Orthogonal Polynomials of Several Variables

Preface to the second edition Preface to the first edition 1. Background 2. Orthogonal polynomials in two variables 3. General properties of orthogonal polynomials in several variables 4. Orthogonal polynomials on the unit sphere 5. Examples of orthogonal polynomials in several variables 6. Root systems and Coxeter groups 7. Spherical harmonics associated with reflection groups 8. Generalized classical orthogonal polynomials 9. Summability of orthogonal expansions 10. Orthogonal polynomials associated with symmetric groups 11. Orthogonal polynomials associated with octahedral groups and applications References Author index Symbol index Subject index.

[1]  Abdelkader Intissar,et al.  Spectral properties of the Cauchy transform on L2(C,e−|z|2λ(z)) , 2006 .

[2]  Claus Müller.  Analysis of Spherical Symmetries in Euclidean Spaces , 1997 .

[3]  Rida T. Farouki,et al.  Construction of orthogonal bases for polynomials in Bernstein form on triangular and simplex domains , 2003, Comput. Aided Geom. Des..

[4]  Yuan Xu,et al.  Asymptotics of the Christoffel Functions on a Simplex in Rd , 1999 .

[5]  Intertwining Operator and h-Harmonics Associated With Reflection Groups , 1998, Canadian Journal of Mathematics.

[6]  Kil Hyun Kwon,et al.  Partial differential equations having orthogonal polynomial solutions , 1998 .

[7]  B. C. Griffiths,et al.  Orthogonal Polynomials on the Multinomial Distribution , 1971 .

[8]  Yuan Xu,et al.  Multivariate Gaussian cubature formulae , 1995 .

[9]  W. N. Bailey,et al.  Generalized hypergeometric series , 1935 .

[10]  V M Badkov,et al.  CONVERGENCE IN THE MEAN AND ALMOST EVERYWHERE OF FOURIER SERIES IN POLYNOMIALS ORTHOGONAL ON AN INTERVAL , 1974 .

[11]  JEAN B. LASSERRE,et al.  The existence of Gaussian cubature formulas , 2011, Journal of Approximation Theory.

[12]  Yuan Xu,et al.  Approximation Theory and Harmonic Analysis on Spheres and Balls , 2013 .

[13]  E. Stein,et al.  Introduction to Fourier Analysis on Euclidean Spaces. , 1971 .

[14]  S. Waldron.  Continuous and Discrete Tight Frames of Orthogonal Polynomials for a Radially Symmetric Weight , 2009 .

[15]  P. K. Suetin.  Orthogonal Polynomials in Two Variables , 1999 .

[16]  F. Dai,et al.  A transference theorem for the Dunkl transform and its applications , 2010 .

[17]  L. Vinet,et al.  The singular and the 2:1 anisotropic Dunkl oscillators in the plane , 2013, 1305.2126.

[18]  Robert V. Moody,et al.  Cubature formulae for orthogonal polynomials in terms of elements of finite order of compact simple Lie groups , 2010, Adv. Appl. Math..

[19]  H. S. M. Coxeter,et al.  The Complete Enumeration of Finite Groups of the Form Ri2=(RiRj)kij=1 , 1935 .

[20]  M. Rösler.  POSITIVITY OF DUNKL'S INTERTWINING OPERATOR , 1997 .

[21]  Yuan Xu,et al.  Reconstruction from Radon projections and orthogonal expansion on a ball , 2007, 0705.1984.

[22]  Yuan Xu,et al.  Lagrange Interpolation on Chebyshev Points of Two Variables , 1996 .

[23]  R. Askey.  Some Basic Hypergeometric Extensions of Integrals of Selberg and Andrews , 1980 .

[24]  Yuan Xu,et al.  Localized Polynomial Frames on the Ball , 2006 .

[25]  Yuan Xu.  Summability of Fourier orthogonal series for Jacobi weight functions on the simplex in $\mathbb{R}^d$ , 1998 .

[26]  Mean Cesàro summability of Laguerre expansions and norm estimates with shifted parameter , 1982 .

[27]  M. Tratnik.  Some multivariable orthogonal polynomials of the Askey tableau—Continuous families , 1991 .

[28]  Christopher D. Sogge,et al.  Oscillatory integrals and spherical harmonics , 1986 .

[29]  María Álvarez de Morales,et al.  A matrix Rodrigues formula for classical orthogonal polynomials in two variables , 2009, J. Approx. Theory.

[30]  Géza Freud,et al.  Orthogonal Polynomials , 1971, Series and Products in the Development of Mathematics.

