Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials
From MaRDI portal
Publication:3689467
DOI10.1090/memo/0319zbMath0572.33012OpenAlexW1965847964WikidataQ56852794 ScholiaQ56852794MaRDI QIDQ3689467
James A. Wilson, Richard Askey
Publication date: 1985
Published in: Memoirs of the American Mathematical Society (Search for Journal in Brave)
Full work available at URL: https://doi.org/10.1090/memo/0319
basic hypergeometric seriesq-ultraspherical polynomialsq-Jacobi polynomialsq-analogue of Meixner-Pollaczek polynomials
Additive difference equations (39A10) Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) (33D45) (q)-gamma functions, (q)-beta functions and integrals (33D05)
Related Items
Coefficients of multiplication formulas for classical orthogonal polynomials, \(C^*\)-algebraic partial compact quantum groups, DAHA-Jones polynomials of torus knots, Note on a \(q\)-contour integral formula, Formulas and identities involving the Askey-Wilson operator, Adjoint difference equation for the Nikiforov-Uvarov-Suslov difference equation of hypergeometric type on non-uniform lattices, Elliptic hypergeometric summations by Taylor series expansion and interpolation, \(q\)-difference equation and \(q\)-polynomials, Extensions of Ramanujan's reciprocity theorem and the Andrews-Askey integral, Spectral properties of a \(q\)-Sturm--Liouville operator, Askey-Wilson type functions with bound states, \(BC_n\)-symmetric abelian functions, Linearization formula for the product of associated \(q\)-ultraspherical polynomials, Askey-Wilson type integrals associated with root systems, Inversions of integral operators and elliptic beta integrals on root systems, Short proofs of summation and transformation formulas for basic hypergeometric series, Multi-element stochastic spectral projection for high quantile estimation, A flexible uncertainty quantification method for linearly coupled multi-physics systems, A weighted \(\ell_1\)-minimization approach for sparse polynomial chaos expansions, Integral representations and holonomic \(q\)-difference structure of Askey-Wilson type functions, Difference Sturm-Liouville problems in the imaginary direction, Ambrosio-Tortorelli segmentation of stochastic images: model extensions, theoretical investigations and numerical methods, The Meixner--Pollaczek polynomials and a system of orthogonal polynomials in a strip, A \(U_q(\widehat{gl}(2|2))_1\)-vertex model: creation algebras and quasi-particles. I, Laplacian operators and Radon transforms on Grassmann graphs, A hypergeometric function transform and matrix-valued orthogonal polynomials, The \(q\)-deformed analogue of the Onsager algebra: beyond the Bethe ansatz approach, Extensions of two \(q\)-series expansions with applications to biorthogonal rational functions, A polynomial chaos approach to the robust analysis of the dynamic behaviour of friction systems, Erratum to ``Tableaux combinatorics for the asymmetric exclusion process and Askey-Wilson polynomials, Bailey's well-poised \(_{6}\psi _{6}\)-series implies the Askey-Wilson integral, On the propagation of statistical model parameter uncertainty in CFD calculations, Bootstrapping and Askey-Wilson polynomials, Selberg integrals, Askey-Wilson polynomials and lozenge tilings of a hexagon with a triangular hole, Analytic solution of the multiloop Baxter equation, On lifting \(q\)-difference operators for a chain of basic hypergeometric polynomials, A new proof of the Askey-Wilson integral via a five-variable Ramanujan's reciprocity theorem, Linear partial \(q\)-difference equations on \(q\)-linear lattices and their bivariate \(q\)-orthogonal polynomial solutions, Prediction of geometric uncertainty effects on fluid dynamics by polynomial chaos and fictitious domain method, Combinatorial proofs of some limit formulas involving orthogonal polynomials, Representations of \(q\)-orthogonal polynomials, Connection relations and characterizations of orthogonal polynomials, Moments for generating functions of Al-Salam-Carlitz polynomials, Askey-Wilson polynomials and the quantum group \(SU_ q(2)\), Elements of q-harmonic analysis, Application of Galerkin method to Kirchhoff plates stochastic bending problem, The Cauchy operator for basic hypergeometric series, Expansions in the Askey-Wilson polynomials, On the structure of Jackson integrals of \(BC_n\) type and holonomic \(q\)-difference equations, A Bernstein type inequality for the Askey-Wilson operator, On the influence of uncertainty in