Luigi Gatteschi

New inequalities for the zeros of Jacobi polynomials

It is shown that certain asymptotic approximations are upper or lower bounds for the zeros

Asymptotics and bounds for the zeros of Laguerre polynomials: a survey

\theta _{n,k} (\alpha ,\beta )

Uniform Approximations for the Zeros of Laguerre Polynomials

of Jacobi polynomials

Sul grado di precisione di formule di quadratura del tipo di tchebycheff

P_n^{(\alpha ,\beta )} (\cos \theta )

Una nuova rappresentazione asintotica dei polinomi ultrasferici

. The procedure for deriving these bounds is based on the Sturm comparison theorem. Numerical examples are given to illustrate the sharpness of the new inequalities.

New Asymptotics for the Zeros of Whittaker's Functions

Some of the work on the construction of inequalities and asymptotic approximations for the zeros λn,k(α), k = 1,2 .... ,n, of the Laguerre polynomial Lnα(x) as v = 4n + 2α + 2 → ∞, is reviewed and discussed. The cases when one or both parameters n and α unrestrictedly diverge are considered. Two new uniform asymptotic representations are presented: the first involves the positive zeros of the Bessel function Jα(x), and the second is in terms of the zeros of the Airy function Ai(x). They hold for k= 1,2 .... , [qn] and for k = [pn], [pn] + 1 ..... n, respectively, where p and q are fixed numbers in the interval (0, 1 ). Numerical results and comparisons are provided which favorably justify the consideration of the new approximations formulas.

Uniform Approximation of Christoffel Numbers for Jacobi Weight

In this paper we obtain two asymptotic formulas for the zeros \( \lambda _{n,k}^{(\alpha )},k = 1,2, \ldots ,n, \) of the Laguerre polynomials \( L_n^{(\alpha )}(x) \), as n → ∞ and α is fixed. These formulas are in terms of the zeros of the Bessel function J (x) and in terms of the zeros of the Airy function Ai(χ). They hold for k — 1, 2, ..., [qn] and for k — [pn], [pn] + 1, ..., n respectively, where p and q are fixed numbers in the interval (0, 1).

On quasi degree quadrature rules

SommarioIn questo lavoro si studiano le due formule di quadratura(1)

Remarks on asymptotics for Jacobi polynomials

\int\limits_{ - 1}^1 {(1 - x^2 )^{\lambda - 1/2} f(x)dx = C_n^{ (\lambda )} \sum\limits_{i = 1}^n f (x_{n,i} ) + R_n \left[ f \right]} ,