Pólya’s property 𝑊 and factorization—A short proof

Abstract

For an n n th order linear differential expression, the equivalence of Pólya’s Property W {\text {W}} and factorization into first order expressions is proven directly and briefly.

Reducibility of Zero Curvature Representations with Application to Recursion Operators

“…However, one encounters the problem of writing the inverse L −1 as an integro-differential operator. In the scalar case, L may be put in the form L = q n Dq n−1 D · · · q 1 Dq 0 , where the coefficients q i are expressible as quotients of wronskians of independent solutions v i of L(v) = 0 (see [31] for a simple derivation of this classical result, equivalent to decomposability into first-order factors; see [44] for the matrix case). In our context, q i are nonlocal functions and finding them is considered to be the most difficult part of the whole procedure.…”

Explicit conditions for the factorization of $n$th order linear differential operators

“…where Wx=yy and vVk=detiJ=1,... ,k[//~v] for k = 2,-■ ■ ,n-l. For a short and elegant proof of this factorization see [13]. A factorization of type (1.2) on an interval (a, b) is known to be equivalent to disconjugacy on (a, b).…”

Exact and Asymptotic Results for Insurance Risk Models with Surplus-dependent Premiums

“…, m + n) for u ∈ R + , such a factorization of T can always be achieved by wellknown techniques described for example in Eqn. (18) of [15]; see also [17] and [23]. Using this factorization, we can break down G in a way similar to (3.3) except that the A σi must be replaced by more complicated operators based on A and s i , similarly the B ρj by suitable operators involving B and r j .…”