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AN ALGEBRAIC INTERPRETATION OF THE SUPER CATALAN NUMBERS

Published online by Cambridge University Press:  06 November 2023

KEVIN LIMANTA*
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales 2052, Australia

Abstract

We extend the notion of polynomial integration over an arbitrary circle C in the Euclidean geometry over general fields $\mathbb {F}$ of characteristic zero as a normalised $\mathbb {F}$-linear functional on $\mathbb {F}[\alpha _1, \alpha _2]$ that maps polynomials that evaluate to zero on C to zero and is $\mathrm {SO}(2,\mathbb {F})$-invariant. This allows us to not only build a purely algebraic integration theory in an elementary way, but also give the super Catalan numbers

$$ \begin{align*} S(m,n) = \frac{(2m)!(2n)!}{m!n!(m+n)!} \end{align*} $$

an algebraic interpretation in terms of values of this algebraic integral over some circle applied to the monomials $\alpha _1^{2m}\alpha _2^{2n}$.

Type
Research Article
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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