Powers of the Catalan Generating Function and Lagrange’s 1770 Trinomial Equation Series  

作　　者：H.W.GOULD 

机构地区：[1]Department of Mathematics, West Virginia University

出　　处：《Journal of Mathematical Research with Applications》2019年第6期603-606,共4页数学研究及应用（英文版）

摘　　要：The Catalan numbers 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862,... are given by C(n)=1/n+1(2n n)for n ≥ 0. They are named for Eugene Catalan who studied them as early as 1838.They were also found by Leonhard Euler(1758), Nicholas von Fuss(1795), and Andreas von Segner(1758). The Catalan numbers have the binomial generating function C(z)=∞∑n=0C(n)z^n=1-√1-4z/2z It is known that powers of the generating function C(z) are given by C^a(z)=∞∑n=0a/a+2n(a+2n n)z^n The above formula is not as widely known as it should be. We observe that it is an immediate, simple consequence of expansions first studied by J. L. Lagrange. Such series were used later by Heinrich August Rothe in 1793 to find remarkable generalizations of the Vandermonde convolution. For the equation x^3-3x + 1 = 0, the numbers 1/2k+1(3k k)analogous to Catalan numbers occur of course. Here we discuss the history of these expansions. and formulas due to L. C. Hsu and the author.The Catalan numbers 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862,... are given by C(n)=■for n ≥ 0. They are named for Eugene Catalan who studied them as early as 1838.They were also found by Leonhard Euler(1758), Nicholas von Fuss(1795), and Andreas von Segner(1758). The Catalan numbers have the binomial generating function ■It is known that powers of the generating function C(z) are given by ■The above formula is not as widely known as it should be. We observe that it is an immediate, simple consequence of expansions first studied by J. L. Lagrange. Such series were used later by Heinrich August Rothe in 1793 to find remarkable generalizations of the Vandermonde convolution. For the equation x^3-3x + 1 = 0, the numbers ■analogous to Catalan numbers occur of course. Here we discuss the history of these expansions. and formulas due to L. C. Hsu and the author.

关 键 词：CATALAN NUMBERS VANDERMONDE convolution LAGRANGE and Bürmann SERIES Rothe's formula(or general identity of Rothe-Hagen) 

分 类 号：O15[理学—数学]