Sieved ultraspherical polynomials

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

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

• 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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