Zeros of Ultraspherical Polynomials Kathy Driver University of Cape - PowerPoint PPT Presentation

Zeros of Ultraspherical Polynomials Kathy Driver University of Cape Town Visiting Vanderbilt University Joint work with Martin Muldoon Midwestern Workshop October 7, 2017 Kathy Driver Zeros of Ultraspherical Polynomials 2017 1 / 15

Ultraspherical Polynomials C ( λ ) n (1 − x ) 2 y ′′ − (2 λ + 1) xy ′ + n ( n + 2 λ ) y = 0 Kathy Driver Zeros of Ultraspherical Polynomials 2017 2 / 15

Ultraspherical Polynomials C ( λ ) n (1 − x ) 2 y ′′ − (2 λ + 1) xy ′ + n ( n + 2 λ ) y = 0 λ = 1 : Chebyshev polynomials of second kind Kathy Driver Zeros of Ultraspherical Polynomials 2017 2 / 15

Ultraspherical Polynomials C ( λ ) n (1 − x ) 2 y ′′ − (2 λ + 1) xy ′ + n ( n + 2 λ ) y = 0 λ = 1 : Chebyshev polynomials of second kind λ = 1 2 : Legendre polynomials Kathy Driver Zeros of Ultraspherical Polynomials 2017 2 / 15

Ultraspherical Polynomials C ( λ ) n (1 − x ) 2 y ′′ − (2 λ + 1) xy ′ + n ( n + 2 λ ) y = 0 λ = 1 : Chebyshev polynomials of second kind λ = 1 2 : Legendre polynomials ⌊ n / 2 ⌋ ( λ ) n − m C ( λ ) � m !( n − 2 m )!(2 x ) n − 2 m ( x ) = n m =0 Kathy Driver Zeros of Ultraspherical Polynomials 2017 2 / 15

Ultraspherical Polynomials C ( λ ) n (1 − x ) 2 y ′′ − (2 λ + 1) xy ′ + n ( n + 2 λ ) y = 0 λ = 1 : Chebyshev polynomials of second kind λ = 1 2 : Legendre polynomials ⌊ n / 2 ⌋ ( λ ) n − m C ( λ ) � m !( n − 2 m )!(2 x ) n − 2 m ( x ) = n m =0 2 of Jacobi polynomials P ( α,β ) Special case α = β = λ − 1 n Kathy Driver Zeros of Ultraspherical Polynomials 2017 2 / 15

Orthogonal Ultraspherical Polynomials C ( λ ) n , λ > − 1 / 2 Kathy Driver Zeros of Ultraspherical Polynomials 2017 3 / 15

Orthogonal Ultraspherical Polynomials C ( λ ) n , λ > − 1 / 2 { C ( λ ) } ∞ n =0 is orthogonal on ( − 1 , 1) for λ > − 1 / 2 n � 1 x k C ( λ ) ( x ) (1 − x 2 ) λ − 1 / 2 dx = 0 for k = 0 , . . . , n − 1 . n − 1 λ > − 1 2 ensures convergence of the integral Kathy Driver Zeros of Ultraspherical Polynomials 2017 3 / 15

Orthogonal Ultraspherical Polynomials C ( λ ) n , λ > − 1 / 2 { C ( λ ) } ∞ n =0 is orthogonal on ( − 1 , 1) for λ > − 1 / 2 n � 1 x k C ( λ ) ( x ) (1 − x 2 ) λ − 1 / 2 dx = 0 for k = 0 , . . . , n − 1 . n − 1 λ > − 1 2 ensures convergence of the integral The zeros of C ( λ ) n − 1 and C ( λ ) are interlacing: n − 1 < x 1 , n < x 1 , n − 1 < x 2 , n < · · · < x n − 1 , n < x n − 1 , n − 1 < x n , n < 1 Kathy Driver Zeros of Ultraspherical Polynomials 2017 3 / 15

