In mathematics, the two families \( c_{n}^{\lambda }(x;k) \) and \( B_{n}^{\lambda }(x;k) \) 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
\( {\displaystyle 2xc_{n}^{\lambda }(x;k)=c_{n+1}^{\lambda }(x;k)+c_{n-1}^{\lambda }(x;k)} \) if n is not divisible by k
\( {\displaystyle 2x(m+\lambda )c_{mk}^{\lambda }(x;k)=(m+2\lambda )c_{mk+1}^{\lambda }(x;k)+mc_{mk-1}^{\lambda }(x;k)} \)
For the sieved ultraspherical polynomials of the second kind the recurrence relations are
\({\displaystyle 2xB_{n-1}^{\lambda }(x;k)=B_{n}^{\lambda }(x;k)+B_{n-2}^{\lambda }(x;k)} \) if n is not divisible by k
\( {\displaystyle 2x(m+\lambda )B_{mk-1}^{\lambda }(x;k)=mB_{mk}^{\lambda }(x;k)+(m+2\lambda )B_{mk-2}^{\lambda }(x;k)} \)
References
Al-Salam, Waleed; Allaway, W. R.; Askey, Richard (1984), "Sieved ultraspherical polynomials", Transactions of the American Mathematical Society, 284 (1): 39–55, doi:10.2307/1999273, ISSN 0002-9947, JSTOR 1999273, MR 0742411
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