[31]  G. Heckman.  An elementary approach to the hypergeometric shift operators of Opdam , 1991 .

[32]  J. F. van Diejen.  Properties of some families of hypergeometric orthogonal polynomials in several variables , 1996, q-alg/9604004.

[33]  Adriano M. Garsia,et al.  Zonal polynomials and domino tableaux , 1992, Discret. Math..

[34]  Poisson and Cauchy kernels for orthogonal polynomials with dihedral symmetry , 1989 .

[35]  G. Lorentz.  Approximation of Functions , 1966 .

[36]  P. Iliev.  Krall-Jacobi commutative algebras of partial differential operators , 2010, 1011.2599.

[37]  L. Littlejohn,et al.  Orthogonal polynomial eigenfunctions of second-order partial differerential equations , 2001 .

[38]  F. Riesz.  FUNCTIONAL ANALYSIS , 2020, Systems Engineering Principles and Practice.

[39]  Kiyosi Itô.  Complex Multiple Wiener Integral , 1952 .

[40]  J. Humphreys.  Reflection groups and coxeter groups , 1990 .

[41]  K. Kwon,et al.  Orthogonal polynomials in two variables and second-order partial differential equations , 1997 .

[42]  M. V. Tratnik,et al.  Some multivariable orthogonal polynomials of the Askey tableau‐discrete families , 1991 .

[43]  Dennis Stanton,et al.  Invariant theory and tableaux , 1990 .

[44]  Franciszek Hugon Szafraniec,et al.  Three term recurrence relation modulo ideal and orthogonality of polynomials of several variables , 2005, J. Approx. Theory.

[45]  Yuan Xu.  Minimal cubature rules and polynomial interpolation in two variables , 2012, J. Approx. Theory.

[46]  C. Dunkl.  Planar harmonic polynomials of type B , 1999, math/9906041.

[47]  Rudolf Lidl,et al.  Generalizations of the classical Chebyshev polynomials to polynomials in two variables , 1982 .

[48]  Yuan Xu,et al.  Hahn, Jacobi, and Krawtchouk polynomials of several variables , 2013, J. Approx. Theory.

[49]  I. G. MacDonald,et al.  Symmetric functions and Hall polynomials , 1979 .

[50]  Konrad Schmüdgen,et al.  Unbounded Operator Algebras and Representation Theory , 1990 .

[51]  Y. Kanjin.  BANACH ALGEBRA RELATED TO DISK POLYNOMIALS , 1985 .

[52]  Intertwining operators for a degenerate double affine Hecke algebra and multivariable orthogonal polynomials , 1997, q-alg/9706019.

[53]  J. Patera,et al.  Orthogonal polynomials of compact simple Lie groups: branching rules for polynomials , 2010, 1007.4431.

[54]  Jasper V. Stokman,et al.  Orthogonal Polynomials of Several Variables , 2001, J. Approx. Theory.

[55]  R. Beerends,et al.  Certain hypergeometric series related to the root system BC , 1993 .

[56]  Lance L. Littlejohn,et al.  Orthogonal polynomial solutions to ordinary and partial differential equations , 1988 .

[57]  J. McGregor,et al.  Linear Growth Models with Many Types and Multidimensional Hahn Polynomials , 1975 .

[58]  Yuan Xu.  Fourier Series and Approximation on Hexagonal and Triangular Domains , 2008, 0802.0316.

[59]  Charles F. Dunkl,et al.  Orthogonal Polynomials with Symmetry of Order Three , 1984, Canadian Journal of Mathematics.

[60]  Christian Berg,et al.  Rotation invariant moment problems , 1991 .

[61]  Juan Manuel Peña,et al.  Three term recurrence for the evaluation of multivariate orthogonal polynomials , 2010, J. Approx. Theory.

[62]  S. Thangavelu.  Lectures on Hermite and Laguerre expansions , 1993 .

[63]  Limit Transitions for BC Type Multivariable Orthogonal Polynomials , 1997, Canadian Journal of Mathematics.

[64]  S. Karlin,et al.  Determinants of orthogonal polynomials , 1962 .

[65]  H. M. Möller,et al.  Kubaturformeln mit minimaler Knotenzahl , 1976 .

[66]  C. Chevalley.  Invariants of Finite Groups Generated by Reflections , 1955 .

[67]  M. Bertran.  Note on Orthogonal Polynomials in v-Variables , 1975 .

[68]  S. Axler,et al.  Harmonic Function Theory , 1992 .

