Ultraspherical and pseudo-ultraspherical polynomials

• Kathy Driver ^[1] ; Martin E. Muldoon ^[2]
  1. [1] University of Cape Town

     University of Cape Town

     City of Cape Town, Sudáfrica

     
  2. [2] York University (Canadá)

     York University (Canadá)

     Canadá

     
• Localización: Jaen journal on approximation, ISSN 1889-3066, ISSN-e 1989-7251, Vol. 10, Nº. 1-2, 2018, págs. 73-80
• Idioma: inglés
• Texto completo no disponible (Saber más ...)
• Resumen

  • The pseudo-ultraspherical polynomial of degree n is defined by C˜(λ) n (x)=(−i) nC(λ) n (ix) where C(λ) n (x) is the ultraspherical polynomial. We discuss the orthogonality of finite sequences of pseudo-ultraspherical polynomials {C˜(λ) n (x)}N n=0 for different values of N that depend on λ. We use an identity linking the zeros of C˜(λ) n (x) and the zeros of C(λ) n (x) where λ = 1/2 − λ − n to prove that for 1 − n<λ< 2 − n and each n ≥ 3, there are two zeros of C˜(λ) n (x) which are purely imaginary and symmetric with respect to the origin. We derive upper and lower bounds for the distance of the two purely imaginary zeros of C˜(λ) n (x) from the origin when 1 − n<λ< 2 − n.