In mathematics, the two families ''c''(''x'';''k'') and ''B''(''x'';''k'') of sieved ultraspherical polynomials, introduced by

Waleed Al-Salam Waleed Al-Salam (born 15 July 1926 in Baghdad, Iraq – died 14 April 1996 in Edmonton, Canada) was a mathematician who introduced Al-Salam–Chihara polynomials, Al-Salam–Carlitz polynomials, q-Konhauser polynomials, and Al-Salam–Ismail po ...

, W.R. Allaway and

Richard Askey Richard Allen Askey (June 4, 1933 – October 9, 2019) was an American mathematician, known for his expertise in the area of special functions. The Askey–Wilson polynomials (introduced by him in 1984 together with James A. Wilson) are on the ...

in 1984, are the archetypal examples of sieved orthogonal polynomials. Their

recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...

s 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

* Orthogonal polynomials {{polynomial-stub