Una nuova rappresentazione asintotica dei polinomi ultrasferici

Luigi Gatteschi’s work on asymptotics of special functions and their zeros

“…If α 2 = β 2 = 1/4, not only the expression in brackets, but also the error term in (7) vanish. In the ultraspherical case α = β, the result (7) is asymptotically in agreement with earlier ones in [11]. There is of course a result analogous to (7) for the zeros x (α,β) n,r of P (α,β) n (x) themselves.…”

“…In the case of Legendre and ultraspherical polynomials, the results obtained in [10][11][12] are somewhat preliminary inasmuch as they cover only limited ranges of zeros. This deficiency is overcome later in [33], though at the expense of sharpness, where Tricomi's method is again applied to ultraspherical polynomials P (λ) n = P (λ−1/2,λ−1/2) n , 0 < λ < 1.…”

“…A slightly different application of the method of Liouville-Steklov, especially if applied successively as suggested by Szegö [55, Section 8.61 (2)] in the case of Legendre polynomials, yields more accurate approximations of ultraspherical polynomials P (λ) n , valid in any compact subinterval of (−1, 1), and of their zeros contained therein [34].…”

“…In [31], this is improved in two ways: First, the method of Liouville-Steklov is refined, with the result that in (20) the number N can be replaced by…”

Fast and Accurate Computation of Gauss--Legendre and Gauss--Jacobi Quadrature Nodes and Weights

“…For nodes near x = 1, there are formulae given by Gatteschi [13], An optimal choice would involve a selection of three of the formulae, but since an accuracy of below the root of machine precision is sufficient for our purposes we simply choose (3.4) when x ≤ 1/2 and (3.7) when x > 1/2.…”