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AN ALGEBRAIC INTERPRETATION OF THE SUPER CATALAN NUMBERS
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
an algebraic interpretation in terms of values of this algebraic integral over some circle applied to the monomials $\alpha _1^{2m}\alpha _2^{2n}$.
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- Research Article
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- © The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.