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AN ALGEBRAIC INTERPRETATION OF THE SUPER CATALAN NUMBERS
Part of:
General field theory
Published online by Cambridge University Press: 06 November 2023
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 
$\alpha _1^{2m}\alpha _2^{2n}$.
MSC classification
- Type
- Research Article
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- Copyright
- © The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
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