[69]  Moshe Dubiner.  Spectral methods on triangles and other domains , 1991 .

[70]  Orthogonal Polynomials and Expansions for a Family of Weight Functions in Two Variables , 2010, 1012.5268.

[71]  Yuan Xu,et al.  Boundedness of projection operators and Cesàro means in weighted $L^p$ space on the unit sphere , 2007, 0706.0756.

[72]  J. G. Van der Corput,et al.  Ungleichungen für Polynome und trigonometrische Polynome , 1935 .

[73]  Charles N. Delzell,et al.  Real Algebraic Geometry and Ordered Structures , 2000 .

[74]  Kendall E. Atkinson,et al.  Solving the Nonlinear Poisson Equation on the Unit Disk , 2005 .

[75]  K. Aomoto,et al.  Jacobi polynomials associated with Selberg integrals , 1987 .

[76]  H. Landau.  Moments in mathematics , 1987 .

[77]  H. Coxeter,et al.  Generators and relations for discrete groups , 1957 .

[78]  Mizan Rahman,et al.  Basic Hypergeometric Series , 1990 .

[79]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[80]  Michael Voit,et al.  Markov Processes Related with Dunkl Operators , 1998 .

[81]  Francisco Marcellán,et al.  Orthogonal Polynomials and their Applications , 1988 .

[82]  M.F.E. de Jeu,et al.  The dunkl transform , 1993 .

[83]  H. Rosengren.  Multivariable orthogonal polynomials and coupling coefficients for discrete series representations , 1999 .

[84]  J. Shohat,et al.  The problem of moments , 1943 .

[85]  Persi Diaconis,et al.  An introduction to multivariate Krawtchouk polynomials and their applications , 2013, 1309.0112.

[86]  D. H. Griffel,et al.  An Introduction to Orthogonal Polynomials , 1979 .

[87]  I. G. MacDonald,et al.  Some Conjectures for Root Systems , 1982 .

[88]  A. Zarzo,et al.  Bivariate second-order linear partial differential equations and orthogonal polynomial solutions , 2011, 1101.2640.

[89]  Yuan Xu.  Unbounded commuting operators and multivariate orthogonal polynomials , 1993 .

[90]  8 Lectures on quantum groups and q-special functions , 1996, q-alg/9608018.

[91]  Yuan Xu,et al.  Discrete Fourier Analysis on Fundamental Domain and Simplex of Ad Lattice in d-Variables , 2010 .

[92]  Hans Volkmer,et al.  Expansions in Products of Heine—Stieltjes Polynomials , 1999 .

[93]  Harmonic Polynomials Associated With Reflection Groups , 2000, Canadian Mathematical Bulletin.

[94]  Jeremy Wade,et al.  Ces\`aro Summability of Fourier Orthogonal Expansions on the Cylinder , 2011, 1111.0363.

[95]  HEINRICH LARCHER.  Notes on orthogonal polynomials in two variables , 1959 .

[96]  C. Dunkl.  Orthogonal polynomials on the hexagon , 1987 .

[97]  J. Geronimo,et al.  Bispectrality of Multivariable Racah–Wilson Polynomials , 2007, 0705.1469.

[98]  Yuan Xu,et al.  Orthogonal polynomials and summability in Fourier orthogonal series on spheres and on balls , 2001, Mathematical Proceedings of the Cambridge Philosophical Society.

[99]  I. G. MacDonald,et al.  Symmetric Functions and Orthogonal Polynomials , 1998 .

[100]  T. Koornwinder,et al.  Spherical harmonics and the product of two Jacobi polynomials , 1971 .

[101]  L. Hua.  Harmonic analysis of functions of several complex variables in the classical domains , 1963 .

[102]  E. Cheney.  Introduction to approximation theory , 1966 .

[103]  Siddhartha Sahi,et al.  A recursion and a combinatorial formula for Jack polynomials , 1996 .

[104]  Yuan Xu,et al.  Sub-exponentially localized kernels and frames induced by orthogonal expansions , 2008, 0809.3421.

[105]  W. C. Connett,et al.  Applications of Hypergroups and Related Measure Algebras , 1995 .

[106]  Shayne Waldron.  Orthogonal polynomials on the disc , 2008, J. Approx. Theory.

[107]  M. Kowalski,et al.  The Recursion Formulas for Orthogonal Polynomials in n Variables , 1982 .

[108]  H. Groemer.  Geometric Applications of Fourier Series and Spherical Harmonics , 1996 .

