A curious polynomial interpolation of Carlitz-Riordan's q-ballot numbers

Frédéric Chapoton, Jiang Zeng


We study a polynomial sequence $C_n(x|q)$ defined as a solution of a $q$-difference equation. This sequence, evaluated at $q$-integers, interpolates Carlitz--Riordan's $q$-ballot numbers. In the basis given by some kind of $q$-binomial coefficients, the coefficients are again some $q$-ballot numbers. We obtain another curious recurrence relation for these polynomials in a combinatorial way.


$q$-Catalan numbers; lattice paths; $q$-ballot numbers

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Contributions to Discrete Mathematics. ISSN: 1715-0868