Laguerre Polynomials and Interlacing of Zeros Kathy Driver - PowerPoint PPT Presentation

Laguerre Polynomials and Interlacing of Zeros Kathy Driver University of Cape Town SANUM Conference Stellenbosch University Kathy Driver Laguerre Polynomials 22-24 March 2016 1 / 16

Laguerre Polynomials and Interlacing of Zeros Kathy Driver Laguerre Polynomials 22-24 March 2016 2 / 16

Laguerre Polynomials and Interlacing of Zeros Joint work with Martin Muldoon Kathy Driver Laguerre Polynomials 22-24 March 2016 2 / 16

Laguerre Polynomials and Interlacing of Zeros Joint work with Martin Muldoon Overview of Talk Kathy Driver Laguerre Polynomials 22-24 March 2016 2 / 16

Laguerre Polynomials and Interlacing of Zeros Joint work with Martin Muldoon Overview of Talk Sharpness of t- interval where zeros of Laguerre polynomials L ( α ) and n L ( α + t ) are interlacing. α > − 1 , t > 0 . Askey Conjecture n − k Kathy Driver Laguerre Polynomials 22-24 March 2016 2 / 16

Laguerre Polynomials and Interlacing of Zeros Joint work with Martin Muldoon Overview of Talk Sharpness of t- interval where zeros of Laguerre polynomials L ( α ) and n L ( α + t ) are interlacing. α > − 1 , t > 0 . Askey Conjecture n − k Breakdown of interlacing of zeros of L ( α ) and L ( α ) n − 1 when − 2 < α < − 1 . n Add one point to restore interlacing. Quasi-orthogonal order 1 case. Kathy Driver Laguerre Polynomials 22-24 March 2016 2 / 16

Laguerre Polynomial L ( α ) n Kathy Driver Laguerre Polynomials 22-24 March 2016 3 / 16

Laguerre Polynomial L ( α ) n Laguerre polynomial L ( α ) defined by n n � ( − x ) k � n + α L ( α ) � n ( x ) = . (1) n + k k ! k =0 Kathy Driver Laguerre Polynomials 22-24 March 2016 3 / 16

Laguerre Polynomial L ( α ) n Laguerre polynomial L ( α ) defined by n n � ( − x ) k � n + α L ( α ) � n ( x ) = . (1) n + k k ! k =0 For α > − 1, sequence { L ( α ) n ( x ) , n = 0 , 1 , . . . } orthogonal with respect to x α e − x on (0 , ∞ ). All n zeros of L α n are real, distinct and positive. ∞ L n ( x ) L m ( x ) x α e − x dx = 0 � for n � = m , 0 ∞ L n L n x α e − x dx � = 0 � 0 α > − 1 necessary for convergence of integral each m , n . Kathy Driver Laguerre Polynomials 22-24 March 2016 3 / 16

Interlacing of zeros Orthogonal sequence { p n } ∞ n =0 , zeros of p n and p n − 1 are interlacing: x 1 , n < x 1 , n − 1 < x 2 , n < x 2 , n − 1 · · · < x n − 1 , n < x n − 1 , n − 1 < x n , n Kathy Driver Laguerre Polynomials 22-24 March 2016 4 / 16

Interlacing of zeros Orthogonal sequence { p n } ∞ n =0 , zeros of p n and p n − 1 are interlacing: x 1 , n < x 1 , n − 1 < x 2 , n < x 2 , n − 1 · · · < x n − 1 , n < x n − 1 , n − 1 < x n , n p , q real polynomials, real, simple, disjoint zeros,deg(p) > deg(q), zeros of p and q interlace if each zero of q lies between two successive zeros of p and at most one zero of q between any two successive zeros of p . Kathy Driver Laguerre Polynomials 22-24 March 2016 4 / 16

Askey Conjecture 1989 Zeros of orthogonal Jacobi polynomials P ( α,β ) and P ( α +2 ,β ) interlace n n Kathy Driver Laguerre Polynomials 22-24 March 2016 5 / 16

Askey Conjecture 1989 Zeros of orthogonal Jacobi polynomials P ( α,β ) and P ( α +2 ,β ) interlace n n Electrostatic interpretation of zeros of Jacobi polynomials, increasing one parameter means increasing charge at one endpoint. Kathy Driver Laguerre Polynomials 22-24 March 2016 5 / 16

Askey Conjecture 1989 Zeros of orthogonal Jacobi polynomials P ( α,β ) and P ( α +2 ,β ) interlace n n Electrostatic interpretation of zeros of Jacobi polynomials, increasing one parameter means increasing charge at one endpoint. Analysis of Mixed TTRR’s gives upper and lower bounds for zeros of classical OP’s Kathy Driver Laguerre Polynomials 22-24 March 2016 5 / 16

Askey Conjecture 1989 Zeros of orthogonal Jacobi polynomials P ( α,β ) and P ( α +2 ,β ) interlace n n Electrostatic interpretation of zeros of Jacobi polynomials, increasing one parameter means increasing charge at one endpoint. Analysis of Mixed TTRR’s gives upper and lower bounds for zeros of classical OP’s Askey Conjecture. Driver, Jordaan and Mbuyi: Zeros of Jacobi P ( α,β ) and n P ( α + k ,β − l ) interlace if k , l ≤ 2 provided β − l remains > − 1 . n Kathy Driver Laguerre Polynomials 22-24 March 2016 5 / 16

