Powers of the Catalan Generating Function and Lagrange's 1770 Trinomial Equation Series
Received:July 15, 2019  Revised:October 10, 2019
Key Word: Catalan numbers   Vandermonde convolution   Lagrange and B\"{u}rmann series   Rothe's formula (or general identity of Rothe-Hagen)
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 Author Name Affiliation H. W. GOULD Department of Mathematics, West Virginia University, PO Box 6310 Morgantown, WV 26506-6310, U. S. A
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The Catalan numbers $1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862,\ldots$ are given by $C(n)=\frac{1}{n+1}\binom{2n}{n}$ for $n\geq 0$. 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 $$\mathbf{C}(z) = \sum_{n=0}^{\infty}C(n)z^n = \frac{1 - \sqrt{1-4z}}{2z}$$ It is known that powers of the generating function $\mathbf{C}(z)$ are given by $$\mathbf{C}^a(z) = \sum_{n=0}^{\infty}\frac{a}{a+2n}\binom{a+2n}{n}z^n.$$ 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 $\frac{1}{2k+1}\binom{3k}{k}$ 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.