Sieved Ultraspherical Polynomials

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.

Recurrence relations

For the sieved ultraspherical polynomials of the first kind the recurrence relations are
:$2xc_n^\lambda(x;k) = c_^\lambda(x;k) + c_^\lambda(x;k)$ if ''n'' is not divisible by ''k''
:$2x(m+\lambda)c_^\lambda(x;k) = (m+2\lambda)c_^\lambda(x;k) + mc_^\lambda(x;k)$

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

References

*{{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

Orthogonal polynomials