Orthogonal polynomials, zeros and electrostatics F. Marcell an - PowerPoint PPT Presentation

Orthogonal polynomials, zeros and electrostatics F. Marcell´ an Universidad Carlos III de Madrid (UC3M) and Instituto de Ciencias Matem´ aticas (ICMAT) OPCOP2017 - Universidad de Cantabria April 19-22, 2017 - (CIEM) Castro Urdiales (OPCOP2017) OP , zeros and electrostatics UC 2017 1 / 32

Outline From classical to semiclassical orthogonal polynomials 1 Electrostatic model for semiclassical OP Electrostatic model for zeros of POPUC 2 Orthogonal and Paraorthogonal polynomials on the unit circle Differential properties for POPUC Electrostatic interpretation Finding the electric field generators: an algorithmic approach Examples: Lebesgue and Chebyshev polynomials Zeros of exceptional orthogonal polynomials 3 Exceptional orthogonal polynomials X m -Laguerre-(I) polynomials X m -Laguerre-(II) polynomials Zeros of Freud-Sobolev type orthogonal polynomials 4 Freud-Sobolev type orthogonal polynomials Electrostatic models for the “even” Q 2 n and “odd” Q 2 n + 1 subsequences (OPCOP2017) OP , zeros and electrostatics UC 2017 2 / 32

From classical to semiclassical OP Let { p n ( x ) } n ≥ 0 be a sequence of polynomials orthonormal (OPRL) with respect to a weight function w ( x ) = exp ( − v ( x )) supported on an interval [ c , d ] ⊂ R , finite or infinite: � d c p m ( x ) p n ( x ) w ( x ) dx = δ m , n . Then { p n } n ≥ 0 satisfies a three-term recurrence relation: xp n ( x ) = a n + 1 p n + 1 ( x )+ b n p n ( x )+ a n p n − 1 ( x ) , n ≥ 0 , a n > 0 , p − 1 = 0 , p 0 = 1 . Under certain assumptions on w , the orthonormal polynomials p n also satisfy a difference-differential relation p ′ n ( x ) = A ( x ; n ) p n − 1 ( x ) − B ( x ; n ) p n ( x ) , where A ( x , n ) , B ( x ; n ) are given in terms of w , a n -s, and the values p n ( c ) and p n ( d ) . A direct consequence of the above is that p n satisfies also the second order linear differential equation p ′′ n ( x ) − 2 R ( x ; n ) p ′ n ( x )+ S ( x ; n ) p n ( x ) = 0 , with R ( x ; n ) = v ′ ( x ) + A ′ ( x ; n ) 2 A ( x ; n ) , 2 S ( x ; n ) = B ′ ( x ; n ) − B ( x ; n ) A ′ ( x ; n ) A ( x ; n ) − B ( x ; n )[ v ′ ( x )+ B ( x ; n )]+ a n a n − 1 A ( x ; n ) A ( x ; n − 1 ) . (OPCOP2017) OP , zeros and electrostatics UC 2017 3 / 32

From classical to semiclassical OP M.E.H. Ismail, An electrostatic model for zeros of general orthogonal polynomials , Pacific J. Math. 193 (2000), 355–369. Ismail considers ϕ ′ ( x ) = R ( x ; n ) , where ϕ ( x ) is the external field ϕ ( x ) = v ( x ) + ln ( k n A ( x ; n )) = ϕ long ( x )+ ϕ short ( x ) 2 2 ϕ ( x ) has two components: ϕ long ( x ) comes from the orthogonality weight w ( x ) = exp ( − v ( x )) , and Ismail called it the long range potential . The other ϕ short ( x ) is called short range potential , allows to give a further generalization of the electrostatic interpretation. Ismail proves that, under certain assumptions, the total energy of the system has a unique point of global minimum, which is located at the vector constituted by the zeros of the orthogonal polynomial p n . A ( x ; n ) is responsible of the creation of “ ghost ” movable charges, as it was shown first in: F.A. Gr¨ unbaum, Variations on a theme of Heine and Stieltjes: An electrostatic interpretation of the zeros of certain polynomials , J. Comput. Appl. Math. 99 (1998), 189–194. (OPCOP2017) OP , zeros and electrostatics UC 2017 4 / 32

Semiclassical OPRL Let us consider here a little bit more general situation. A generalized weight function (or linear functional) is semiclassical if it satisfies the Pearson equation D ( φ w ) = ψ w , where φ , ψ are polynomials, with degree of ψ ≥ 1, and D is the “derivative” operator (in the usual, but also possibly in a distributional sense). It is well known that for such a weight the corresponding orthogonal polynomials (called also semiclassical ) satisfy a differential equation of the type referred above, where the coefficients R ( x ; n ) and S ( x ; n ) are rational functions. The classical-type orthogonal polynomials considered by Gr¨ unbaum in the above work are an example of a semiclassical family, but there are many more. F. Marcell´ an, A. Mart´ ınez-Finkelshtein, P . Mart´ ınez-Gonz´ alez, Electrostatic models for zeros of polynomials: old, new, and some open problems , J. Comput. Appl. Math. 207 (2007), no. 2, 258–272. (OPCOP2017) OP , zeros and electrostatics UC 2017 5 / 32

