Resumen de Ultraspherical and pseudo-ultraspherical polynomials

Kathy Driver, Martin E. Muldoon

• 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.