{i\theta}\Big|^\gamma d\theta\Bigg\}^\frac{1}{\gamma}. \] In this paper, we shall obtain a result which generalizes and sharpens the above inequality by obtaining a bound that depends upon the location of all the zeros of $P(z)$ rather than just on the location of the zero of largest modulus.

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.

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 differential expansion is one of the key structures reflecting group theory properties of colored knot polynomials, which also becomes an important tool for evaluation of non-trivial Racah matrices. This makes highly desirable its extension from knots to links, which, however, requires knowledge of the $6j$-symbols, at least, for the simplest triples of non-coincident representations. Based on the recent achievements in this direction, we conjecture a shape of the differential expansion for symmetrically-colored links and provide a set of examples. Within this study, we use a special framing that is an unusual extension of the topological framing from knots to links. In the particular cases of Whitehead and Borromean rings links, the differential expansions are different from the previously discovered., 11 pages

International audience; We prove a general expansion formula in Askey-Wilson polynomials using Bailey transform and Bressoud inversion. As applications, we give new proofs and generalizations of some recent results of Ismail-Stanton and Liu. Moreover, we prove a new q-beta integral formula involving Askey-Wilson polynomials, which includes the Nassrallah-Rahman integral as a special case. We also give a bootstrapping proof of Ismail-Stanton's recent generating function of Askey-Wilson polynomials.

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.

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 note we study the connection between orthogonal polynomials on an ellipse and orthogonal Laurent polynomials on the unit circle relative to some multiplicative measures and then establish the recurrence relations for orthogonal polynomials on an ellipse. The matrix representation of the operator of multiplication by coordinate function is obtained

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].

We investigate the mutual location of the zeros of two families of orthogonal polynomials. One of the families is orthogonal with respect to the measure d@m(x), supported on the interval (a,b) and the other with respect to the measure |x-c|^@t|x-d|^@cd@m(x), where c and d are outside (a,b). We prove that the zeros of these polynomials, if they are of equal or consecutive degrees, interlace when either 0

This article is a survey of the recent studies jointly with Lies Boelen, Christophe Smet, Walter Van Assche and Lun Zhang (KULeuven, Belgium) on semi-classical continuous and discrete orthogonal polynomials and, in particular, on the connection of their recurrence coefficients to the solutions of the Painleve equations. After recalling some basic facts about the Painleve equations, we discuss continuous and discrete orthogonal polynomials and explain their connection.

We continue the study of a “compound model of a generalized oscillator” and related elementary 3-symmetric Chebyshev polynomials. For these polynomials, we obtain second-order differential equations which are of Fuchs type and have 13 singular points. In the considered simplest case, the obtained results give us an answer to a more general question: What changes in the differential equations for polynomials of the Askey–Wilson scheme when the Jacobi matrix related to these polynomials is perturbed by a diagonal matrix with a complex diagonal? Bibliography: 8 titles.

The linearization problem is the problem of finding the coefficients C"k(m,n) in the expansion of the product P"n(x)Q"m(x) of two polynomial systems in terms of a third sequence of polynomials R"k(x),P"n(x)Q"m(x)=@?k=0n+mC"k(m,n)R"k(x). The polynomials P"n, Q"m and R"k may belong to three different polynomial families. In the case P=Q=R, we get the (standard) linearization or Clebsch-Gordan-type problem. If Q"m(x)=1, we are faced with the so-called connection problem. In this paper, we compute explicitly, in a more general setting and using an algorithmic approach, the connection and linearization coefficients of the Askey-Wilson orthogonal polynomial families. We find our results by an application of computer algebra. The major algorithmic tool for our development is a refined version of q-Petkovsek@?s algorithm published by Horn (2008), Horn et al. (2012) and implemented in Maple by Sprenger (2009), Sprenger and Koepf (2012) (in his package qFPS.mpl) which finds the q-hypergeometric term solutions of q-holonomic recurrence equations. The major ingredients which make this application non-trivial are*the use of appropriate operators D"x and S"x; *the use of an appropriate basis B"n(a,x) for these operators; *and a suitable characterization of the classical orthogonal polynomials on a non-uniform lattice which was developed very recently (Foupouagnigni et al., 2011). Without this preprocessing the relevant recurrence equations are not accessible, and without the mentioned algorithm the solutions of these recurrence equations are out of reach. Furthermore, we present an algorithm to deduce the inversion coefficients for the basis B"n(a,x) in terms of the Askey-Wilson polynomials. Our results generalize and extend known results, and they can be used to deduce the connection and linearization coefficients for any family of classical orthogonal polynomial (including very classical orthogonal polynomials and classical orthogonal polynomials on non-uniform lattices), using the fact that from the Askey-Wilson polynomials, one can deduce, by specialization and/or by limiting processes, any other family of classical orthogonal polynomials. As illustration, we give explicitly the connection coefficients of the continuous q-Hahn, q-Racah and Wilson polynomials.