Askey Conjecture 1989 Zeros of orthogonal Jacobi polynomials P ( α,β ) and P ( α +2 ,β ) interlace n n Electrostatic interpretation of zeros of Jacobi polynomials, increasing one parameter means increasing charge at one endpoint. Analysis of Mixed TTRR’s gives upper and lower bounds for zeros of classical OP’s Askey Conjecture. Driver, Jordaan and Mbuyi: Zeros of Jacobi P ( α,β ) and n P ( α + k ,β − l ) interlace if k , l ≤ 2 provided β − l remains > − 1 . n Ismail, Dimitrov and Rafaeli: Askey Conjecture sharp. Kathy Driver Laguerre Polynomials 22-24 March 2016 5 / 16

Zeros of Laguerre polynomials, different parameters Kathy Driver Laguerre Polynomials 22-24 March 2016 6 / 16

Zeros of Laguerre polynomials, different parameters Kerstin Jordaan- KD 2011 Indag Math Kathy Driver Laguerre Polynomials 22-24 March 2016 6 / 16

Zeros of Laguerre polynomials, different parameters Kerstin Jordaan- KD 2011 Indag Math Parameter difference = integer, assume no common zeros n and L α + k Zeros L α n − 2 interlace, k ∈ { 1 , 2 , 3 , 4 } Kathy Driver Laguerre Polynomials 22-24 March 2016 6 / 16

Zeros of Laguerre polynomials, different parameters Kerstin Jordaan- KD 2011 Indag Math Parameter difference = integer, assume no common zeros n and L α + k Zeros L α n − 2 interlace, k ∈ { 1 , 2 , 3 , 4 } At least one ”gap interval”, no zero of L α + t n − 2 , changes with t . Markov monotonicity argument breaks down Kathy Driver Laguerre Polynomials 22-24 March 2016 6 / 16

Zeros of Laguerre polynomials, different parameters Kerstin Jordaan- KD 2011 Indag Math Parameter difference = integer, assume no common zeros n and L α + k Zeros L α n − 2 interlace, k ∈ { 1 , 2 , 3 , 4 } At least one ”gap interval”, no zero of L α + t n − 2 , changes with t . Markov monotonicity argument breaks down Conjecture Kerstin Jordaan -KD 2011 n and L α + t Zeros L α n − 2 interlace for 0 ≤ t ≤ 4 if the two polynomials have no common zeros Kathy Driver Laguerre Polynomials 22-24 March 2016 6 / 16

Sharp interval. Zeros of Laguerre polynomials Martin Muldoon and KD Journal of Approximation Theory 2013 Kathy Driver Laguerre Polynomials 22-24 March 2016 7 / 16

Sharp interval. Zeros of Laguerre polynomials Martin Muldoon and KD Journal of Approximation Theory 2013 Each t , 0 ≤ t ≤ 2 k , excluding t for which common zeros occur, zeros of L ( α ) and L ( α + t ) interlace n n − k Kathy Driver Laguerre Polynomials 22-24 March 2016 7 / 16

Sharp interval. Zeros of Laguerre polynomials Martin Muldoon and KD Journal of Approximation Theory 2013 Each t , 0 ≤ t ≤ 2 k , excluding t for which common zeros occur, zeros of L ( α ) and L ( α + t ) interlace n n − k Interval 0 ≤ t ≤ 2 k is largest possible for which interlacing holds each n , α, k Kathy Driver Laguerre Polynomials 22-24 March 2016 7 / 16

Proofs of interlacing and sharpness results t interval 0 ≤ t ≤ 2 k largest possible for interlacing of zeros of L ( α ) n , L ( α + t ) n − k , each n ∈ N , 0 < k ≤ n − 2 , and fixed α ≥ 0 , (excluding t for which common zeros occur) Kathy Driver Laguerre Polynomials 22-24 March 2016 8 / 16

Proofs of interlacing and sharpness results t interval 0 ≤ t ≤ 2 k largest possible for interlacing of zeros of L ( α ) n , L ( α + t ) n − k , each n ∈ N , 0 < k ≤ n − 2 , and fixed α ≥ 0 , (excluding t for which common zeros occur) Interlacing proof involves monotonicity properties of common zeros of L ( α ) n and L ( α + t ) and Sturm Comparison Theorem. n − k Kathy Driver Laguerre Polynomials 22-24 March 2016 8 / 16

Proofs of interlacing and sharpness results t interval 0 ≤ t ≤ 2 k largest possible for interlacing of zeros of L ( α ) n , L ( α + t ) n − k , each n ∈ N , 0 < k ≤ n − 2 , and fixed α ≥ 0 , (excluding t for which common zeros occur) Interlacing proof involves monotonicity properties of common zeros of L ( α ) n and L ( α + t ) and Sturm Comparison Theorem. n − k Sharpness Inequalities satisfied by zeros of L ( α ) and zeros of Bessel function n Asymptotic behaviour of zeros of L ( α ) using Airy function n Kathy Driver Laguerre Polynomials 22-24 March 2016 8 / 16

Zeros of L 0 6 ( x ) and L t 5 ( x ) as a function of t Kathy Driver Laguerre Polynomials 22-24 March 2016 9 / 16

Zeros of L 0 6 ( x ) and L t 4 ( x ) as a function of t Kathy Driver Laguerre Polynomials 22-24 March 2016 10 / 16

Zeros of L 0 6 ( x ) and L t 3 ( x ) as a function of t Kathy Driver Laguerre Polynomials 22-24 March 2016 11 / 16

Laguerre sequences { L ( α ) n } ∞ n =0 , − 2 < α < − 1 As α decreases below − 1, one positive zero of L ( α ) leaves (0 , ∞ ) each n time α passes through − 1 , − 2 , . . . , − n . Kathy Driver Laguerre Polynomials 22-24 March 2016 12 / 16