The combinatorics of associated Hermite polynomials

We develop a combinatorial model of the associated Hermite polynomials and their moments, and prove their orthogonality with a sign-reversing involution. We find combinatorial interpretations of the moments as complete matchings, connected complete matchings, oscillating tableaux, and rooted maps and show weight-preserving bijections between these objects. Several identities, linearization formulas, the moment generating function, and a second combinatorial model are also derived. The associated Hermite polynomials are a sequence of orthogonal polynomials considered by Askey and Wimp in [AW84], who analytically derived a number of results about these polynomials. They are also treated in [Ism05, Section 5.6]. In section 1 we provide a combinatorial interpretation of these polynomials, their moments, and describe an involution that proves the orthogonality and L norms of the polynomials with respect to those moments. Then in section 2 we shall describe several linearization formulas involving associated Hermite polynomials. We finish with weight-preserving bijections between a number of classes of combinatorial objects whose generating functions all yield the moments of the associated Hermites, and a second combinatorial model for the polynomials. We will assume that the reader is familiar with Viennot’s general combinatorial theory of orthogonal polynomials [Vie83, Vie85] and with the combinatorics of Hermite polynomials; see [AGV82, dSCV85, LY89] and also [Vie83, §II.6]. In this paper we use [n] to mean the set of integers 1 to n, inclusive, and write [n] ⊔ [m] for the disjoint union of two such sets. 1. Definition and orthogonality The associated Hermite polynomials may be defined by shifting the recurrence relation for the usual Hermite polynomials, which is Hn+1(x) = xHn(x)− nHn−1(x), to (1.1) Hn+1(x; c) = xHn(x; c)− (n− 1 + c)Hn−1(x; c), with H0(x) = H0(x; c) = 1 and polynomials with negative indices equal to zero. Askey and Wimp use a different normalization than we do; one obtains our normalization from plugging x/ √ 2 and c− 1 into their associated Hermites and dividing by (√ 2 )n . The usual Hermite polynomial Hn+1(x) is the generating function for incomplete matchings of [n + 1], in which fixed points have weight x and edges have weight −1; that combinatorial interpretation can be derived from the recurrence relation as follows: the vertex n+1 may be fixed Date: 16 May 2008. 2000 Mathematics Subject Classification. Primary: 05E35; Secondary: 33C45.