On the zeros of Meixner and Meixner-Pollaczek polynomials Alta - PowerPoint PPT Presentation

On the zeros of Meixner and Meixner-Pollaczek polynomials Alta Jooste University of Pretoria SANUM 2016, University of Stellenbosch March 22, 2016

Introduction Background Meixner polynomials Meixner-Pollaczek polynomials 1 Introduction 2 Background 3 Meixner polynomials Quasi-orthogonal Meixner polynomials 4 Meixner-Pollaczek polynomials Quasi-orthogonal Meixner-Pollaczek polynomials Alta Jooste On the zeros of Meixner and Meixner-Pollaczek polynomials

Introduction Background Meixner polynomials Meixner-Pollaczek polynomials Orthogonal polynomials To define families of orthogonal polynomials, we use a scalar product � b � f , g � := f ( x ) g ( x ) d φ ( x ) , a positive measure d φ ( x ) supported on [ a , b ], a , b ∈ R . A sequence of real polynomials { p n } N n =0 , N ∈ N ∪ {∞} , is orthogonal on ( a , b ) with respect to d φ ( x ) if � p n , p m � = 0 for m = 0 , 1 , . . . , n − 1 . Alta Jooste On the zeros of Meixner and Meixner-Pollaczek polynomials

Introduction Background Meixner polynomials Meixner-Pollaczek polynomials Orthogonal polynomials If d φ ( x ) is absolutely continuous and d φ ( x ) = w ( x ) dx , � b p n ( x ) p m ( x ) w ( x ) dx = 0 for m = 0 , 1 , . . . , n − 1 a { p n } is orthogonal on ( a , b ) w.r.t. the weight w ( x ) > 0 . If the weight is discrete and w j = w ( j ) , j ∈ L ⊂ Z , � p n ( j ) p m ( j ) w j = 0 for m = 0 , 1 , . . . , n − 1 j ∈ L and the sequence { p n } is discrete orthogonal . In the classical case: L = { 0 , 1 , . . . , N } . Alta Jooste On the zeros of Meixner and Meixner-Pollaczek polynomials

Introduction Background Meixner polynomials Meixner-Pollaczek polynomials Properties of orthogonal polynomials (i) Three-term recurrence relation ( x − B n ) p n − 1 ( x ) = A n p n ( x ) + C n p n − 2 ( x ) , n ≥ 1 p − 1 ( x ) = 0; A n , B n , C n ∈ R ; A n − 1 C n > 0 , n = 1 , 2 , . . . ; (ii) p n has n real, distinct zeros in ( a , b ); (iii) Classic interlacing of zeros The zeros of p n and p n − 1 separate each other: a < x n , 1 < x n − 1 , 1 < x n , 2 < · · · < x n − 1 , n − 1 < x n , n < b . Alta Jooste On the zeros of Meixner and Meixner-Pollaczek polynomials

Introduction Background Meixner polynomials Meixner-Pollaczek polynomials Orthogonality and quasi-orthogonality Polynomials are orthogonal for specific values of their parameters, e.g. Jacobi polynomials ( P α,β ): n orthogonal on [ − 1 , 1] w.r.t w ( x ) = (1 − x ) α (1 + x ) β for α, β > − 1. Deviation from restricted values of the parameters results in zeros departing from interval of orthogonality Question: Do polynomials with ”shifted” parameters retain some form of orthogonality that explains the amount of zeros that remain in the interval of orthogonality? Alta Jooste On the zeros of Meixner and Meixner-Pollaczek polynomials

Introduction Background Meixner polynomials Meixner-Pollaczek polynomials Orthogonality and quasi-orthogonality Polynomials are orthogonal for specific values of their parameters, e.g. Jacobi polynomials ( P α,β ): n orthogonal on [ − 1 , 1] w.r.t w ( x ) = (1 − x ) α (1 + x ) β for α, β > − 1. Deviation from restricted values of the parameters results in zeros departing from interval of orthogonality Question: Do polynomials with ”shifted” parameters retain some form of orthogonality that explains the amount of zeros that remain in the interval of orthogonality? Alta Jooste On the zeros of Meixner and Meixner-Pollaczek polynomials

Introduction Background Meixner polynomials Meixner-Pollaczek polynomials Quasi-orthogonality (Riesz, 1923) A sequence of polynomials { R n } N n =0 is quasi-orthogonal of order k with respect to w ( x ) on [ a , b ] if � b � = 0 m = 0 , 1 , . . . , n − k − 1 for x m R n ( x ) w ( x ) dx � = 0 for m = n − k . a Note that n = k + 1 , k + 2 , . . . . Alta Jooste On the zeros of Meixner and Meixner-Pollaczek polynomials

Introduction Background Meixner polynomials Meixner-Pollaczek polynomials Preliminary results (Shohat, Brezinski et al) Lemma 1 Let { p n } be orthogonal on [ a , b ] with respect to w ( x ). A necessary and sufficient condition for a polynomial R n to be quasi-orthogonal of order k on [ a , b ] with respect to w ( x ), is that R n ( x ) = c 0 p n ( x ) + c 1 p n − 1 ( x ) + · · · + c k p n − k ( x ) where the c i ’s are numbers which can depend on n and c 0 c k � = 0. Lemma 2 If { R n } are real polynomials that are quasi-orthogonal of order k with respect to w ( x ) on an interval [ a , b ], then at least ( n − k ) zeros of R n ( x ) lie in the interval [ a , b ]. Alta Jooste On the zeros of Meixner and Meixner-Pollaczek polynomials

