Lattice paths and branched continued fractions. III. Generalizations of the Laguerre, rook and Lah polynomials

TL;DR

This paper introduces a new family of polynomials generalizing Laguerre, rook, and Lah polynomials, proves their total positivity properties, and connects them to branched continued fractions and multiple orthogonal polynomials.

Contribution

It defines a new triangular array of polynomials related to Laguerre digraphs, proves their total positivity, and links them to branched continued fractions and orthogonal polynomials.

Findings

The polynomials are totally positive under certain conditions.

The row-generating polynomials are Hankel-totally positive.

Connections to branched continued fractions and Bessel function-based weights.

Abstract

We introduce a triangular array $\widehat{\sf L}^{(\alpha)}$ of 5-variable homogeneous polynomials that enumerate Laguerre digraphs (digraphs in which each vertex has out-degree 0 or 1 and in-degree 0 or 1) with separate weights for peaks, valleys, double ascents, double descents, and loops. These polynomials generalize the classical Laguerre polynomials as well as the rook and Lah polynomials. We show that this triangular array is totally positive and that the sequence of its row-generating polynomials is Hankel-totally positive, under suitable restrictions on the values given to the indeterminates. This implies, in particular, the coefficientwise Hankel-total positivity of the monic unsigned univariate Laguerre polyomials. Our proof uses the method of production matrices as applied to exponential Riordan arrays. Our main technical lemma concerns the total positivity of a large class of quadridiagonal production matrices; it generalizes the tridiagonal comparison theorem. In some cases these polynomials are given by a branched continued fraction. Our constructions are motivated in part by recurrences for the multiple orthogonal polynomials associated to weights based on modified Bessel functions of the first kind $I_\alpha$.