computational simulations of a high-speed jet flow from an aircraft exhaust, Symmetric and nonsymmetric Koornwinder polynomials in the \(q \to 0\) limit, Nonsymmetric Askey-Wilson polynomials and \(Q\)-polynomial distance-regular graphs, A bijection from staircase tableaux to inversion tables, giving some Eulerian and Mahonian statistics, The classical hyperbolic Askey-Wilson dynamics without bound states, Fractional powers of a difference operator, A robust and efficient stepwise regression method for building sparse polynomial chaos expansions, The subconstituent algebra of an association scheme. I, On an involution on the Dyck paths, Another homogeneous \(q\)-difference operator, Asymmetric simple exclusion process with open boundaries and Koornwinder polynomials, Dependence of polynomial chaos on random types of forces of KdV equations, A continuous extension of a \(q\)-analogue of the \(9\)-\(j\) symbols and its orthogonality, Some expansions in basic Fourier series and related topics, Multivariable Askey-Wilson function and bispectrality, The Askey scheme as a four-manifold with corners, On the structure and probabilistic interpretation of Askey-Wilson densities and polynomials with complex parameters, Hyperbolic beta integrals, Bispectrality of multivariable Racah-Wilson polynomials, Linear quadratic regulation of systems with stochastic parameter uncertainties, The continuous symmetric Hahn polynomials found in Ramanujan's lost notebook, New orthogonality relations for the continuous and the discrete \(q\)-ultraspherical polynomials, Markov processes with free-Meixner laws, Askey-Wilson polynomials, quadratic harnesses and martingales, Orthogonality of \(q\)-polynomials for non-standard parameters, SAMBA: sparse approximation of moment-based arbitrary polynomial chaos, A well-posed and stable stochastic Galerkin formulation of the incompressible Navier-Stokes equations with random data, The vector-valued big \(q\)-Jacobi transform, Fixed point theory approach to boundary value problems for second-order difference equations on non-uniform lattices, Uncertainty quantification for systems of conservation laws, On a family of integrals that extend the Askey-Wilson integral, Structure of Jackson integrals of \(BC_n\) type, Branching formula for Macdonald-Koornwinder polynomials, The symmetric \(D_{\omega}\)-semi-classical orthogonal polynomials of class one, Finite-dimensional irreducible modules of the universal Askey-Wilson algebra, Jacobi functions as limit cases of q-ultraspherical polynomials, Central limit theorems for random walks on \({\mathbb{N}}_ 0\) that are associated with orthogonal polynomials, Dunkl shift operators and Bannai-Ito polynomials, Stochastic design optimization: application to reacting flows, Rational interpolation to solutions of Riccati difference equations on elliptic lattices, On Hecke algebras associated with elliptic root systems and the double affine Hecke algebras., An identity of Andrews and the Askey-Wilson integral, Expansion of analytic functions in \(q\)-orthogonal polynomials, Elliptic polynomials orthogonal on the unit circle with a dense point spectrum, A \(q\)-analogue of the 9-\(j\) symbols and their orthogonality, Generalized n-gons and Chebyshev polynomials, Modèles combinatoires pour les polynômes de Meixner, Asymptotics for the \({}_ 4F_ 3\) polynomials, Wilson polynomials and some continued fractions of Ramanujan, An algorithmic proof theory for hypergeometric (ordinary and ``\(q\)) multisum/integral identities, Modeling uncertainty in flow simulations via generalized polynomial chaos., Applications of \(q\)-Taylor theorems, Macdonald polynomials of type \(C_n\) with one-column diagrams and deformed Catalan numbers, Algebraic Heun operator and band-time limiting, Géza Freud, orthogonal polynomials and Christoffel functions. A case study, Josef Meixner: his life and his orthogonal polynomials, The combinatorics of \(q\)-Charlier polynomials, The variation of zeros of certain orthogonal polynomials, Connection and inversion coefficients for basic hypergeometric polynomials, An integral representation and some transformation properties of q-Bessel functions, A \(q\)-analogue of Appell's \(F_ 1\) function, its integral representation and transformations, Limits of some q-Laguerre polynomials, The combinatorics of q-Hermite polynomials and the Askey-Wilson integral, On sieved orthogonal polynomials. IV. Generating functions, Generalized Meixner-Pollaczek polynomials, On