An Arithmetic Metric

What is the distance between 11 (a prime number) and 12 (a highly composite number)? If your answer is 1, then ask yourself "is this reasonable?" In this work, we will introduce a distance between natural numbers based on their arithmetic properties, instead of their position on the real line

Some observations on a Kapteyn series

We study the Kapteyn series $t^{n}\mathrm{J}_{n}(nz)$. We find a series representation in powers of $z$ and analyze its radius of convergence

An application of Kapteyn series to a problem from queueing theory

We obtain exact solutions of a problem arising from queueing theory using properties of Kapteyn series.Comment: 2 page

Asymptotic analysis of the Krawtchouk polynomials by the WKB method

We analyze the Krawtchouk polynomials K(n,x,N,p,q) asymptotically. We use singular perturbation methods to analyze them for N large with appropriate scalings of the two variables x and n. In particular, the WKB method and asymptotic matching are used. We obtain asymptotic approximations valid in the whole domain [0,N]x[0,N] involving some special functions. We give numerical examples showing the accuracy of our formulas.Comment: 41 pages, 14 figure

Asymptotic analysis of generalized Hermite polynomials

We analyze the polynomials $H_{n}^{r}(x)$ considered by Gould and Hopper, which generalize the classical Hermite polynomials. We present the main properties of $H_{n}^{r}(x)$ 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.Comment: 28 pages, 6 figure

Fisher information of orthogonal polynomials I

Following the lead of J. Dehesa and his collaborators, we compute the Fisher information of the Meixner-Pollaczek, Meixner, Krawtchouk and Charlier polynomials.Comment: 12 page

Asymptotic analysis of the Bell polynomials by the ray method

We analyze the Bell polynomials $B_{n}(x)$ asymptotically as $n\to\infty$. We obtain asymptotic approximations from the differential-difference equation which they satisfy, using a discrete version of the ray method. We give some examples showing the accuracy of our formulas.Comment: 7 pages, 1 figur

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

We analyze the Charlier polynomials C(n,x)and their zeros asymptotically for large 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.Comment: 29 pages, 9 figure

Variations on a Theme by James Stirling

We present the history and previous approaches to the proof of Stirling's series. We use a different procedure, based on the asymptotic analysis of the difference equation $\Gamma(z+1)=z\Gamma(z)$. The method reproduces Stirling's series very easily and can be extended to use in more complicated difference equations.Comment: 16 page

Mehler-Heine type formulas for Charlier and Meixner polynomials

We derive Mehler--Heine type asymptotic formulas for Charlier and Meixner polynomials, and also for their associated families. These formulas provide good approximations for the polynomials in the neighborhood of $x=0,$ and determine the asymptotic limit of their zeros as the degree $n$ goes to infinity.Comment: 25 pages, 2 figure