The zero distribution of the Tricomi-Carlitz polynomials

✍ Scribed by W.M.Y. Goh; J. Wimp

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 407 KB
• Volume: 33
• Category: Article
• ISSN: 0898-1221
• DOI: 10.1016/s0898-1221(96)00224-6

✦ Synopsis

In this paper, we improve our previous results concerning the asymptotic distribution of the zeros of the Tricomi-Carlitz polynomials. Our approach is a probabilistic one, motivated by the fact that the central limit theorem is simply a version of the weak law of laxge numbers whose distribution is a &function. Also, our argument requires an extension of the Grommer-Hamburger theorem. This generalization is appropriate to situations where uniform convergence on compact subsets is difficult to establish, and is accomplished by means of an approximation-theoretic tool, namely, the closure of a special set of rational functions.

Keywords--Orthogonal polynomials, Zero distribution, Saddle point method, Stieltjes transforms, Uniform convergence, Theorems of Grommer and Hamburger.