[109]  Yuan Xu,et al.  ORTHOGONAL POLYNOMIALS AND CUBATURE FORMULAE ON SPHERES AND ON BALLS , 1998 .

[110]  H. Rosengren.  Multilinear Hankel forms of higher order and orthogonal polynomials , 1998 .

[111]  E. Opdam.  Harmonic analysis for certain representations of graded Hecke algebras , 1995 .

[112]  Eric M. Opdam,et al.  Some applications of hypergeometric shift operators , 1989 .

[113]  M. Putinar,et al.  Solving moment problems by dimensional extension , 1999 .

[114]  R. A. Silverman,et al.  Special functions and their applications , 1966 .

[115]  Lidia Fernández,et al.  Orthogonal polynomials in two variables as solutions of higher order partial differential equations , 2011, J. Approx. Theory.

[116]  Rudolf Lidl,et al.  Chebyshev polynomials in several variables. , 1972 .

[117]  Rudolf Lidl,et al.  A class of orthogonal polynomials ink variables , 1982 .

[118]  R. Askey.  Orthogonal Polynomials and Special Functions , 1975 .

[119]  T. H. Baker,et al.  Nonsymmetric Jack polynomials and integral kernels , 1996 .

[120]  Yuan Xu,et al.  Orthogonal polynomials and cubature formulae on spheres and on simplices , 1998 .

[121]  S. Ole Warnaar,et al.  The importance of the Selberg integral , 2007, 0710.3981.

[122]  Mourad E. H. Ismail,et al.  Analytic properties of complex Hermite polynomials , 2015 .

[123]  Yuan Xu.  Recurrence formulas for multivariate orthogonal polynomials , 1994 .

[124]  Yuan Xu,et al.  On multivariate orthogonal polynomials , 1993 .

[125]  Multicomponent Calogero Model of $B_N$-Type Confined in Harmonic Potential , 1995, cond-mat/9508012.

[126]  Yuan Xu,et al.  Cubature Formulae and Polynomial Ideals , 1999 .

[127]  Margit Rosler.  Generalized Hermite Polynomials and the Heat Equation for Dunkl Operators , 1997 .

[128]  von F. Zernike.  Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode , 1934 .

[129]  I. Cherednik.  A unification of Knizhnik-Zamolodchikov and Dunkl operators via affine Hecke algebras , 1991 .

[130]  J. R. Higgins.  Completeness and basis properties of sets of special functions: Bibliography , 1977 .

[131]  On a Two-Variable Class of Bernstein–Szegő Measures , 2007, 0710.0035.

[132]  Orthogonal Polynomials of Types A and B and Related Calogero Models , 1997, q-alg/9710015.

[133]  Borislav Bojanov,et al.  Numerical integration over a disc. A new Gaussian quadrature formula , 1998, Numerische Mathematik.

[134]  A. G. Constantine,et al.  Generalized Jacobi Polynomials as Spherical Functions of the Grassmann Manifold , 1974 .

[135]  Lidia Fernández,et al.  Orthogonal polynomials in several variables for measures with mass points , 2009, Numerical Algorithms.

[136]  Yuan Xu,et al.  On two-dimensional definite orthogonal systems and a lower bound for the number of nodes of associated cubature formulae , 1995 .

[137]  Teresa E. Pérez,et al.  Weighted Sobolev orthogonal polynomials on the unit ball , 2013, J. Approx. Theory.

[138]  P. Verlinden.  Common Zeros of Polynomials in Several Variables and Higher Dimensional Quadrature, Pitman Research Notes in Mathematics Series 312, Y. Xu, Longman Scientific & Technical, Essex (U.K.), 1994 , 1995 .

[139]  Yuan Xu,et al.  On zeros of multivariate quasi-orthogonal polynomials and Gaussian cubature formulae , 1994 .

[140]  A new scalar product for nonsymmetric Jack polynomials , 1996, q-alg/9608013.

[141]  Charles F. Dunkl,et al.  Orthogonal polynomials on the sphere with octahedral symmetry , 1984 .

[142]  Henry Jack,et al.  I.—A class of symmetric polynomials with a parameter , 1970, Proceedings of the Royal Society of Edinburgh. Section A. Mathematical and Physical Sciences.

[143]  K. Atkinson,et al.  Spherical Harmonics and Approximations on the Unit Sphere: An Introduction , 2012 .

[144]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[145]  Robert C. Griffiths,et al.  Multivariate Jacobi and Laguerre polynomials, infinite-dimensional extensions, and their probabilistic connections with multivariate Hahn and Meixner polynomials , 2008, 0809.1431.

