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Ballot matrix as Catalan matrix power and related identities

https://doi.org/10.1016/j.dam.2011.10.016Get rights and content
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Abstract

We use an analytical approach to find the kth power of the Catalan matrix. Precisely, it is proven that the power of the Catalan matrix is a lower triangular Toeplitz matrix which contains the well-known ballot numbers. A result from [H. S. Wilf, Generatingfunctionology, Academic Press, New York, 1990, Free download available from http://www.math.upenn.edu/~wilf/Downld.html.], related to the generating function for Catalan numbers, is extended to the negative integers. Three interesting representations for Catalan numbers by means of the binomial coefficients and the hypergeometric functions are obtained using relations between Catalan matrix powers.

Keywords

Ballot number
Ballot matrix
Catalan number
Catalan matrix
Generalized hypergeometric function
Generating function

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