summation formulas due to Plana, Lindelöf and Abel, and related Gauss-Christoffel rules. I, Association schemes and quadratic transformations for orthogonal polynomials, MacDonald-Koornwinder moments and the two-species exclusion process, Zeros and orthogonality of the Askey-Wilson polynomials for \(q\) a root of unity, Rational spectral transformations and orthogonal polynomials, Balanced \(_ 3\phi_ 2\) summation theorems for \(U(n)\) basic hypergeometric series, Generalized bibasic hypergeometric series and their \(U(n)\) extensions, Norm behavior and zero distribution for orthogonal polynomials with nonsymmetric weights, Parameter augmentation for basic hypergeometric series. II, Some combinatorics of the hypergeometric series, Combinatorial proofs of symmetry formulas for the generalized hypergeometric series, Product formulas and convolution structure for Fourier-Bessel series, Matrix-valued orthogonal polynomials related to the quantum analogue of \((\mathrm{SU}(2) \times \mathrm{SU}(2), \text{diag})\), Orbit uncertainty propagation and sensitivity analysis with separated representations, Model order reduction of random parameter-dependent linear systems, \(q\)-Taylor theorems, polynomial expansions, and interpolation of entire functions., Free martingale polynomials., Matrix valued Jacobi polynomials, A generalization of a theorem of Bochner, Elliptic beta integrals and modular hypergeometric sums: An overview., An inverse to the Askey-Wilson operator., Transmutation kernels for the little \(q\)-Jacobi function transform., Leonard pairs from 24 points of view., Theory of \(n\)th-order linear general quantum difference equations, Spectral analysis of two doubly infinite Jacobi matrices with exponential entries, Asymptotic and generating relations for the q-Jacobi and //\(4Phi_ 3\) polynomials, Wiman-Valiron theory for a polynomial series based on the Wilson operator, Quasi-orthogonality with applications to some families of classical orthogonal polynomials., A holonomic systems approach to special functions identities, Vertex-IRF transformations, dynamical quantum groups and harmonic analysis, Pseudoisotropic random walks on free groups and semigroups, Associated Wilson polynomials, Solutions of convex Bethe ansatz equations and the zeros of (basic) hypergeometric orthogonal polynomials, The one-quarter class of orthogonal polynomials, A note on polynomial chaos expansions for designing a linear feedback control for nonlinear systems, Error analysis of the Wiener-Askey polynomial chaos with hyperbolic cross approximation and its application to differential equations with random input, Nevanlinna theory of the Askey-Wilson divided difference operator, New results involving a certain class of \(q\)-orthogonal polynomials, Classical partition functions and the \(U(n+1)\) Rogers-Selberg identity, Askey-Wilson functions of the first and second kind: Series and integral representations of \(C^ 2_ n(x ;\beta{}|{}q)+D^ 2_ n(x ;\beta{}|{}q)\), On the Askey-Wilson polynomials, Orthogonal polynomials on systems of non-uniform lattices from compatibility conditions, Quantum affine algebras and holonomic difference equations, \(q\)-Bessel, arbres, et chemins valués. (\(q\)-Bessel functions, trees, and weighted paths), Attribute grammars are useful for combinatorics, An extension of the Borel-Weil construction to the quantum group \(U_ q(n)\), An extension of Askey-Wilson's \(q\)-beta integral and its applications, Structure relations of classical orthogonal polynomials in the quadratic and \(q\)-quadratic variable, An involution on the Dyck paths, Extremal measures for a system of orthogonal polynomials, Enumeration of skew Ferrers diagrams and basic Bessel functions, Determinants with orthogonal polynomial entries, A relativistic hypergeometric function, An asymptotic formula for the Koornwinder polynomials, The symmetric Meixner-Pollaczek polynomials with real parameter, Uncertainty analysis for the steady-state flows in a dual throat nozzle, Variational calculus on \(q\)-nonuniform lattices, Conditional moments of \(q\)-Meixner processes, Parallelogram polyominos with border and Bessel functions, Rogers's \(q\)-ultraspherical polynomials on a quantum \(2\)-sphere, Lie algebras and recurrence relations. III: \(q\)-analogs and quantized algebras, Some remarks on the construction of quantum symmetric spaces, On some classes of polynomials orthogonal on arcs of the unit circle connected with symmetric orthogonal polynomials