A curious polynomial interpolation of Carlitz-Riordan's q-ballot numbers
Keywords:$q$-Catalan numbers, lattice paths, $q$-ballot numbers
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.
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