sieved ultraspherical polynomials

In mathematics, the two families c{{su|b=n|p=λ}}(x;k) and B{{su|b=n|p=λ}}(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

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

{{Citation | last1=Al-Salam | first1=Waleed | last2=Allaway | first2=W. R. | last3=Askey | first3=Richard | authorlink3=Richard Askey| title=Sieved ultraspherical polynomials | doi=10.2307/1999273 | mr=742411 | year=1984 | journal=Transactions of the American Mathematical Society | issn=0002-9947 | volume=284 | issue=1 | pages=39–55| jstor=1999273 | doi-access=free }}