on an interval, On some limit cases of Askey-Wilson polynomials, Some formulas involving q-Jacobi and related polynomials, Contiguous relations, basic hypergeometric functions, and orthogonal polynomials. I, An easy proof of the Askey-Wilson integral and applications of the method, On the polynomials orthogonal on regular polygons, Deformations of \(W\)-algebras associated to simple Lie algebras, A multidimensional generalization of the Barnes integral and the continuous Hahn polynomial, \(q\)-KP hierarchy, bispectrality and Calogero-Moser systems, On the rate of convergence of the laws of Markov chains associated with orthogonal polynomials, The factorization method for the Askey-Wilson polynomials, A duality of MacDonald-Koornwinder polynomials and its application to integral representations, One-dimensional double Hecke algebra and Gaussian sums, The Terwilliger algebra of a 2-homogeneous bipartite distance-regular graph, Extremal problems, inequalities, and classical orthogonal polynomials., \(q\)-coherent pairs and \(q\)-orthogonal polynomials, Fourier transforms on the quantum \(\text{SU}(1,1)\) group. (With an appendix by Mizan Rahman)., Macdonald polynomials and algebraic integrability, Inverse operators, \(q\)-fractional integrals, and \(q\)-Bernoulli polynomials, An expansion formula for the Askey-Wilson function, A product formula for orthogonal polynomials associated with infinite distance-transitive graphs, Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos, A product formula for the continuous q-Jacobi polynomials, An elementary approach to \(6j\)-symbols (classical, quantum, rational, trigonometric, and elliptic), Short proofs of the elliptic beta integrals, Elliptic grids, rational functions, and the Padé interpolation, Quadratic transformations for a function that generalizes \(_{2}F_{1}\) and the Askey-Wilson polynomials, LU factorizations, \(q = 0\) limits, and \(p\)-adic interpretations of some \(q\)-hypergeometric orthogonal polynomials, Some systems of multivariable orthogonal \(q\)-Racah polynomials, Method of separation of the variables for basic analogs of equations of mathematical physics, Monic bivariate polynomials on quadratic and \(q\)-quadratic lattices, On polynomial chaos expansion via gradient-enhanced \(\ell_1\)-minimization, A bijective proof of a Touchard-Riordan formula, On classical orthogonal polynomials, The quantum group of plane motions and basic Bessel functions, Elliptic Racah polynomials, \(q\)-difference equations for Askey-Wilson type integrals via \(q\)-polynomials, Expansions in Askey-Wilson polynomials via Bailey transform, A Littlewood-Richardson rule for Koornwinder polynomials, Spectra, eigenvectors and overlap functions for representation operators of \(q\)-deformed algebras, Bispectrality and biorthogonality of the rational functions of \(q\)-Hahn type, A new generalization of some quantum integral inequalities for quantum differentiable convex functions, On homogeneous second order linear general quantum difference equations, \(q\)-ultraspherical polynomials for \(q\) a root of unity, Contiguous relations, continued fractions and orthogonality: An \(_ 8\varphi_ 7\) model, Special nonuniform lattice (snul) orthogonal polynomials on discrete dense sets of points, Spectral analysis and the Haar functional on the quantum \(SU(2)\) group, On a general \(q\)-Fourier transformation with nonsymmetric kernels, The \(q\)-version of a theorem of Bochner, Contiguous relations, basic hypergeometric functions, and orthogonal polynomials. III: Associated continuous dual \(q\)-Hahn polynomials, Diagonalization of certain integral operators. II, Yet another basic analogue of Graf's addition formula, \(U(n)\) very-well-poised \(_{10}\phi_ 9\) transformations, Some generating functions for the associated Askey-Wilson polynomials, Octabasic Laguerre polynomials and permutation statistics, On the Askey-Wilson \(q\)-beta integral, Orthogonality of Askey-Wilson polynomials with respect to a measure of Ramanujan type, Solution of the 3D density-driven groundwater flow problem with uncertain porosity and permeability, Some combinatorial and analytical identities, On the complex difference equation of hypergeometric type on non-uniform lattices, Multi-fidelity non-intrusive polynomial chaos based on regression, Elliptic extension of Gustafson's \(q\)-integral of type \(G_2\), On exponential and trigonometric functions on nonuniform lattices, Spectral representation of stochastic