  We prove that every multiplier sequence for the Legendre basis which can be
interpolated by a polynomial has the form $\{h(k^2+k)\}_{k=0}^{\infty}$, where
$h\in\mathbb{R}[x]$. We also prove that a non-trivial collection of polynomials
of a certain form interpolate multiplier sequences for the Legendre basis, and
we state conjectures on how to extend these results.
