Characterizing properties of generalized Freud polynomials Abey - PowerPoint PPT Presentation

Characterizing properties of generalized Freud polynomials Abey Kelil Supervisor: Prof. Kerstin Jordaan University of Pretoria, South Africa Joint work with Prof. Peter Clarkson University of Kent, UK 22 - 24 March, 2016 SANUM Stellenboch University Characterizing properties of generalized Freud polynomials 1 / 22

Outline Outline Semi-classical orthogonal polynomials 1 Semi-classical Laguerre polynomials Symmetrization The link to Painlev´ e equations 2 Generalized Freud polynomials 3 Properties of generalized Freud polynomials 4 Moments Differential-difference equation Second order linear ODE Recurrence coefficient Characterizing properties of generalized Freud polynomials 2 / 22

Outline Extract from Digital Library of Mathematical Functions § 18.32 OP’s 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] and Nevai [1986]. For a uniform asymptotic expansion in terms of Airy functions for the OP’s in the case x 4 see Bo and Wong [1999]. Characterizing properties of generalized Freud polynomials 3 / 22

Semi-classical orthogonal polynomials Orthogonal polynomial sequences Given { µ n } ∈ R , we define the moment functional L : x n → µ n on the linear space of polynomials P . Assume µ 0 = L (1) = 1. The inner product �· , ·� for the functional L is given by � P m ( x ) , P n ( x ) � = L ( P m ( x ) P n ( x )) Monic polynomials { P n ( x ) } ∞ n =0 orthogonal w.r.t. a moment functional L related to an absolutely continuous Borel measure µ on R ; d µ ( x ) = w ( x ) dx ; w ( x ) > 0 : � L ( P m ( x ) P n ( x )) = � P m , P n � = P m ( x ) P n ( x ) d µ ( x ) = h n δ mn , R where the normalization constant h n > 0 and δ mn is the Kronecker delta. Characterizing properties of generalized Freud polynomials 4 / 22

Semi-classical orthogonal polynomials Monic orthogonal polynomials P n ( x ) satisfy P − 1 ( x ) = 0 , P 0 ( x ) = 1 , P n +1 ( x ) = ( x − α n ) P n ( x ) − β n P n − 1 ( x ) , α n = � xP n , P n � � P n , P n � � P n , P n � ∈ R ; β n = � P n − 1 , P n − 1 � > 0 , β 0 = 1 , n ∈ N 0 , and the constant: n h n = � P n , P n � = � P n � 2 = � β j . j =1 To construct P n ( x ) for L : � � µ 0 µ 1 · · · µ n � � � � · · · µ 1 µ 2 µ n +1 � � 1 . . . � ... � . . . , ∆ n := det( µ i + j ) n P n ( x ) = i , j =0 > 0 . � � . . . ∆ n − 1 � � � � µ n − 1 µ n · · · µ 2 n − 1 � � � x n � 1 · · · x � � Characterizing properties of generalized Freud polynomials 5 / 22

Semi-classical orthogonal polynomials Classical orthogonal polynomials Classical weights satisfy Pearson’s equation d dx ( σ w ) = τ w , (2.1) with deg( σ ) ≤ 2 and deg( τ ) = 1 , and bcs : σ ( x ) w ( x ) = 0 for x = a and x = b . p n w ( x ) σ ( x ) τ ( x ) interval exp ( − x 2 ) Hermite 1 − 2 x ( −∞ , ∞ ) x α exp ( − x ) , α > − 1 Laguerre x 1 + α − x (0 , ∞ ) (1 − x ) α (1 + x ) β 1 − x 2 Jacobi β − α − (2 + α + β ) x [ − 1 , 1] p n ’s are solutions of Lp n = λ n p n where L is a second order differential operator (Sturm-Liouville) [Bochner, 1929] Structural relation: n − r +1 ′ = � σ ( x ) ( p n ( x )) A n , j p j ( x ) , r = deg( σ ) (2.2) j = n − 1 (2.2) together with xp n = a n +1 p n +1 + b n p n + a n p n − 1 , yields a first order recurrence equation for the recurrence coefficients a n and b n , which can be solved explicitly. Characterizing properties of generalized Freud polynomials 6 / 22