field data using sparse polynomial chaos expansions, Stochastic collocation approach with adaptive mesh refinement for parametric uncertainty analysis, A \(q\)-series expansion formula and the Askey-Wilson polynomials, The \(c\)-function expansion of a basic hypergeometric function associated to root systems, Spectral decomposition and matrix-valued orthogonal polynomials, Connection coefficients for basic Harish-Chandra series, Finite-dimensional irreducible modules of the universal Askey-Wilson algebra at roots of unity, Integral representations of the Wilson polynomials and the continuous dual Hahn polynomials, The endless beta integrals, Branching rules for Koornwinder polynomials with one column diagrams and matrix inversions, Moments of Askey-Wilson polynomials, Bispectral extensions of the Askey-Wilson polynomials, Askey-Wilson polynomials by means of a \(q\)-Selberg type integral, Variants of the Rogers-Ramanujan identities, Inversion problems in the \(q\)-Hahn tableau, Algorithms for \(q\)-hypergeometric summation in computer algebra, Some limit transitions between BC type orthogonal polynomials interpreted on quantum complex Grassmannians, Multivariable \(q\)-Racah polynomials, Nevanlinna theory for Jackson difference operators and entire solutions of \(q\)-difference equations, Two-species asymmetric simple exclusion process with open boundaries, Propagating uncertainties in large-scale hemodynamics models via network uncertainty quantification and reduced-order modeling, Wiman-Valiron theory for a polynomial series based on the Askey-Wilson operator, A domain decomposition method for the Navier-Stokes equations with stochastic input, Applications of a generalized \(q\)-difference equation, Segmentation of stochastic images using level set propagation with uncertain speed, A \(q\)-summation formula, the continuous \(q\)-Hahn polynomials and the big \(q\)-Jacobi polynomials, Numerical comparison of three stochastic methods for nonlinear PN junction problems, A note on moment integrals and some applications, An extension of the \(q\)-beta integral with applications, An extended class of orthogonal polynomials defined by a Sturm-Liouville problem, On the parametric uncertainty quantification of the Rothermel's rate of spread model, Computation of Al-Salam Carlitz and Askey-Wilson moments using Motzkin paths, Limit fluctuations for density of asymmetric simple exclusion processes with open boundaries, An aspect of optimal regression design for LSMC, Pitfalls in the frequency response represented onto polynomial chaos for random SDOF mechanical systems, Proper orthogonal decomposition method for multiscale elliptic PDEs with random coefficients, A \(q\)-translation approach to Liu's calculus, A general inverse matrix series relation and associated polynomials. II, Fractional sum and fractional difference on non-uniform lattices and analogue of Euler and Cauchy beta formulas, Connection relations for \(q\)-Taylor polynomial bases, Moments of random matrices and hypergeometric orthogonal polynomials, \(q\)-non uniform difference calculus and classical integral inequalities, A topological origin of quantum symmetric pairs, Stochastic Darboux transformations for quasi-birth-and-death processes and urn models, Robust optimization of nonlinear energy sinks used for mitigation of friction-induced limit cycle oscillations, Double affine Hecke algebra of rank 1 and orthogonal polynomials on the unit circle, Matrix valued Hermite polynomials, Burchnall formulas and non-abelian Toda lattice, \(q\)-type Lidstone expansions and an interpolation problem for entire functions, Numerical solutions of stochastic PDEs driven by arbitrary type of noise, On matrix product ansatz for asymmetric simple exclusion process with open boundary in the singular case, DAHA and skein algebra of surfaces: double-torus knots, Propagation of uncertainties in density-driven flow, A characterization theorem for semi-classical orthogonal polynomials on non-uniform lattices, A tribute to Dick Askey, Explicit matrix inverses for lower triangular matrices with entries involving Jacobi polynomials, Generalizations of Ramanujan's reciprocity formula and the Askey-Wilson integral, Koornwinder polynomials and the \(XXZ\) spin chain, A general matrix series inversion pair and associated polynomials, Connection problems for quantum affine KZ equations and integrable lattice models, Meromorphic functions of finite \(\varphi \)-order and linear