A generalized sextic Freud weight

ABSTRACT We discuss the recurrence coefficients of orthogonal polynomials with respect to a generalized sextic Freud weight with parameters and . We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of generalized hypergeometric functions . We derive a nonlinear discrete as well as a system of differential equations satisfied by the recurrence coefficients and use these to investigate their asymptotic behaviour. We conclude by highlighting a fascinating connection between generalized quartic, sextic, octic and decic Freud weights when expressing their first moments in terms of generalized hypergeometric functions.

[1]  Yang Chen,et al.  On semiclassical orthogonal polynomials associated with a Freud‐type weixght , 2020, Mathematical Methods in the Applied Sciences.

[2]  Arieh Iserles,et al.  Orthogonal Systems with a Skew-Symmetric Differentiation Matrix , 2019, Foundations of Computational Mathematics.

[3]  W. Assche,et al.  Three-fold symmetric Hahn-classical multiple orthogonal polynomials , 2018, Analysis and Applications.

[4]  Xiang-Ke Chang,et al.  Moment modification, multipeakons, and nonisospectral generalizations , 2018, Journal of Differential Equations.

[5]  Yang Chen,et al.  Asymptotic gap probability distributions of the Gaussian unitary ensembles and Jacobi unitary ensembles , 2018, 1801.00521.

[6]  Kerstin Jordaan,et al.  Properties of generalized Freud polynomials , 2016, J. Approx. Theory.

[7]  W. Assche.  Orthogonal Polynomials and Painlevé Equations , 2017 .

[8]  Francisco Marcellán,et al.  On polynomials associated with an Uvarov modification of a quartic potential Freud-like weight , 2015, Appl. Math. Comput..

[9]  P. Clarkson,et al.  A Generalized Freud Weight , 2015, 1510.03772.

[10]  József Szabados,et al.  A family of nonlinear difference equations: Existence, uniqueness, and asymptotic behavior of positive solutions , 2013, J. Approx. Theory.

[11]  H. Wang,et al.  Monotonicity of zeros of polynomials orthogonal with respect to an even weight function , 2014 .

[12]  P. Clarkson,et al.  The Relationship Between Semiclassical Laguerre Polynomials and the Fourth Painlevé Equation , 2013, 1301.4134.

[13]  Kerstin Jordaan,et al.  Convexity of the zeros of some orthogonal polynomials and related functions , 2008, J. Comput. Appl. Math..

[14]  M. Ismail,et al.  Classical and Quantum Orthogonal Polynomials in One Variable: Bibliography , 2005 .

[15]  Yang Chen,et al.  Painlevé IV and degenerate Gaussian unitary ensembles , 2006, math-ph/0606064.

[16]  Rene F. Swarttouw,et al.  Orthogonal Polynomials , 2005, Series and Products in the Development of Mathematics.

[17]  W. Assche.  Discrete Painlev\'e equations for recurrence coefficients of orthogonal polynomials , 2005, math/0512358.

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

[19]  A. S. Fokas,et al.  The Isomonodromy Approach to Matrix Models in 2 D Quantum Gravity , 2004 .

[20]  Steven B. Damelin,et al.  Asymptotics of recurrence coefficients for orthonormal polynomials on the line - Magnus's method revisited , 2004, Math. Comput..

[21]  P. Clarkson,et al.  The second Painlevé equation, its hierarchy and associated special polynomials , 2003 .

[22]  Peter A. Clarkson,et al.  Painlevé equations: nonlinear special functions , 2003 .

[23]  Doron S. Lubinsky,et al.  Orthogonal Polynomials for Exponential Weights , 2001 .

[24]  Mourad E. H. Ismail,et al.  An electrostatics model for zeros of general orthogonal polynomials , 2000 .

[25]  Nalini Joshi,et al.  The discrete first, second and thirty-fourth Painlevé hierarchies , 1999 .

[26]  A. Magnus.  Symmetries and Integrability of Difference Equations: Freud's equations for orthogonal polynomials as discrete Painlevé equations , 1999 .

