In mathematics, the two families c(x;k) and B(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.
For the sieved ultraspherical polynomials of the first kind the recurrence relations are
| λ(x;k) | |
| 2xc | |
| n |
=
| λ(x;k) | |
| c | |
| n+1 |
+
| λ(x;k) | |
| c | |
| n-1 |
| λ(x;k) | |
| 2x(m+λ)c | |
| mk |
=
| λ(x;k) | |
| (m+2λ)c | |
| mk+1 |
+
| λ(x;k) | |
| mc | |
| mk-1 |
For the sieved ultraspherical polynomials of the second kind the recurrence relations are
| λ(x;k) | |
| 2xB | |
| n-1 |
=
| λ(x;k) | |
| B | |
| n |
+
| λ(x;k) | |
| B | |
| n-2 |
| λ(x;k) | |
| 2x(m+λ)B | |
| mk-1 |
=
| λ(x;k) | |
| mB | |
| mk |
| λ(x;k) | |
| +(m+2λ)B | |
| mk-2 |