Askey-Wilson divided difference equations, A new generalization of q-Hermite-Hadamard type integral inequalities for p, (p-s) and modified (p-s)-convex functions, Boundary confining dualities and Askey-Wilson type \(q\)-beta integrals, Design of linear parameter varying quadratic regulator in polynomial chaos framework, Gaussian active learning on multi-resolution arbitrary polynomial chaos emulator: concept for bias correction, assessment of surrogate reliability and its application to the carbon dioxide benchmark, An algebraic geometric foundation for a classification of second-order superintegrable systems in arbitrary dimension, Interview with Xavier Viennot, Interview with Lauren K. Williams, Dual addition formulas: the case of continuous \(q\)-ultraspherical and \(q\)-Hermite polynomials, Moments of orthogonal polynomials and exponential generating functions, Surrogate-accelerated Bayesian framework for high-temperature thermal diffusivity characterization, \(q\)-fractional integral operators with two parameters, A multiple \(q\)-translation formula and its implications, From the asymmetric simple exclusion processes to the stationary measures of the KPZ fixed point on an interval, Proofs of some partition identities conjectured by Kanade and Russell, A fully Bayesian sparse polynomial chaos expansion approach with joint priors on the coefficients and global selection of terms, Global sensitivity analysis using multi-resolution polynomial chaos expansion for coupled Stokes-Darcy flow problems, 3d exceptional gauge theories and boundary confinement, The general inversion pair and its applications, Spectrum of a q‐deformed Schrödinger equation by means of the variational method, Stationary measure for the open KPZ equation, Some recent progress on the stationary measure for the open KPZ equation, Unnamed Item, Unnamed Item, Unnamed Item, Preface, Enumeration of skew Ferrers diagrams, Elliptic hypergeometric function and $6j$-symbols for the SL(2,$\mathbb{C}$) group, Positivity of Gibbs states on distance-regular graphs, Indeterminate moment problem associated with continuous dual q-Hahn polynomials, Charting the 𝑞-Askey scheme, Remarks on a generalizedq-difference equation, Sieved Ultraspherical Polynomials, Center of the universal Askey-Wilson algebra at roots of unity, Mathematical Properties of Polynomial Dimensional Decomposition, Askey–Wilson operator on entire functions of exponential type, Limit theorems for isotropic random walks on triangle buildings, Distributed Model Predictive Control of Linear Systems with Stochastic Parametric Uncertainties and Coupled Probabilistic Constraints, Laws of large numbers for polynomial hypergroups and some applications, The structure relation for Askey-Wilson polynomials, On the families of polynomials forming a part of the Askey–Wilson scheme and their probabilistic applications, Coupling coefficients for tensor product representations of quantum SU(2), Stable analysis of large-size signals and images by Racah's discrete orthogonal moments, Unnamed Item, A \(\beta\)-convolution theorem associated with the general quantum difference operator, Charting the \(q\)-Askey scheme. II. The \(q\)-Zhedanov scheme, Specializing Koornwinder polynomials to Macdonald polynomials of type \(B\), \(C\), \(D\) and \(BC\), Expansions over Legendre polynomials and infinite double series identities, Comparison of the performance and reliability between improved sampling strategies for polynomial chaos expansion, Multivariate Polynomial Chaos Expansions with Dependent Variables, An Askey-Wilson algebra of rank 2, Orthogonal polynomials with a resolvent-type generating function, Nonterminating transformations and summations associated with some \(q\)-Mellin-Barnes integrals, On Jacobi and continuous Hahn polynomials, A generating function and formulae defining the first-associated Meixner–Pollaczek polynomials, The importance of the Selberg integral, Askey–Wilson polynomials and a double $q$-series transformation formula with twelve parameters, A characterization of Askey-Wilson polynomials, CONTINUOUS ASSOCIATION SCHEMES AND HYPERGROUPS, Discrete semi-classical orthogonal polynomials of class one on quadratic lattices, Dualities in the q‐Askey Scheme and Degenerate DAHA, Generalizations of Rodrigues type formulas for hypergeometric difference equations on nonuniform lattices, A new approach to the theory of classical hypergeometric polynomials, Ladder operators for Szegö polynomials and related