Elin Ottergren: Multiplier sequences for generalized Laguerre bases

Consider a linear operator acting diagonally on the standard monomial basis. Pólya and Schur completely characterized all such operators perserving real-rootedness in their famous article from 1914. The corresponding eigenvalues are from then on known as multiplier sequences. In 2009 Borcea & Brändén gave a complete characterization for general linear operators preserving real-rootedness (and stability) via the symbol. Using these results we were recently able to characterize multiplier sequences with respect to the Laguerre polynomials as a basis for R[x].  

  arxiv.org/abs/1210.7151