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

Abstract

A good portion of Gatteschi's research publications-about 65%-is devoted to asymptotics of special functions and their zeros. Most prominently among the special functions studied figure classical orthogonal polynomials, notably Jacobi polynomials and their special cases, Laguerre polynomials, and Hermite polynomials by implication. Other important classes of special functions dealt with are Bessel functions of the first and second kind, Airy functions, and confluent hypergeometric functions, both in Tricomi's and Whittaker's form. This work is reviewed here, and organized along methodological lines.

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

“…Both (3.6) and (3.7) can be generalised to roots of more general Jacobi polynomials. These results, and many others, can be found in a survey of Gatteschi's work [19]. Lether [31] investigates which of the above approximations to use for each k, and an empirical rule suggested by Yakimiw [51] is to use (3.6) whenk ≤ ⌈0.063(n + 33)(n − 1.5)/n⌉ and (3.4) otherwise.…”

“…The first term when evaluating the derivative at the nodes is then given by 19) and by the same argument as above for the interior formula, we require an O(1) the term contained in the square parentheses to obtain an O(1) relative error in weights. Since, for a fixed k, θk and sin θk are O(n −1 ), we must then demand an O(n −1 ) error in the numerator of this term.…”

On a conjectured inequality for the largest zero of Jacobi polynomials

“…The asymptotic behavior for large degree n of the zeros of Jacobi polynomials (ordered decreasingly) has been studied extensively by L. Gatteschi (see also the paper [2] in this issue). A particular result that is relevant to us holds for the kth zero, where k is fixed, and specialized to the case k = 1 reads as follows [1, Theorem 4.1]: If α > −1 and β > −1 (actually, β could be arbitrary real), then…”

Evaluation of q-gamma function and q-analogues by iterative algorithms

“…Considering the series in (8), an estimate of the remainder of the infinite product in (7) is available. To see this we write (7) in the form…”