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We introduce a triangular array $\widehat{\sf L}^{(\alpha)}$ of 5-variablehomogeneous polynomials that enumerate Laguerre digraphs (digraphs in whicheach vertex has out-degree 0 or 1 and in-degree 0 or 1) with separate weightsfor peaks, valleys, double ascents, double descents, and loops. Thesepolynomials generalize the classical Laguerre polynomials as well as the rookand Lah polynomials. We show that this triangular array is totally positive andthat the sequence of its row-generating polynomials is Hankel-totally positive,under suitable restrictions on the values given to the indeterminates. Thisimplies, in particular, the coefficientwise Hankel-total positivity of themonic unsigned univariate Laguerre polyomials. Our proof uses the method ofproduction matrices as applied to exponential Riordan arrays. Our maintechnical lemma concerns the total positivity of a large class ofquadridiagonal production matrices; it generalizes the tridiagonal comparisontheorem. In some cases these polynomials are given by a branched continuedfraction. Our constructions are motivated in part by recurrences for themultiple orthogonal polynomials associated to weights based on modified Besselfunctions of the first kind $I_\alpha$.

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