Neha Patil (Editor)

Sieved ultraspherical polynomials

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In mathematics, the two families cλ
n
(x;k) and Bλ
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 ) 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 )

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 ) 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 )

References

Sieved ultraspherical polynomials Wikipedia


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