[27]  Robert Vein,et al.  Determinants and their applications in mathematical physics , 1998 .

[28]  A. Ramani,et al.  From continuous Painlevé IV to the asymmetric discrete Painlevé I , 1998 .

[29]  Yang Chen,et al.  Ladder operators and differential equations for orthogonal polynomials , 1997 .

[30]  Nicolai A. Kudryashov,et al.  The first and second Painlevé equations of higher order and some relations between them , 1997 .

[31]  Hrushikesh Narhar Mhaskar,et al.  Introduction to the theory of weighted polynomial approximation , 1997, Series in approximations and decompositions.

[32]  P. Clarkson,et al.  Bäcklund Transformations and Solution Hierarchies for the Third Painlevé Equation , 1995 .

[33]  W. H. Reid,et al.  Integral representations for products of Airy functions , 1995 .

[34]  Alphonse P. Magnus,et al.  Painleve´-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials , 1995 .

[35]  Athanassios S. Fokas,et al.  Discrete Painlevé equations and their appearance in quantum gravity , 1991 .

[36]  A. Kitaev,et al.  Mathematical Aspects of the Non-Perturbative 2d Quantum Gravity , 1990 .

[37]  A. Draux.  On quasi-orthogonal polynomials , 1990 .

[38]  G. Parisi,et al.  A non-perturbative ambiguity free solution of a string model , 1990 .

[39]  H. Mhaskar,et al.  A proof of Freud's conjecture for exponential weights , 1988 .

[40]  P. Maroni,et al.  Prolégomènes à l'étude des polynômes orthogonaux semi-classiques , 1987 .

[41]  Rong-Chyu Sheen.  Plancherel-Rotach-type asymptotics for orthogonal polynomials associated with exp(- x 6 /6) , 1987 .

[42]  Kazuo Okamoto.  Studies on the Painleve equations II , 1987 .

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

[44]  Attila Máté,et al.  Asymptotics for the Zeros of Orthogonal Polynomials associated with Infinite Intervals , 1986 .

[45]  A. Magnus.  On Freud's equations for exponential weights , 1986 .

[46]  Géza Freud,et al.  On the greatest zero of an orthogonal polynomial , 1986 .

[47]  T. Zaslavsky,et al.  Asymptotic Expansions of Ratios of Coefficients of Orthogonal Polynomials with Exponential Weights , 1985 .

[48]  Alphonse P. Magnus,et al.  A Proof of Freud Conjecture About the Orthogonal Polynomials Related To [x]rho-exp[-x2m], for Integer-m , 1985 .

[49]  Y. Murata,et al.  Rational solutions of the second and the fourth Painlevé equations , 1985 .

[50]  E. Hendriksen,et al.  Semi-classical orthogonal polynomials , 1985 .

[51]  E. Rakhmanov,et al.  ON ASYMPTOTIC PROPERTIES OF POLYNOMIALS ORTHOGONAL ON THE REAL AXIS , 1984 .

[52]  Paul Nevai,et al.  Orthogonal polynomials and their derivatives, I , 1984 .

[53]  Rong-Chyu Sheen.  Orthogonal polynomials associated with EXP(-x⁶/6) / , 1984 .

[54]  P. Nevai.  Two of My Favorite Ways of Obtaining Asymptotics for Orthogonal Polynomials , 1984 .

[55]  A. Newell,et al.  Monodromy- and spectrum-preserving deformations I , 1980 .

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

[57]  Mark J. Ablowitz,et al.  Exact Linearization of a Painlevé Transcendent , 1977 .

[58]  E. Pavlikova.  Higher monotonicity properties of certain Sturm-Liouville functions. V , 1977, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[59]  Mark Kac,et al.  On an Explicitly Soluble System of Nonlinear Differential Equations Related to Certain Toda Lattices , 1975 .

[60]  Martin E. Muldoon,et al.  Higher monotonicity properties of certain Sturm-Liouville functions , 1963, Canadian Journal of Mathematics.

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

[62]  J. Shohat.  A differential equation for orthogonal polynomials , 1939 .