The aim of this work is to describe new explicit recursive formulas for (renormalized) Macdonald polynomials, using Pieri-like formulas.

In this paper, we continue to investigate the value distribution of some difference polynomials, which can be considered as difference counterparts of some classic results of Hayman.

We study the monotonicity of zeros in connection with perturbed recurrence coefficients of polynomials satisfying certain three-term recurrence relations of Frobenius-type. These recurrence relations are the key ingredient for the tridiagonal approach developed by Delsarte and Genin to solve the standard linear prediction problem. As a particular case, we consider the Askey para-orthogonal polynomials on the unit circle, 2F1(−n,a+bi;2a;1−z),a,b∈R, extending a recent result about the monotonicity of their zeros with respect to the parameter b. Finally, the consequences of our results in the theory of orthogonal polynomials on the real line are discussed. The author thanks the reviewers for their kind comments and careful reading of the work which contributed to its improvement and presentation. The research of the author is supported by the Portuguese Government through the Fundação para a Ciência e a Tecnologia (FCT) under the grant SFRH/BPD/101139/2014, and partially supported by the Brazilian Government through the CNPq under theproject 470019/2013-1 and the Dirección General de Investigación Científica y Técnica, Ministerio de Economía y Competitividad of Spain under the project MTM2012-36732-C03-01.

The subject of special functions is often presented as a collection of disparate results, rarely organized in a coherent way. This book emphasizes general principles that unify and demarcate the subjects of study. The authors' main goals are to provide clear motivation, efficient proofs, and original references for all of the principal results. The book covers standard material, but also much more. It shows how much of the subject can be traced back to two equations - the hypergeometric equation and confluent hypergeometric equation - and it details the ways in which these equations are canonical and special. There is extended coverage of orthogonal polynomials, including connections to approximation theory, continued fractions, and the moment problem, as well as an introduction to new asymptotic methods. There are also chapters on Meijer G-functions and elliptic functions. The final chapter introduces Painlevé transcendents, which have been termed the 'special functions of the twenty-first century'.

This paper is a survey on combinatorics of moments of orthogonal polynomials and linearization coefficients. This area was started by the seminal work of Flajolet followed by Viennot at the beginning of the 1980s. Over the last 30 years, several tools were conceived to extract the combinatorics and compute these moments. A survey of these techniques is presented, with applications to polynomials in the Askey scheme.

We introduce the notion of "hypergeometric" polynomials with respect to Newtonian bases. These polynomials are eigenfunctions ($L P_n(x) = \lambda_n P_n(x)$) of some abstract operator $L$ which is 2-diagonal in the Newtonian basis $\varphi_n(x)$: $L \varphi_n(x) = \lambda_n \varphi_n(x) + \tau_n(x) \varphi_{n-1}(x)$ with some coefficients $\lambda_n$, $\tau_n$. We find the necessary and sufficient conditions for the polynomials $P_n(x)$ to be orthogonal. For the special cases where the sets $\lambda_n$ correspond to the classical grids, we find the complete solution to these conditions and observe that it leads to the most general Askey-Wilson polynomials and their special and degenerate classes.

These are the open problems presented at the 13th International Symposium on Orthogonal Polynomials, Special Functions and Applications (OPSFA13), Gaithersburg, Maryland, on June 4, 2015., Comment: Section 3 corrected

We give a general method of characterizing symmetric orthogonal polynomials through a certain type of connection relations. This method is applied to Al-Salam-Chihara, Askey-Wilson, and Meixner-Pollaczek polynomials. This characterization technique unifies and extends some previous characterization results of Lasser and Obermaier and Ismail and Obermaier. Along the way we explicitly evaluate the connection coefficients in the expansion of D"q^2p"n in terms of {p"k}, where D"q is the Askey-Wilson operator and {p"k} are general Askey-Wilson polynomials. As a limiting case we derive the corresponding connection coefficients in the expansion of W^2W"n in terms of {W"k}, where W is the Wilson operator and {W"k} are general Wilson polynomials. Using the connection relation for Askey-Wilson polynomials, we obtain a characterization for the two-parameter symmetric Askey-Wilson polynomials. The connection relations between D^mP"n^(^@a^,^@b^), D:=d/dx and {P"k^(^@a^,^@b^)} are also derived.

Macdonald polynomials are orthogonal polynomials associated to root systems, and in the type A case, the symmetric Macdonald polynomials are a common generalization of Schur functions, Macdonald spherical functions, and Jack polynomials. We use the combinatorics of alcove walks to calculate products of monomials and intertwining operators of the double affine Hecke algebra. From this, we obtain a product formula for Macdonald polynomials of general Lie type.

We prove that a relatively general even function f(x) (satisfying a vanishing condition, and also some analyticity and growth conditions) on the real line can be expanded in terms of a certain function series closely related to the Wilson functions introduced by Groenevelt in 2003. The coefficients in the expansion of f will be inner products in a suitable Hilbert space of f and some polynomials closely related to Wilson polynomials (these are well-known hypergeometric orthogonal polynomials).