Abstract:   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 Λ_R denote the algebra of symmetric functions with real coefficients. If is the homomorphism from Λ_R to R defined by (h_n) = 1/((t)_nn!) for some t > 0, then for any Schur function s_λ, the value (s_λ) 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:  Primary 05E05; Secondary 26C10

  Keywords:   symmetric function, Schur function, multiplier sequence, totally positive sequence

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