biorthogonal rational functions, Stieltjes’ theorem for classical discrete orthogonal polynomials, An Introduction to Special Functions with Some Applications to Quantum Mechanics, Lectures on affine Hecke algebras and Macdonald’s conjectures, Limit relations involving 2-orthogonal polynomials, The Combinatorics of Laguerre, Charlier, and Hermite Polynomials, Numerical simulations for two-dimensional incompressible Navier-Stokes equations with stochastic boundary conditions, Contiguous relations, continued fractions and orthogonality, Generalized Bochner theorem: Characterization of the Askey-Wilson polynomials, The Associated Askey-Wilson Polynomials, Turán inequalities for symmetric Askey-Wilson polynomials, The bi-Poisson process: A quadratic harness, A Right Inverse of the Askey-Wilson Operator, A Paley-Wiener theorem for the Askey-Wilson function transform, The Algebraic Method in Quadrature for Uncertainty Quantification, A generalization of Selberg’s beta integral, Two linear transformations each tridiagonal with respect to an eigenbasis of the other, A quadratic formula for basic hypergeometric series related to Askey-Wilson polynomials, Determination of nodes in numerical integration rules using difference equation, Theta hypergeometric integrals, Ted Chihara and his work on orthogonal polynomials, Solving connection and linearization problems within the Askey scheme and its \(q\)-analogue via inversion formulas, Commuting difference operators with polynomial eigenfunctions, From \(p\)-adic to real Grassmannians via the quantum, Quadratic harnesses, $q$-commutations, and orthogonal martingale polynomials, Addition theorems via continued fractions, Ruijsenaars' hypergeometric function and the modular double of \(\mathcal {U}_{q}(\mathfrak {sl}_{2}(\mathbb C))\), Orthogonal Polynomials, Wilson function transforms related to Racah coefficients, On $BC$ type basic hypergeometric orthogonal polynomials, Stochastic finite element methods for partial differential equations with random input data, A Model Reduction Method for Multiscale Elliptic Pdes with Random Coefficients Using an Optimization Approach, Quadrature rules from finite orthogonality relations for Bernstein-Szegö polynomials, Unnamed Item, Adaptive multi-element polynomial chaos with discrete measure: algorithms and application to SPDEs, Quadratic Transformation Formulas for Basic Hypergeometric Series, On the Askey–Wilson polynomials and a $q$-beta integral, Bispectral commuting difference operators for multivariable Askey-Wilson polynomials, Double affine Hecke algebras and generalized Jones polynomials, Solutions of the Al-Salam–Chihara and allied moment problems, An Eigenvalue Problem for the Associated Askey–Wilson Polynomials, Unnamed Item, A determinant formula associated with the elliptic hypergeometric integrals of type BC n, Elements of the q-Askey scheme in the algebra of symmetric functions, Elements of a function analytic approach to probability, Uncertainty quantification in chemical systems, Error Estimation of Polynomial Chaos Approximations in Transient Structural Dynamics, Numerical simulations for two-dimensional stochastic incompressible Navier-Stokes equations, Gradient system for the roots of the Askey-Wilson polynomial, Recurrence equations and their classical orthogonal polynomial solutions on a quadratic or a q-quadratic lattice, The q‐Bannai–Ito algebra and multivariate (−q)‐Racah and Bannai–Ito polynomials, A Projection Formula for the Askey-Wilson Polynomials and an Application, Bounded Littlewood identities, A family of quantum projective spaces and related $q$-hypergeometric orthogonal polynomials, Formulae for Askey-Wilson moments and enumeration of staircase tableaux, Multiple big q-Jacobi polynomials, Adaptive Wick--Malliavin Approximation to Nonlinear SPDEs with Discrete Random Variables, Properties of some families of hypergeometric orthogonal polynomials in several variables, An operator approach to the Al-Salam–Carlitz polynomials, The spectral matrices of Toda solitons and the fundamental solution of some discrete heat equations., Two operator identities and their applications to terminating basic hypergeometric series and \(q\)-integrals, A bispectral q-hypergeometric basis for a class of quantum integrable models, On the second-order holonomic equation for Sobolev-type orthogonal polynomials, Degenerate Sklyanin algebras, Askey–Wilson polynomials and Heun operators, The Askey–Wilson algebra and its avatars