In  Olver and Townsend presented a fast spectral method for solving linear differential equations using bases of ultraspherical polynomials
, which has subsequently been exploited in Chebfun  and ApproxFun .
The ultraspherical polynomials have received considerable attention in recent decades, from both theoretical and practical points of view (see, e.g., ).
Some Properties of Ultraspherical Polynomials and Their Shifted Ones.
It should be noted here that the ultraspherical polynomials [C.sup.([lambda]).sub.n](x) are normalized such that [C.sup.([lambda]).sub.n](1) = 1.
An extension of (1.1) to ultraspherical polynomials [P.sup.([lambda]).sub.n] [P.sup.([lambda]-1/2, [lambda]-1/2).sub.n], 0 < [lambda] < 1, is due to Lorch , and a further extension to Jacobi polynomials [P.sup.([alpha], [beta].sub.n]) with [absolute value of [alpha]] [less than or equal to] 1/2, [absolute value of [beta]] [less than or equal to] 1/2 to Baratella .
LORCH, Inequalities for ultraspherical polynomials and the gamma function, J.
ISMAIL, A generalization of ultraspherical polynomials
, in Studies in Pure Mathematics, P.
BALAZS, Weighted (0, 2) -interpolation on the zeros of the ultraspherical polynomials, (in Hungarian: Sulyozott (0, 2)-interpolacio ultraszferikus polinom gyokein), MTA IILoszt.
TER AN, Notes on Interpolation I, On some interpolatorical properties of the ultraspherical polynomials, Acta Math.
 --, Zeros of ultraspherical polynomials
and the Hilbert-Klein formulas, J.
are solutions of the differential equations [L.sub.[alpha]] [p.sup.([alpha]).sub.n] = n(n + 2[alpha] + 1)[p.sup.([alpha]).sub.n], for n [member of] [N.sub.0] , where the Laplace operator is defined as