Diego Dominici

The Inverse of the Cumulative Standard Normal Probability Function

Some properties of the inverse of the function

Asymptotic analysis of the Hermite polynomials from their differential–difference equation

N\lpar x\rpar = \lpar1/\sqrt{2\pi}\rpar \vint_{-\infty}^{x}e^{-t^2/2}\,\hbox{d}t

Nested derivatives: a simple method for computing series expansions of inverse functions

are studied. Its derivatives, integrals and asymptotic behavior are presented.

Generalized Heine's identity for complex Fourier series of binomials

We analyze the Hermite polynomials H n (x) and their zeros asymptotically, as n → ∞ We obtain asymptotic approximations from the differential–difference equation which they satisfy, using the ray method. We give numerical examples showing the accuracy of our formulas.

Asymptotic analysis of the Askey-scheme I: from Krawtchouk to Charlier

We give an algorithm to compute the series expansion for the inverse of a given function. The algorithm is extremely easy to implement and gives the first N terms of the series. We show several examples of its application in calculating the inverses of some special functions.

Discrete semiclassical orthogonal polynomials of class one

In his treatise, Heine (Heine 1881 In Theorie und Anwendungen) gave an identity for the Fourier series of the function , with , and z>1, in terms of associated Legendre functions of the second kind . In this paper, we generalize Heine’s identity for the function , with , and , in terms of . We also compute closed-form expressions for some .

Asymptotic analysis of generalized Hermite polynomials

We analyze the Charlier polynomials Cn(χ) and their zeros asymptotically as n → ∞. We obtain asymptotic approximations, using the limit relation between the Krawtchouk and Charlier polynomials, involving some special functions. We give numerical examples showing the accuracy of our formulas.

A new Kapteyn series

We study discrete semiclassical orthogonal polynomials of class s D 1. By considering particular solutions of the Pearson equation, we obtain five canonical families of such polynomials. We also consider limit relations between these and other families of orthogonal polynomials.

A remarkable identity involving Bessel functions

We analyze the polynomials Hnr(x) considered by Gould and Hopper, which generalize the classical Hermite polynomials. We present the main properties of Hnr(x) and derive asymptotic approximations for large values of n from their differential-difference equation, using a discrete ray method. We give numerical examples showing the accuracy of our formulas.

On Taylor series and Kapteyn series of the first and second type

We study the Kapteyn series . We find a series representation in powers of z and analyze its radius of convergence.