On certain properties of a perturbed Freud-type weight Abey Kelil - PowerPoint PPT Presentation

On certain properties of a perturbed Freud-type weight Abey Kelil University of Pretoria AIMS-Volkswagen Stiftung Workshop on Introduction to Orthogonal Polynomials and Applications Douala, Cameroon October 09, 2018 1

Plan of the talk 1 Introduction 2 Background 3 A class of perturbed Freud type weight A ‘Generalized Freud’ Weight 4 Conclusions & Future perspectives 2

Introduction Orthogonal polynomials on the real line Let P = span { x k : k ∈ N 0 } be the linear space of polynomials with real coefficients and consider the inner product �· , ·� µ : P × P → R : � b � f , g � µ = f ( x ) g ( x ) d µ ( x ) , f , g ∈ P , supp( µ ) = [ a , b ] ⊆ R . a Let { P n ( x ) } ∞ n =0 be a monic orthogonal polynomial sequence with respect to this inner product: � b � P m , P n � µ = P n ( x ) P m ( x ) d µ ( x ) = ζ n δ mn , ζ n > 0 . a If µ is absolutely continuous; i.e., d µ ( x ) = w ( x ) dx , then � b x k w ( x ) dx < ∞ , µ k = k = 0 , 1 , 2 , . . . , a are said to be moments for a positive weight function w ( x ). 3

Introduction Theorem (Three-term recurrence) Let { P n ( x ) } ∞ n =0 be a sequence of monic orthogonal polynomials on [ a , b ] relative to an inner product �· , ·� w , P 1 ( x ) = x − � xP 0 ( x ) , P 0 ( x ) � w P 0 ( x ) = 1 , . � P 0 ( x ) , P 0 ( x ) � w Then { P n ( x ) } ∞ n =0 satisfies the recursive scheme xP n ( x ) = P n +1 ( x ) + α n P n ( x ) + β n P n − 1 ( x ) , (1) where � � α n = � xP n , P n � = 1 1 x P 2 n ( x ) w ( x ) dx ; β n = x P n ( x ) P n − 1 w ( x ) dx > 0 . � P n � 2 ζ n ζ n − 1 R R Given the positive measure µ (the weight w ), what are the recurrence coefficients? For classical orthogonal polynomials (Jacobi, Hermite, Laguerre), their recurrence coefficients are explicit (cf. Chihara, Szeg´ o, Rainville). 4

Background Semi-classical orthogonal polynomials Classical orthogonal polynomials are characterized by their weight function, which satisfy Pearson’s equation ′ [ σ ( x ) w ( x )] = τ ( x ) w ( x ) , (2) where σ, τ ∈ P with deg( σ ) ≤ 2 and deg( τ ) = 1 and boundary conditions ( σ w ) ( a ) = 0 = ( σ w ) ( b ), whereas semi-classical orthogonal polynomials have (2) with deg( σ ) > 2 or deg( τ ) � = 1 (Hendriksen and van Rossum, 1977). Weight function w ( x ) Parameters σ ( x ) τ ( x ) x λ exp ( − x 2 + tx ) 1 + λ + tx − 2 x 2 Semi-classical Laguerre λ > − 1 x 4 x 4 − tx 2 ) exp ( − 1 − 2 tx − x 3 Freud x , t ∈ R 1 | x | 2 λ +1 exp ( − x 4 + tx 2 ) 2 λ + 2 − 2 tx 2 − x 4 Generalized Freud λ > 0 , x , t ∈ R x These polynomials satisfy a structural relation (Maroni, 1985) � n + r r = deg( σ ) , � ′ σ ( x ) P n +1 ( x ) = A n , j P j ( x ) , . s = max { deg( σ ) − 2 , deg( τ ) − 1 } j = n − s In this case, the recurrence coefficients are usually not explicit and they obey non-linear recurrence relations. 5

Background Certain semi-classical weights obey non-linear recurrence Equations The weight w ( x ) = exp( − x 4 ) on R (cf. Nevai, 1983): Since α n = 0 (symmetry) in the ttrr (1), the coefficient β n obeys � ∞ −∞ x 2 exp( − x 4 ) dx = Γ( 3 4 ) 4 β n ( β n − 1 + β n + β n +1 ) = n ; β 0 = 0 , β 1 = 4 ) . � ∞ Γ( 1 −∞ exp( − x 4 ) dx The semiclassical Laguerre w ν ( x ) = x ν exp( − x 2 + tx ) , ν > − 1 , x ∈ R + : (W. Van Assche, L. Boelen (2011) and ( P. Clarkson, K. H. Jordaan (2014))  (2 α n − t ) (2 α n − 1 − t ) = (2 β n − n ) (2 β n − n − ν ) ,  β n  2 β n + 2 β n +1 − α n (2 α n − t ) = 2 n + 1 + ν. � − x 4 + tx 2 � The Freud weight w ( x ) = exp , x ∈ R (Freud (1976)): n = − 2 t + 4 [ β n +1 + β n + β n − 1 ] , β 0 = 0 , (3) β n (3) is known as Shohat-Freud’s (‘String’ or Discrete Painlev´ e) equation. � n � 1 / 2 � � 24 n 2 + O ( n − 4 ) 1 Asymptotic behavior : β n = 1 + ( Nevai, 1984). 12 6