Semi-classical orthogonal polynomials Semi-classical orthogonal polynomials Semi-classical weights satisfy Pearson’s equation (2.1) with deg( σ ) > 2 or deg( τ ) > 1. [Hendriksen, van Rossum, 1977] weight w ( x ) parameters σ ( x ) τ ( x ) exp ( − x 4 ) − 4 x 3 - - 1 exp ( − 1 3 x 3 + tx ) t − x 2 Airy t > 0 1 x λ exp ( − x 2 + tx ) 1 + λ + tx − 2 x 2 Semi-classical Laguerre λ > − 1 x 4 x 4 − tx 2 ) − 2 tx − x 3 exp ( − 1 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 p n does not satisfy Sturm-Liouville differential equation. Structural relation n − r +1 � r = deg( σ ) , ′ � σ ( x ) p n ( x ) = A n , j p j ( x ) , . (2.3) s = max { deg( σ ) − 1 , deg( τ ) } j = n − s (2.3) and xp n ( x ) = a n +1 p n +1 ( x ) + b n p n ( x ) + a n p n − 1 ( x ) , n ≥ 0 , yield second or higher order (non-linear) equations for the recurrence coefficients a n and b n . Example: w ( x ) = exp( − x 4 ) on R [Nevai, 1983]: b n = 0 (symmetry); 1 = Γ( 3 4 ) � ∞ 4 a 2 a 2 n + a 2 n + a 2 a 2 � � = n , n ≥ 2 , a 0 = 1 , 4 ) , where t z − 1 e − t dt . Γ( z ) = n n Γ( 1 0 Characterizing properties of generalized Freud polynomials 7 / 22

The link to Painlev´ e equations 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. Some Discrete Painlev´ e eqns: 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 � x 2 n − κ 2 � � x 2 n − µ 2 � (d- P IV ) ( x n +1 + x n ) ( x n + x n − 1 ) = ( x n + z n ) 2 − γ 2 The continuous fourth Painlev´ e equation ( P IV ) � 2 d 2 q dz 2 = 1 � d q + 3 2 q 3 + 4 zq 2 + 2( z 2 − A ) q + B q , (3.1) 2 q dz where A and B are constants, which are expressed in terms of parabolic cylinder (Hermite-Weber) functions. Characterizing properties of generalized Freud polynomials 8 / 22

The link to Painlev´ e equations Semi-classical Laguerre Theorem (LB-WVA, 2012) The coefficients α n ( t ) and β n ( t ) in the three-term recurrence L ( ν ) n +1 ( x ; t ) = ( x − α n ) L ( ν ) n ( x ; t ) − β n L ( ν ) n − 1 ( x ; t ); associated with the the semi-classical Laguerre w ν ( x ) = x ν exp( − x 2 + tx ) , ν > − 1 , x ∈ R + are: (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. For explicit formulations of α n and β n , see [CJ, 2014]. Characterizing properties of generalized Freud polynomials 9 / 22

The link to Painlev´ e equations Discrete Painlev´ e and more semi-classical weights Question: What semi-classical weights are related to discrete Painlev´ e equations? Which discrete Painlev´ e equations do we obtain? w ( x ) = | x | ̺ exp( − x 4 ) , ̺ > − 1 on R is related to (d- P I ). [Magnus, 1986]. w ( x ) = x α exp( − x 2 ) , α > − 1 on R + is related to (d- P IV ) [Sonin-type]. w ( x ; t ) = x α exp( − x 2 + tx ) , α > − 1 on R + is related to ( P IV ) [GF-WVA-LZ, 2011]. W λ ( x ) = | x | 2 λ +1 exp( − x 4 + tx 2 ) , λ > − 1 , t , x ∈ R related to (d- P I ) and continuous ( P IV ) [LB-WVA, 2011, GF-WVA-LZ, 2012]. Characterizing properties of generalized Freud polynomials 10 / 22

The link to Painlev´ e equations The recurrence coefficient related to Painlev´ e IV Theorem. (LB-WVA, 2011; GF-WVA-LZ, 2012) The recurrence coefficients β n ( t ; λ ) in the three term recurrence xS n ( x ; t ) = S n +1 ( x ; t ) + β n ( t ; λ ) S n − 1 ( x ; t ) associated with the weight W λ satisfy the equation � 2 d 2 β n � d β n 1 2 A n ) β n + B n 8 t 2 − 1 2 β 3 n − t β 2 + 3 n + ( 1 dt 2 = (3.2) , 2 β n dt 16 β n where the parameters A n and B n are given by A 2 n = − 2 λ − n − 1 , A 2 n +1 = λ − n , B 2 n = − 2 n 2 , B 2 n +1 = − 2( λ + n + 1) 2 . Further β n ( t ) satisfies the non-linear difference equation 2 t + 2 n + (2 λ + 1)[1 − ( − 1) n ] β n +1 + β n + β n − 1 = 1 , –discrete P I ( dP I ). 8 β n Remark: (3.2) ≡ P IV via the transformation β n ( t ; λ ) = 1 2 w ( z ) , with z = − 1 2 t. Hence β 2 n ( t ; λ ) = 1 z ; − 2 λ − n − 1 , − 2 n 2 � β 2 n +1 ( t ; λ ) = 1 z ; λ − n , − 2( λ + n + 1) 2 � � � 2 w ; 2 w , with z = − 1 2 t, where w ( z ; A , B ) satisfies P IV ( 3.1). Characterizing properties of generalized Freud polynomials 11 / 22

The link to Painlev´ e equations Our interest: What more can be said about properties of polynomials orthogonal with respect to W ( x ) = | x | 2 λ +1 exp − x 4 + tx 2 � � ? Characterizing properties of generalized Freud polynomials 12 / 22