[146]  P. Etingof.  A uniform proof of the Macdonald-Mehta-Opdam identity for finite Coxeter groups , 2009, 0903.5084.

[147]  E. Stein.  Singular Integrals and Di?erentiability Properties of Functions , 1971 .

[148]  Integrals of polynomials associated with tableaux and the Garsia-Haiman conjecture , 1998 .

[149]  Yuan Xu,et al.  Orthogonal polynomials and partial differential equations on the unit ball , 2007 .

[150]  Kendall E. Atkinson,et al.  Evaluating polynomials over the unit disk and the unit ball , 2013, Numerical Algorithms.

[151]  C. Dunkl.  Reflection groups and orthogonal polynomials on the sphere , 1988 .

[152]  Lance L. Littlejohn,et al.  Sobolev orthogonal polynomials in two variables and second order partial differential equations , 2006 .

[153]  V. Maiorov.  On Best Approximation by Ridge Functions , 1999 .

[154]  T. N. L. Patterson,et al.  Construction of Algebraic Cubature Rules Using Polynomial Ideal Theory , 1978 .

[155]  Lidia Fernández,et al.  Classical orthogonal polynomials in two variables: a matrix approach , 2005, Numerical Algorithms.

[156]  Yuan Xu.  Solutions of three-term relations in several variables , 1994 .

[157]  B. Logan,et al.  Optimal reconstruction of a function from its projections , 1975 .

[158]  Yuan Xu,et al.  Convolution operator and maximal function for the Dunkl transform , 2004, math/0403049.

[159]  András Kroó,et al.  Christoffel functions and universality on the boundary of the ball , 2013 .

[160]  C. Dunkl.  Cube group invariant spherical harmonics and Krawtchouk polynomials , 1981 .

[161]  A. Wünsche.  Generalized Zernike or disc polynomials , 2005 .

[162]  R. Eier,et al.  Tschebyscheffpolynome in einer und zwei Variablen , 1974 .

[163]  Yuan Xu.  Orthogonal Polynomials on the Ball and the Simplex for Weight Functions with Reflection Symmetries , 2001 .

[164]  Charles F. Dunkl,et al.  Topics in harmonic analysis , 1971 .

[165]  Yuan Xu,et al.  On bivariate Gaussian cubature formulae , 1994 .

[166]  D. Jackson.  Formal properties of orthogonal polynomials in two variables , 1936 .

[167]  P. Petrushev.  Approximation by ridge functions and neural networks , 1999 .

[168]  András Kroó,et al.  Christoffel Functions and Universality in the Bulk for Multivariate Orthogonal Polynomials , 2013, Canadian Journal of Mathematics.

[169]  Francesco Calogero,et al.  Solution of the One‐Dimensional N‐Body Problems with Quadratic and/or Inversely Quadratic Pair Potentials , 1971 .

[170]  Multivariable big and little q -Jacobi polynomials , 1997 .

[171]  H. L. Krall,et al.  Orthogonal polynomials in two variables , 1967 .

[172]  C. F. Dunkl,et al.  Dunkl operators for complex reflection groups , 2001 .

[173]  Morris Weisfeld,et al.  Orthogonal polynomials in several variables , 1959, Numerische Mathematik.

[174]  Intertwining operators and polynomials associated with the symmetric group , 1998 .

[175]  Mourad E. H. Ismail,et al.  Theory and Applications of Special Functions , 2005 .

[176]  Lidia Fernández,et al.  New steps on Sobolev orthogonality in two variables , 2010, J. Comput. Appl. Math..

[177]  Yuan Xu,et al.  Decomposition of spaces of distributions induced by tensor product bases , 2009, 0902.2601.

[178]  H. Engels,et al.  Numerical Quadrature and Cubature , 1980 .

[179]  Yuan Xu,et al.  Decomposition of Spaces of Distributions Induced by Hermite Expansions , 2007, 0705.0318.

[180]  Pencho Petrushev,et al.  Localized Tight Frames on Spheres , 2006, SIAM J. Math. Anal..

[181]  Jeremy Wade.  A discretized Fourier orthogonal expansion in orthogonal polynomials on a cylinder , 2010, J. Approx. Theory.

[182]  P. Appell,et al.  Fonctions hypergéométriques et hypersphériques : polynomes d'Hermite , 1926 .

[183]  Analysis on root systems , 1987 .

[184]  Yuan Xu,et al.  Summability of Fourier orthogonal series for Jacobi weight on a ball in ℝ , 1999 .

