Sieved ultraspherical polynomials

In mathematics, the two families c^λ
_n(x;k) and B^λ
_n(x;k) of sieved ultraspherical polynomials, introduced by Waleed Al-Salam, W.R. Allaway and Richard Askey in 1984, are the archetypal examples of sieved orthogonal polynomials. Their recurrence relations are a modified (or "sieved") version of the recurrence relations for ultraspherical polynomials.

Recurrence relations

For the sieved ultraspherical polynomials of the first kind the recurrence relations are

$2x{c}_n^{\lambda }(x;k)=c_{n+1}^{\lambda }(x;k)+c_{n-1}^{\lambda }(x;k)$ if n is not divisible by k

$2x(m+\lambda )c_{mk}^{\lambda }(x;k)=(m+2\lambda )c_{mk+1}^{\lambda }(x;k)+m{c}_{mk-1}^{\lambda }(x;k)$

For the sieved ultraspherical polynomials of the second kind the recurrence relations are

$2x{B}_{n-1}^{\lambda }(x;k)=B_n^{\lambda }(x;k)+B_{n-2}^{\lambda }(x;k)$ if n is not divisible by k

$2x(m+\lambda )B_{mk-1}^{\lambda }(x;k)=m{B}_{mk}^{\lambda }(x;k)+(m+2\lambda )B_{mk-2}^{\lambda }(x;k)$