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Sieved ultraspherical polynomials

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In mathematics, the two families c
n(x;k) and B
n(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

2 x c n λ ( x ; k ) = c n + 1 λ ( x ; k ) + c n − 1 λ ( x ; k ) {\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
2 x ( m + λ ) c m k λ ( x ; k ) = ( m + 2 λ ) c m k + 1 λ ( x ; k ) + m c m k − 1 λ ( x ; 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

2 x B n − 1 λ ( x ; k ) = B n λ ( x ; k ) + B n − 2 λ ( x ; k ) {\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
2 x ( m + λ ) B m k − 1 λ ( x ; k ) = m B m k λ ( x ; k ) + ( m + 2 λ ) B m k − 2 λ ( x ; 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


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