Background The link to Painlev´ e equations Some history : The first non-linear recurrence equation - Shohat (1930’s) and Laguerre , Freud (late 70’s) and very recently recognized as discrete Painlev´ e equations by Fokas, Its, and Kitaev . Work by Magnus (relation between discrete and continuous Painlev´ e equations), Witte , Clarkson , Van Assche , Nijhoff , Spicer , Chen and Ismail extended theory with some more examples. The Painlev´ e equations are a chapter in ‘ DLMF ’. d 2 q P II ( α ) : d 2 q dt 2 = 6 q 2 + z , z ∈ C ( P I ); dz 2 = 2 q 3 + zq + α, � dq � 2 P III ( α, β, γ, δ ) : d 2 q dz 2 = 1 − 1 dq dz + 1 z ( α q 2 + β ) + γ q 3 + δ q , q dz z � dq � 2 P IV ( α, β ) : d 2 q dz 2 = 1 + 3 2 q 3 + 4 zq 2 + 2( z 2 − α ) q + β (4) q , 2 q dz where α, β, γ, δ are constants and z ∈ C . Some discrete Painlev´ e equations: x n +1 + x n + x n − 1 = z n + γ ( − 1) n (d- P I ) + σ x n x n +1 + x n − 1 = x n z n + γ (d- P II ) 1 − x 2 n � n − κ 2 � � n − µ 2 � x 2 x 2 (d- P IV ) ( x n +1 + x n ) ( x n + x n − 1 ) = ( x n + z n ) 2 − γ 2 7

Background Certain semi-classical weights giving rise to (discrete) Painlev´ e equations w ( x ) = | x | ̺ exp( − x 4 ) , ̺ > − 1 on R is related to dP I (Magnus, 1986). 4 x 4 + tx 2 ) , t ∈ R on R is related to dP I (Magnus, 1995). w ( x ; t ) = exp( − 1 In the continuous sense, w ( x ) = x λ exp( − x 2 + tx ) on R + is related to P IV (Galina et. al, Clarkson et.al). w ( x , t ) = (1 − x ) α (1 + x ) β exp( − tx ) , α, β > − 1 , x ∈ [ − 1 , 1] , t ∈ R related to P V , (Basor, Chen and Ehrhardt (2009)). w ( x , t ) = x α exp( − x − t / x ) , α > − 1 , x ∈ R + , related to P III , Chen & Its (2010). Some observations Solutions of Painlev´ e are sometimes not directly the recurrence coefficients, but functions of these, with extra terms and/or changes of variable. For e.g. (after Clarkson & Jordaan), if w ( x ; t ) = x λ exp � − x 2 + tx � , λ > − 1 then the function q n ( z ) = 2 α n ( t ) + t , with z = 1 2 t satisfies P IV in z & ( A , B ) = (2 n + λ + 1 , − 2 λ 2 ). The solutions obtained from deforming OPs are typically not ‘generic’, but with very specific values of the parameters . Special function solutions of P IV are expressed in terms of parabolic cylinder functions. 8

Background Ladder relations help to obtain nonlinear equations Assume the weight w vanishes at the endpoints of [ a , b ] ⊆ R . The ladder operators for the polynomials P n ( z ) (cf. Chen & Ismail, Van Assche et.al) are given by � d � dz + B n ( z ) � P n ( z ) = β n A n ( z ) P n − 1 ( z ) (5) � d dz − B n ( z ) − ν ′ ( z ) � P n − 1 ( z ) = − A n − 1 ( z ) P n ( z ) where ν ( x ) = − log w ( x ) , since w ( x ) > 0 , x ∈ [ a , b ] ⊆ R and the coefficients in (5) are given by � � � ∞ ν ′ ( z ) − ν ′ ( y ) A n ( z ) = 1 P 2 n ( y ) w ( y ) dy , h n z − y −∞ � ∞ � � 1 ν ′ ( z ) − ν ′ ( y ) B n ( z ) = P n ( y ) P n − 1 ( y ) w ( y ) dy , h n − 1 z − y −∞ Note : We can calculate A n ( z ) and B n ( z ) without explicitly knowing the polynomials other than the weight function. 9

Background Compatibility conditions (cf. Chen & Ismail, Magnus, Van Assche et.al), For the ladder operators, the associated compatibility conditions are Lemma The functions A n ( z ) and B n ( z ) satisfy B n +1 ( z ) + B n ( z ) = ( z − α n ) A n ( z ) − ν ′ ( z ) 1 + ( z − α n ) ( B n +1 ( z ) − B n ( z )) = β n +1 A n +1 ( z ) − β n A n − 1 ( z ) , valid for all z ∈ C ∪ {∞} . n − 1 � The functions A n ( z ) , B n ( z ) and A k ( z ) satisfy the identity k =0 n − 1 � B 2 n ( z ) + ν ′ ( z ) B n ( z ) + A k ( z ) = β n A n ( z ) A n − 1 ( z ) . k =0 10

A class of perturbed Freud type weight Extract from Digital Library of Mathematical Functions § 18.32 Orthogonal polynomials with Respect to Freud Weights A Freud weight is a weight function of the form 18.32.1 w ( x ) = exp ( − Q ( x )) , −∞ < x < ∞ where Q ( x ) is real, even, non-negative, and continuously differentiable. Of special interest are the cases Q ( x ) = x 2 m , m = 1 , 2 , . . . . No explicit expressions for the corresponding OP’s are available . However, for asymptotic approximations in terms of elementary functions for the OP’s, and also for their largest zeros, see Levin and Lubinsky [2001]. For a uniform asymptotic expansion in terms of Airy functions for the OP’s in the case x 4 see Bo and Wong [1999]. Our interest: Can we obtain concise formulations for the recurrence coefficients as well as the polynomials that are orthogonal with respect to the generalized Freud weight? What more non-linear recurrence relations can one obtain? 11