On Legendre Multiplier Sequences

Abstract

In this paper we give a complete characterization of linear, quadratic, and geometric Legendre multiplier sequences. We also prove that all Legendre multiplier sequences must be Hermite multiplier sequences, and describe the relationship between the Legendre and generalized Laguerre multiplier sequences. We conclude with a list of open questions for further research. 26C10, 30C15

The nonexistence of cubic Legendre multiplier sequences

“…We now reprove the non-existence of linear Legendre multiplier sequences (see [1,Proposition 2]). Although the result is known, our proof is novel, and has the promise of being suitable for use when investigating Q-multiplier sequences in larger generality.…”

“…Some of the work was completed while the first author was on sabbatical leave at the University of Hawai'i at Manoa, whose support he gratefully acknowledges. line of inquiry by settling a conjecture on the non-existence of cubic Legendre multiplier sequences (Open problem (1) in [1]). In addition, we give a new proof of the non-existence of linear Legendre multiplier sequences, which is more methodical than the educated hunt for test polynomials whose zeros fail to remain real after having been acted on by a linear sequence.…”

“…In Section 2 we present a number of known results which are relevant to our investigations. Section 3 exhibits a new proof of the non-existence of linear Legendre multiplier sequences (Proposition 2 in [1]) using a theorem of Borcea and Brändén. Our method exploits the fact that one does not need to have full knowledge of all coefficient polynomials T k (x) of a linear operator T = ∞ k=0 T k (x)D k in order to decide whether or not T is reality preserving.…”

“…In 1914, Pólya and Schur completely characterized classical multiplier sequences. Their seminal theorem maintains relevance in the setting of Legendre multiplier sequences, since every Legendre multiplier sequence must also be a classical multiplier sequence (see [1,Theorem 8] together with [11,Proposition 118]). We note that if {γ k } ∞ k=0 is a classical multiplier sequence, then either…”

Quadratic hyperbolicity preservers and multiplier sequences

“…(1) f and g have interlacing zeros with form (1) or (4) in Definition 3 and the leading coefficients of f and g are of the same sign, or (2) f and g have interlacing zeros with form (2) or (3) in Definition 3, and the leading coefficients of f and g are of opposite sign. We will say that the zero polynomial is in proper position with any other hyperbolic polynomial f and write 0 ≪ f or f ≪ 0.…”