[185]  V. Man'ko,et al.  Group Theoretical Methods in Physics , 1987 .

[186]  Grigori Olshanski,et al.  Asymptotics of Jack polynomials as the number of variables goes to infinity , 1997 .

[187]  G. C. Shephard,et al.  Finite Unitary Reflection Groups , 1954, Canadian Journal of Mathematics.

[188]  P. R. Milch.  A multi-dimensional linear growth birth and death process , 1965 .

[189]  C. Dunkl,et al.  Vector-Valued Jack Polynomials from Scratch , 2010, 1009.2366.

[190]  Jack polynomials with prescribed symmetry and hole propagator of spin Calogero-Sutherland model , 1998, cond-mat/9806095.

[191]  George Gasper,et al.  Banach Algebras for Jacobi Series and Positivity of a Kernel , 1972 .

[192]  Tom H. Koornwinder,et al.  Product Formulas and Associated Hypergroups for Orthogonal Polynomials on the Simplex and on a Parabolic Biangle , 1997 .

[193]  Paul Neval,et al.  Ge´za Freud, orthogonal polynomials and Christoffel functions. A case study , 1986 .

[194]  Paul M. Terwilliger,et al.  The Rahman polynomials and the Lie algebra sl_3(C) , 2010, 1006.5062.

[195]  Mihai Putinar,et al.  A Dilation Theory Approach to Cubature Formulas. II , 2000 .

[196]  Yuan Xu,et al.  Block Jacobi matrices and zeros of multivariate orthogonal polynomials , 1994 .

[197]  M. V. Tratnik,et al.  Multivariable Meixner, Krawtchouk, and Meixner–Pollaczek polynomials , 1989 .

[198]  T. Koornwinder.  Two-Variable Analogues of the Classical Orthogonal Polynomials , 1975 .

[199]  Manuel Mañas,et al.  Multivariate orthogonal polynomials and integrable systems , 2014, Advances in Mathematics.

[200]  C. Berg.  The multidimensional moment problem and semi-groups , 1987 .

[201]  Orthogonal polynomials and a Dirichlet problem related to the Hilbert transform , 1984 .

[202]  S. Helgason.  Groups and geometric analysis , 1984 .

[203]  A. Nussbaum.  Quasi-analytic vectors , 1965 .

[204]  J. Radon,et al.  Zur mechanischen Kubatur , 1948 .

[205]  Yuan Xu,et al.  Constructing cubature formulae by the method of reproducing kernel , 2000, Numerische Mathematik.

[206]  N. Vilenkin.  Special Functions and the Theory of Group Representations , 1968 .

[207]  M. Noumi.  Macdonald's Symmetric Polynomials as Zonal Spherical Functions on Some Quantum Homogeneous Spaces , 1995, math/9503224.

[208]  P. Petrushev,et al.  Decomposition of Triebel-Lizorkin and Besov spaces in the context of Laguerre expansions , 2008, 0804.4648.

[209]  L. Vinet,et al.  The Dunkl oscillator in the plane: I. Superintegrability, separated wavefunctions and overlap coefficients , 2012, 1212.4459.

[210]  R. J. Beerends.  Chebyshev polynomials in several variables and the radial part of the Laplace-Beltrami operator , 1991 .

[211]  A. Ghanmi.  Operational formulae for the complex Hermite polynomials Hp, q(z, z¯) , 2013 .

[212]  Intertwining operators associated to the group , 1995 .

[213]  Generalized Power Series Expansions for a Class of Orthogonal Polynomials in Two Variables , 1978 .

[214]  A. Bonami,et al.  Sommes de Cesàro et multiplicateurs des développements en harmoniques sphériques , 1973 .

[215]  Rodrigues Formula for the Nonsymmetric Multivariable Hermite Polynomial , 1999, cond-mat/9905178.

[216]  Yuan Xu.  Christoffel functions and Fourier series for multivariate orthogonal polynomials , 1995 .

[217]  Yuan Xu,et al.  Cesàro means of Jacobi expansions on the parabolic biangle , 2008, J. Approx. Theory.

[218]  Luc Vinet,et al.  Algebraic methods and q-special functions , 1999 .

[219]  Lars Vretare,et al.  Formulas for Elementary Spherical Functions and Generalized Jacobi Polynomials , 1984 .

[220]  Harold Exton,et al.  Multiple hypergeometric functions and applications , 1979 .