Orthogonal Ultraspherical Polynomials C ( λ ) n , λ > − 1 / 2 { C ( λ ) } ∞ n =0 is orthogonal on ( − 1 , 1) for λ > − 1 / 2 n � 1 x k C ( λ ) ( x ) (1 − x 2 ) λ − 1 / 2 dx = 0 for k = 0 , . . . , n − 1 . n − 1 λ > − 1 2 ensures convergence of the integral The zeros of C ( λ ) n − 1 and C ( λ ) are interlacing: n − 1 < x 1 , n < x 1 , n − 1 < x 2 , n < · · · < x n − 1 , n < x n − 1 , n − 1 < x n , n < 1 The zeros of C ( λ ) ( x ) and (1 − x 2 ) C ( λ ) n − 1 ( x ) are interlacing n Kathy Driver Zeros of Ultraspherical Polynomials 2017 3 / 15

Zeros of C ( λ ) n , λ < − 1 / 2 λ < − 1 / 2 Kathy Driver Zeros of Ultraspherical Polynomials 2017 4 / 15

Zeros of C ( λ ) n , λ < − 1 / 2 λ < − 1 / 2 Is the sequence { C ( λ ) } ∞ n =0 orthogonal for any value(s) of λ < − 1 / 2? n Kathy Driver Zeros of Ultraspherical Polynomials 2017 4 / 15

Zeros of C ( λ ) n , λ < − 1 / 2 λ < − 1 / 2 Is the sequence { C ( λ ) } ∞ n =0 orthogonal for any value(s) of λ < − 1 / 2? n If the sequence is orthogonal for some λ < − 1 2 , what is the measure of orthogonality? Kathy Driver Zeros of Ultraspherical Polynomials 2017 4 / 15

Zeros of C ( λ ) n , λ < − 1 / 2 λ < − 1 / 2 Is the sequence { C ( λ ) } ∞ n =0 orthogonal for any value(s) of λ < − 1 / 2? n If the sequence is orthogonal for some λ < − 1 2 , what is the measure of orthogonality? Are the n zeros of C ( λ ) all real? Distinct? All in ( − 1 , 1)? n Kathy Driver Zeros of Ultraspherical Polynomials 2017 4 / 15

Zeros of C ( λ ) n , λ < − 1 / 2 λ < − 1 / 2 Is the sequence { C ( λ ) } ∞ n =0 orthogonal for any value(s) of λ < − 1 / 2? n If the sequence is orthogonal for some λ < − 1 2 , what is the measure of orthogonality? Are the n zeros of C ( λ ) all real? Distinct? All in ( − 1 , 1)? n If the zeros of C ( λ ) are all real and distinct n Kathy Driver Zeros of Ultraspherical Polynomials 2017 4 / 15

Zeros of C ( λ ) n , λ < − 1 / 2 λ < − 1 / 2 Is the sequence { C ( λ ) } ∞ n =0 orthogonal for any value(s) of λ < − 1 / 2? n If the sequence is orthogonal for some λ < − 1 2 , what is the measure of orthogonality? Are the n zeros of C ( λ ) all real? Distinct? All in ( − 1 , 1)? n If the zeros of C ( λ ) are all real and distinct n are the zeros of C ( λ ) n − 1 and C ( λ ) interlacing? n Kathy Driver Zeros of Ultraspherical Polynomials 2017 4 / 15

Behaviour of zeros of C ( λ ) as λ decreases below − 1 n 2 Kathy Driver Zeros of Ultraspherical Polynomials 2017 5 / 15

Behaviour of zeros of C ( λ ) as λ decreases below − 1 n 2 2 , two zeros of C ( λ ) As λ decreases below − 1 leave ( − 1 , 1) through − 1 and n 1 . Each time λ decreases below − 1 / 2 , − 3 / 2 , − 5 / 2 , ... two more zeros leave ( − 1 , 1) . When λ reaches 1 / 2 − [ n / 2] , no zeros of C ( λ ) remain in ( − 1 , 1) n Kathy Driver Zeros of Ultraspherical Polynomials 2017 5 / 15

