Proof of a positivity conjecture on Schur functions

In the study of Zeilberger’s 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 Lassalle’s conjecture by using the Laguerre–Pólya-Schur theory of multiplier sequences. AMS Classification 2010: Primary 05E05; Secondary 26C10.