Abstract
We give new sufficient conditions for a sequence of multivariate polynomials to be real stable. As applications, we obtain several known results, such as the real stability of multivariate Eulerian polynomials, multivariate Bell polynomials and multivariate polynomials over Stirling permutations, in a unified manner. And we also show some new results, such as the real stability of multivariate Lah polynomials and multivariate polynomials over Jacobi-Stirling permutations, and the proper position property of multivariate orthogonal polynomials and multivariate matching polynomials. Finally, we obtain a multivariate generalization of Fisk's result by presenting a 2×2 matrix, which preserves the proper position property.
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