Outline From classical to semiclassical orthogonal polynomials 1 Electrostatic model for semiclassical OP Electrostatic model for zeros of POPUC 2 Orthogonal and Paraorthogonal polynomials on the unit circle Differential properties for POPUC Electrostatic interpretation Finding the electric field generators: an algorithmic approach Examples: Lebesgue and Chebyshev polynomials Zeros of exceptional orthogonal polynomials 3 Exceptional orthogonal polynomials X m -Laguerre-(I) polynomials X m -Laguerre-(II) polynomials Zeros of Freud-Sobolev type orthogonal polynomials 4 Freud-Sobolev type orthogonal polynomials Electrostatic models for the “even” Q 2 n and “odd” Q 2 n + 1 subsequences (OPCOP2017) OP , zeros and electrostatics UC 2017 6 / 32

Orthogonal and Paraorthogonal polynomials on the unit circle Given an infinitely supported probability measure µ on the unit circle, one defines the OPUC { Φ n ( z ; µ ) } n ≥ 0 as the monic sequence of monic polynomials satisfying the Szeg˝ o recursion: Φ n + 1 ( z ; µ ) = z Φ n ( z ; µ ) − ¯ α n Φ ∗ n ( z ; µ ) , where α n ∈ D := { z : | z | < 1 } and Φ ∗ n ( z ) = z n Φ n ( 1 / ¯ z ) . To each µ on the unit circle, we can associate a sequence { α n } n ≥ 0 of corresponding Verblunsky coefficients . For the above µ , and a complex number β of modulus 1, one can defines paraorthogonal polynomials (POPUC) { Φ n ( z ; β ; µ ) } n ≥ 0 as the sequence of polynomials given by Φ n ( z ; β ; µ ) := z Φ n − 1 ( z ; µ ) − ¯ β Φ ∗ n − 1 ( z ; µ ) . All the zeros of Φ n ( z ; β ; µ ) are simple and lie on the unit circle . (OPCOP2017) OP , zeros and electrostatics UC 2017 7 / 32

Paraorthogonal polynomials on the unit circle If τ � = β are distinct complex numbers of modulus 1, then the zeros of Φ n ( z ; β ; µ ) and Φ n ( z ; τ ; µ ) strictly interlace on the unit circle, i.e.if x and y are two zeros of Φ n ( z ; β ; µ ) and [ x , y ] is the arc of the unit circle that runs from x to y in the counter-clockwise direction, then [ x , y ] \{ x , y } contains a zero of Φ n ( z ; τ ; µ ) . Paraorthogonal polynomials are not orthogonal polynomials, but they often serve as an appropriate analog of OPRL in settings where the real line is replaced by the unit circle. The basic reference in this section will be: B. Simanek, An electrostatic interpretation of the zeros of paraorthogonal polynomials on the unit circle. SIAM J. Math. Anal. 48 (3) (2016), 2250–2268. (OPCOP2017) OP , zeros and electrostatics UC 2017 8 / 32

Theorem 1 (B. Simanek, 2016) Suppose d µ ( θ ) = w ( θ ) d θ 2 π is a probability measure on the unit circle, where w is continuous on [ 0 , 2 π ] (mod 2 π ) and differentiable on ( 0 , 2 π ) and let { α n } ∞ n = 0 be the corresponding sequence of Verblunsky coefficients. If β ∈ C , then the POPUC polynomial y ( z ) = Φ n ( z ; β ) defined above solves the following differential equation on any domain including infinity or zero on which the coefficients are meromorphic: n ( z ; β ; β ) − h ′ � 1 − n � 0 = y ′′ ( z )+ y ′ ( z ) h n ( z ; β ; β ) z � W [ h n ( z ; β ; β ) , h n ( z ; − β ; β )] − 1 � z (( n + zG n ( z )) G n ( z )+ J n ( z )( D n ( z ) − n α n − 1 )) + y ( z ) , 2 ¯ β zh n ( z ; β ; β ) where n − 1 ( e i θ ) | 2 w ′ ( θ ) | ϕ ∗ � 2 π d θ G n ( z ) := i 2 π , ( z − e i θ ) 0 ϕ ∗ n − 1 ( e i θ ) 2 w ′ ( θ ) ϕ n − 1 ( e i θ ) 2 w ′ ( θ ) � 2 π � 2 π d θ ( z − e i θ ) e i ( n − 2 ) θ d θ D n ( z ) := − iz 2 π , J n ( z ) := i 2 π , ( z − e i θ ) e in θ 0 0 y α n − 1 )+ zG n ( z )+ ¯ h n ( z ; x ; y ) := ¯ x ( n ( 1 − ¯ yD n ( z )) − z ( J n ( z ) − ¯ yG n ( z )) , and W [ f , g ] denotes the Wronskian of f and g . (OPCOP2017) OP , zeros and electrostatics UC 2017 9 / 32

Theorem 2 (B. Simanek, 2016) Suppose µ is as in the above Theorem and τ � = β are complex numbers. The polynomials u ( z ) := Φ n ( z ; β , µ ) and υ ( z ) := Φ n ( z ; τ , µ ) solve the following system of differential equations on any domain containing infinity or zero where the coefficients are meromorphic: � h n ( z ; β ; β ) � h n ( z ; τ ; β ) � � u ′ ( z ) = υ ( z ) − u ( z ) z (¯ β − ¯ τ ) z (¯ β − ¯ τ ) � h n ( z ; β ; τ ) � h n ( z ; τ ; τ ) � � υ ′ ( z ) = υ ( z ) − u ( z ) . z (¯ z (¯ β − ¯ τ ) β − ¯ τ ) More precisely, we need to know exactly the coefficient of n − d / 2 to estimate the above expressions correctly. The main restriction in the applicability of the above two Theorems is the requirement that the measure is absolutely continuous and the weight is a continuous and differentiable function. (OPCOP2017) OP , zeros and electrostatics UC 2017 10 / 32