Introduction Background Meixner polynomials Meixner-Pollaczek polynomials Meixner polynomials (Josef Meixner, 1934) n ( − n ) k ( − x ) k (1 − 1 c ) k � M n ( x ; β, c ) = ( β ) n ( β ) k k ! k =0 β, c ∈ R , β � = − 1 , − 2 , . . . , − n + 1 , c � = 0 . ( ) k is the Pochhammer symbol ( a ) k = a ( a + 1) ... ( a + k − 1) , k ≥ 1 ( a ) 0 = 1 when a � = 0 Alta Jooste On the zeros of Meixner and Meixner-Pollaczek polynomials

Introduction Background Meixner polynomials Meixner-Pollaczek polynomials Meixner polynomials For 0 < c < 1 , β > 0, ∞ c j ( β ) j � M m ( j ; β, c ) M n ( j ; β, c ) = 0 , m = 1 , 2 , . . . , n − 1 , j ! j =0 hence the zeros are real, distinct and in (0 , ∞ ) . c j ( β ) j constant on ( j , j + 1) , j = 0 , 1 , 2 , . . . ; j ! zeros are separated by mass points j = 0 , 1 , 2 , . . . . Alta Jooste On the zeros of Meixner and Meixner-Pollaczek polynomials

Introduction Background Meixner polynomials Meixner-Pollaczek polynomials Difference equation Meixner polynomials satisfy the difference equation: � � c ( x + β ) M n ( x +1; β, c ) = n ( c − 1)+ x +( x + β ) c M n ( x ; β, c ) − xM n ( x − 1; β, c ) . Krasikov, Zarkh (2009) : Suppose p n ( x ) satisfies p n ( x + 1) = 2 A ( x ) p n ( x ) − B ( x ) p n ( x − 1) and B ( x ) > 0 for x ∈ ( a , b ), then M ( p n ) > 1 . M ( p n ) ≡ minimum distance between the zeros of p n ( x ). True for Hahn, Meixner, Krawtchouk and Charlier polynomials; Hahn polynomials: Levit (1967); Krawtchouk polynomials: Chihara and Stanton (1990). Alta Jooste On the zeros of Meixner and Meixner-Pollaczek polynomials

Introduction Background Meixner polynomials Meixner-Pollaczek polynomials Difference equation Meixner polynomials satisfy the difference equation: � � c ( x + β ) M n ( x +1; β, c ) = n ( c − 1)+ x +( x + β ) c M n ( x ; β, c ) − xM n ( x − 1; β, c ) . Krasikov, Zarkh (2009) : Suppose p n ( x ) satisfies p n ( x + 1) = 2 A ( x ) p n ( x ) − B ( x ) p n ( x − 1) and B ( x ) > 0 for x ∈ ( a , b ), then M ( p n ) > 1 . M ( p n ) ≡ minimum distance between the zeros of p n ( x ). True for Hahn, Meixner, Krawtchouk and Charlier polynomials; Hahn polynomials: Levit (1967); Krawtchouk polynomials: Chihara and Stanton (1990). Alta Jooste On the zeros of Meixner and Meixner-Pollaczek polynomials

Introduction Background Meixner polynomials Meixner-Pollaczek polynomials As a consequence: Zeros of p n ( x − 1) , p n ( x ) and p n ( x + 1) interlace. 3000 2000 1000 5 10 15 � 1000 Zeros of M 4 ( x − 1 , 5; 0 . 45) , M 4 ( x , 5; 0 . 45) and M 4 ( x + 1 , 5; 0 . 45). Alta Jooste On the zeros of Meixner and Meixner-Pollaczek polynomials

Introduction Background Meixner polynomials Meixner-Pollaczek polynomials Quasi-orthogonal Meixner polynomials Jordaan, Tookos, AJ (2011) Let 0 < β < 1 , 0 < c < 1 . By iterating the recurrence relation M n ( x ; β − 1 , c ) = M n ( x ; β, c ) − nM n − 1 ( x ; β, c ) , we obtain M n ( x ; β − k , c ) = c 0 M n ( x ; β, c )+ c 1 M n − 1 ( x ; β, c )+ · · · + c k M n − k ( x ; β, c ) and M n ( x ; β − k , c ) is quasi-orthogonal of order k for k ∈ { 1 , 2 , . . . n − 1 } ; at least n − k zeros remain in (0 , ∞ ). To obtain relations necessary to prove our results, we use a Maple program by Vidunas. Alta Jooste On the zeros of Meixner and Meixner-Pollaczek polynomials

Introduction Background Meixner polynomials Meixner-Pollaczek polynomials Quasi-orthogonal Meixner polynomials Quasi-orthogonality of order 1 Theorem: If 0 < c < 1 and 0 < β < 1, then the smallest zero of M n ( x ; β − 1 , c ) is negative. 25 20 15 10 5 2 4 6 8 � 5 Zeros of M 3 ( x , 0 . 4; 0 . 6) and M 3 ( x , 0 . 4 − 1; 0 . 6). Interlacing results between the zeros of Quasi-orthogonal Meixner and Meixner polynomials were studied in 2015 [Driver, AJ, submitted 2015] Alta Jooste On the zeros of Meixner and Meixner-Pollaczek polynomials

Introduction Background Meixner polynomials Meixner-Pollaczek polynomials Quasi-orthogonal Meixner polynomials Quasi-orthogonality of order 1 Theorem: If 0 < c < 1 and 0 < β < 1, then the smallest zero of M n ( x ; β − 1 , c ) is negative. 25 20 15 10 5 2 4 6 8 � 5 Zeros of M 3 ( x , 0 . 4; 0 . 6) and M 3 ( x , 0 . 4 − 1; 0 . 6). Interlacing results between the zeros of Quasi-orthogonal Meixner and Meixner polynomials were studied in 2015 [Driver, AJ, submitted 2015] Alta Jooste On the zeros of Meixner and Meixner-Pollaczek polynomials