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
 * $$2xc_n^\lambda(x;k) = c_{n+1}^\lambda(x;k) + c_{n-1}^\lambda(x;k)$$ if n is not divisible by k
 * $$2x(m+\lambda)c_{mk}^\lambda(x;k) = (m+2\lambda)c_{mk+1}^\lambda(x;k) + mc_{mk-1}^\lambda(x;k)$$

For the sieved ultraspherical polynomials of the second kind the recurrence relations are
 * $$2xB_{n-1}^\lambda(x;k) = B_{n}^\lambda(x;k) + B_{n-2}^\lambda(x;k)$$ if n is not divisible by k
 * $$2x(m+\lambda)B_{mk-1}^\lambda(x;k) = mB_{mk}^\lambda(x;k) +(m+2\lambda)B_{mk-2}^\lambda(x;k) $$