Proof of a positivity conjecture on Schur functions

In the study of Zeilbergerʼs conjecture on an integer sequence related to the Catalan numbers, Lassalle proposed the following conjecture. Let (t)n denote the rising factorial, and let ΛR denote the algebra of symmetric functions with real coefficients. If φ is the homomorphism from ΛR to R defined by φ(hn)=1/((t)nn!) for some t>0, then for any Schur function sλ, the value φ(sλ) is positive. In this paper, we provide an affirmative answer to Lassalleʼs conjecture by using the Laguerre–Pólya–Schur theory of multiplier sequences.