[221]  Shayne Waldron.  On the Bernstein-Bézier form of Jacobi polynomials on a simplex , 2006, J. Approx. Theory.

[222]  J. Dieudonne,et al.  Special Functions and Linear Representations of Lie Groups , 1980 .

[223]  Yuan Xu,et al.  Asymptotics for orthogonal polynomials and Christoffel functions on a ball , 1996 .

[224]  Mourad E. H. Ismail,et al.  Classical and Quantum Orthogonal Polynomials in One Variable , 2005 .

[225]  The Calogero-Sutherland model and polynomials with prescribed symmetry , 1996, solv-int/9609010.

[226]  G. Gasper.  Orthogonality of Certain Functions with Respect to Complex Valued Weights , 1981, Canadian Journal of Mathematics.

[227]  Charles F. Dunkl,et al.  Integral Kernels with Reflection Group Invariance , 1991, Canadian Journal of Mathematics.

[228]  Yuan Xu,et al.  Orthogonal Polynomials for a Family of Product Weight Functions on the Spheres , 1997, Canadian Journal of Mathematics.

[229]  C. Dunkl.  Boundary Value Problems for Harmonic Functions on the Heisenberg Group , 1986, Canadian Journal of Mathematics.

[230]  R. Stanley.  Some combinatorial properties of Jack symmetric functions , 1989 .

[231]  Gian-Carlo Rota,et al.  Theory and application of special functions , 1977 .

[232]  G. Folland.  Spherical harmonic expansion of the Poisson-Szegő kernel for the ball , 1975 .

[233]  Plamen Iliev,et al.  Meixner polynomials in several variables satisfying bispectral difference equations , 2011, Adv. Appl. Math..

[234]  C. Dunkl.  AN INTERTWINING OPERATOR FOR THE GROUP B 2 , 2006, Glasgow Mathematical Journal.

[235]  Arthur Erdélyi.  Axially Symmetric Potentials and Fractional Integration , 1965 .

[236]  G. Alexits.  Approximation theory , 1983 .

[237]  W. Miller,et al.  Families of orthogonal and biorthogonal polynomials on the N -sphere , 1991 .

[238]  Mourad E. H. Ismail,et al.  Special functions, q-series, and related topics , 1997 .

[239]  Yuan Xu,et al.  A family of Sobolev orthogonal polynomials on the unit ball , 2005, J. Approx. Theory.

[240]  Yuan Xu.  On discrete orthogonal polynomials of several variables , 2004, Adv. Appl. Math..

[241]  P. Forrester,et al.  The Calogero-Sutherland Model and Generalized Classical Polynomials , 1996, solv-int/9608004.

[242]  Yuan Xu,et al.  Discrete orthogonal polynomials and difference equations of several variables , 2007 .

[243]  M. Gekhtman,et al.  On the orthogonal polynomials in several variables , 1994 .

[244]  Yuan Xu,et al.  Discrete Fourier Analysis, Cubature, and Interpolation on a Hexagon and a Triangle , 2007, SIAM J. Numer. Anal..

[245]  B. Braaksma,et al.  Jacobi Polynomials as Spherical Harmonics , 1968 .

[246]  R. Marr,et al.  On the reconstruction of a function on a circular domain from a sampling of its line integrals , 1974 .

[247]  V. Ignatenko.  ON INVARIANTS OF FINITE GROUPS GENERATED BY REFLECTIONS , 1984 .

[248]  Hans Joachim Schmid,et al.  Two-Dimensional Minimal Cubature Formulas and Matrix Equations , 1995, SIAM J. Matrix Anal. Appl..

[249]  George Gasper,et al.  Positive Sums of the Classical Orthogonal Polynomials , 1977 .

[250]  Charles F. Dunkl,et al.  Differential-difference operators associated to reflection groups , 1989 .

[251]  Yuan Xu.  Second order difference equations and discrete orthogonal polynomials of two variables , 2004 .

[252]  Robert C. Griffiths,et al.  A transition density expansion for a multi-allele diffusion model , 1979, Advances in Applied Probability.

[253]  William Douglas Withers,et al.  Generalized Chebyshev polynomials associated with affine Weyl groups , 1988 .

[254]  Allal Ghanmi,et al.  A class of generalized complex Hermite polynomials , 2008 .

[255]  A. Stroud.  Approximate calculation of multiple integrals , 1973 .

[256]  Yuan Xu,et al.  Cesàro Means of Orthogonal Expansions in Several Variables , 2007, 0705.2477.

