ASYMPTOTIC NORMALITY OF THE COEFFICIENTS OF THE MORGAN-VOYCE POLYNOMIALS

Abstract

We study arithmetic and asymptotic properties of polynomials provided by Qn(x):=x∑k=1nkQn-k(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q_n(x):= x \\sum _{k=1}^n k \\, Q_{n-k}(x)$$\\end{document} with initial value Q0(x)=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q_0(x)=1$$\\end{document}. The coefficients satisfy a central limit theorem and a local limit theorem involving Fibonacci numbers. We apply the methods of Berry and Esseen, Harper, Bender, and Canfield.