Behaviour of zeros of C ( λ ) as λ decreases below − 1 n 2 2 , two zeros of C ( λ ) As λ decreases below − 1 leave ( − 1 , 1) through − 1 and n 1 . Each time λ decreases below − 1 / 2 , − 3 / 2 , − 5 / 2 , ... two more zeros leave ( − 1 , 1) . When λ reaches 1 / 2 − [ n / 2] , no zeros of C ( λ ) remain in ( − 1 , 1) n As λ decreases below the negative integer − [( n + 1) / 2] , two zeros of C ( λ ) n join the imaginary axis and another pair joins the imaginary axis each time λ decreases through successive negative integers. Kathy Driver Zeros of Ultraspherical Polynomials 2017 5 / 15

Behaviour of zeros of C ( λ ) as λ decreases below − 1 n 2 2 , two zeros of C ( λ ) As λ decreases below − 1 leave ( − 1 , 1) through − 1 and n 1 . Each time λ decreases below − 1 / 2 , − 3 / 2 , − 5 / 2 , ... two more zeros leave ( − 1 , 1) . When λ reaches 1 / 2 − [ n / 2] , no zeros of C ( λ ) remain in ( − 1 , 1) n As λ decreases below the negative integer − [( n + 1) / 2] , two zeros of C ( λ ) n join the imaginary axis and another pair joins the imaginary axis each time λ decreases through successive negative integers. For λ < 1 − n , C ( λ ) ( x ) has n distinct pure imaginary zeros. n Kathy Driver Zeros of Ultraspherical Polynomials 2017 5 / 15

Behaviour of zeros of C ( λ ) as λ decreases below − 1 n 2 2 , two zeros of C ( λ ) As λ decreases below − 1 leave ( − 1 , 1) through − 1 and n 1 . Each time λ decreases below − 1 / 2 , − 3 / 2 , − 5 / 2 , ... two more zeros leave ( − 1 , 1) . When λ reaches 1 / 2 − [ n / 2] , no zeros of C ( λ ) remain in ( − 1 , 1) n As λ decreases below the negative integer − [( n + 1) / 2] , two zeros of C ( λ ) n join the imaginary axis and another pair joins the imaginary axis each time λ decreases through successive negative integers. For λ < 1 − n , C ( λ ) ( x ) has n distinct pure imaginary zeros. n D-Duren 2000. A specific number of zeros of C ( λ ) collide at the endpoints n 1 and − 1 of the interval of orthogonality each time λ decreases through the next successive negative half-integer. The location and kinematics of the zeros at each stage of the process are known. Kathy Driver Zeros of Ultraspherical Polynomials 2017 5 / 15

Zeros of C ( λ ) n , − 3 / 2 < λ < − 1 / 2 − 3 / 2 < λ < − 1 / 2 Kathy Driver Zeros of Ultraspherical Polynomials 2017 6 / 15

Zeros of C ( λ ) n , − 3 / 2 < λ < − 1 / 2 − 3 / 2 < λ < − 1 / 2 2 for which all n zeros of C ( λ ) Only λ − range apart from λ > − 1 are real n Kathy Driver Zeros of Ultraspherical Polynomials 2017 6 / 15

Zeros of C ( λ ) n , − 3 / 2 < λ < − 1 / 2 − 3 / 2 < λ < − 1 / 2 2 for which all n zeros of C ( λ ) Only λ − range apart from λ > − 1 are real n Sequence { C ( λ ) } ∞ n =0 quasi-orthogonal order 2 on ( − 1 , 1) n Kathy Driver Zeros of Ultraspherical Polynomials 2017 6 / 15

Zeros of C ( λ ) n , − 3 / 2 < λ < − 1 / 2 − 3 / 2 < λ < − 1 / 2 2 for which all n zeros of C ( λ ) Only λ − range apart from λ > − 1 are real n Sequence { C ( λ ) } ∞ n =0 quasi-orthogonal order 2 on ( − 1 , 1) n Quasi-orthogonality:Riesz (Moment Problem), Fejer, Shohat, Chihara... � 1 x k C ( λ ) ( x ) (1 − x 2 ) λ +1 / 2 dx = 0 for k = 0 , . . . , n − 3 . n − 1 Kathy Driver Zeros of Ultraspherical Polynomials 2017 6 / 15

Zeros of C ( λ ) n , − 3 / 2 < λ < − 1 / 2 Kathy Driver Zeros of Ultraspherical Polynomials 2017 7 / 15