[257]  Donald St. P. Richards,et al.  Hypergeometric functions on domains of positivity, Jack polynomials, and applications : proceedings of an AMS special session held March 22-23, 1991 in Tampa, Florida , 1992 .

[258]  C. Berg,et al.  Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions , 1984 .

[259]  Yuan Xu,et al.  Summability of orthogonal expansions of several variables , 2003, J. Approx. Theory.

[260]  Mizan Rahman,et al.  A Probablistic Origin for a New Class of Bivariate Polynomials , 2008, 0812.3879.

[261]  E. Görlich,et al.  A convolution structure for Laguerre series , 1982 .

[262]  M. Kowalski,et al.  Orthogonality and Recursion Formulas for Polynomials in n Variables , 1982 .

[263]  Yuan Xu,et al.  Sobolev orthogonal polynomials on a simplex , 2011, 1111.2994.

[264]  Yuan Xu,et al.  Fejér means for multivariate fourier series , 1996 .

[265]  C. Dunkl,et al.  Singular polynomials for finite reflection groups , 1994 .

[266]  B. Sutherland.  Exact results for a quantum many body problem in one-dimension , 1971 .

[267]  R. Griffiths.  Orthogonal polynomials on the negative multinomial distribution , 1975 .

[268]  Luc Lapointe,et al.  Exact operator solution of the Calogero-Sutherland model , 1995, q-alg/9509003.

[269]  Yuan Xu,et al.  Monomial orthogonal polynomials of several variables , 2005, J. Approx. Theory.

[270]  M. Rosier.  Biorthogonal decompositions of the Radon transform , 1995 .

[271]  Yet another proof of the addition formula for Jacobi polynomials , 1977 .

[272]  Hajime Tanaka,et al.  (n+1, m+1)-hypergeometric functions associated to character algebras , 2004 .

[273]  A. D. Shishkin.  Some properties of special classes of orthogonal polynomials in two variables , 1997 .

[274]  Yuan Xu.  Sobolev orthogonal polynomials defined via gradient on the unit ball , 2008, J. Approx. Theory.

[275]  H. M. Möller,et al.  Invariant Integration Formulas for the n-Simplex by Combinatorial Methods , 1978 .

[276]  Charles F. Dunkl,et al.  Computing with Differential-difference Operators , 1999, J. Symb. Comput..

[277]  Yuan Xu.  Weighted Approximation of Functions on the Unit Sphere , 2003 .

[278]  Tom H. Koornwinder,et al.  Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators. III , 1974 .

[279]  R. Littler,et al.  Polynomials biorthogonal to Appell's polynomials , 1974, Bulletin of the Australian Mathematical Society.

[280]  P. Sally,et al.  Harmonic Analysis on Reductive Groups , 1991 .

[281]  Yuan Xu,et al.  Spectral Approximation on the Unit Ball , 2013, SIAM J. Numer. Anal..

[282]  M. Tratnik.  Multivariable biorthogonal Hahn polynomials , 1989 .

[283]  T. MacRobert.  Higher Transcendental Functions , 1955, Nature.

[284]  Yuan Xu,et al.  Integration of the intertwining operator for $h$-harmonic polynomials associated to reflection groups , 1997 .

[285]  Ronald Cools,et al.  On cubature formulae of degree 4k+1 attaining Möller's lower bound for integrals with circular symmetry , 1992 .

[286]  E. Opdam,et al.  Root systems and hypergeometric functions. I , 1988 .

[287]  Orthogonal Symmetric Polynomials Associated with the Quantum Calogero Model , 1995 .

[288]  Orthogonality of the Hi-Jack Polynomials Associated with the Calogero Model , 1997 .

[289]  Plamen Iliev,et al.  A Lie-theoretic interpretation of multivariate hypergeometric polynomials , 2011, Compositio Mathematica.

[290]  Yuan Xu.  Rodrigues type formula for orthogonal polynomials on the unit ball , 2005 .

[291]  T. Constantinescu.  Orthogonal polynomials in several variables. I , 2002 .

[292]  I. Sprinkhuizen-Kuyper.  Orthogonal polynomials in two variables : a further analysis of the polynomials orthogonal over a region bounded by two lines and a parabola , 1976 .

[293]  Yuan Xu.  Complex versus real orthogonal polynomials of two variables , 2013, 1307.7819.

[294]  Francisco Marcellán,et al.  Two variable orthogonal polynomials and structured matrices , 2006, SIAM J. Matrix Anal. Appl..

[295]  C. T. Benson,et al.  Finite Reflection Groups , 1985 .