A curious $q$-analogue of Hermite polynomials

Johann Cigler (1) , Jiang Zeng (2)

1 Fakultät für Mathematik [Wien]
2 ICJ - Institut Camille Jordan

Two well-known $q$-Hermite polynomials are the continuous and discrete $q$-Hermite polynomials. In this paper we consider a new family of $q$-Hermite polynomials and prove several curious properties about these polynomials. One striking property is the connection with $q$-Fibonacci and $q$-Lucas polynomials. The latter relation yields a generalization of the Touchard-Riordan formula.

Domaines: Combinatoire [math.CO]; Analyse classique [math.CA]

Johann Cigler, Jiang Zeng. A curious $q$-analogue of Hermite polynomials. J. Combin. Theory Ser. A, 2011, 118 (1), pp.9--26. ⟨hal-00863436⟩