2018/07/09 by Pétréolle, Mathias, Sokal, Alan D., Zhu, Bao-Xuan · 1 citation
#05A15 (Primary) #05A19 #05A20 #05C30 #05E05 #15B05 #15B48 #30B70 #30E05 #33C05 #33C20 #33D05 #44A60 (Secondary) #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1807.03271
We define an infinite sequence of generalizations, parametrized by an integer m ≥ 1, of the Stieltjes--Rogers and Thron--Rogers polynomials; they arise as the power-series expansions of some branched continued fractions, and as the generating polynomials for m-Dyck and m-Schröder paths with height-dependent weights. We prove that all of these sequences of polynomials are coefficientwise Hankel-totally positive, jointly in all the (infinitely many) indeterminates. We then apply this theory to prove the coefficientwise Hankel-total positivity for combinatorially interesting sequences of polynomials. Enumeration of unlabeled ordered trees and forests gives rise to multivariate Fuss--Narayana polynomials and Fuss--Narayana symmetric functions. Enumeration of increasing (labeled) ordered trees and forests gives rise to multivariate Eulerian polynomials and Eulerian symmetric functions, which include the univariate mth-order Eulerian polynomials as specializations. We also find branched continued fractions for ratios of contiguous hypergeometric series r Fs for arbitrary r and s, which generalize Gauss' continued fraction for ratios of contiguous 2 F1; and for s=0 we prove the coefficientwise Hankel-total positivity. Finally, we extend the branched continued fractions to ratios of contiguous basic hypergeometric series r ϕs.