The Alternating Compositions of Weighted Differential Operators Yield The Weights' Wronskian With Which Constant?

TL;DR

This paper derives explicit formulas for the constants arising from alternating compositions of weighted differential operators, revealing a new integer sequence related to Wronskian determinants.

Contribution

It provides a novel explicit expression for the constants in compositions of differential operators, extending known cases to higher orders with exact and approximate values.

Findings

Exact values for the constants at p=4, 5, 6

Expression of the constants in terms of signed sums over specific permutations

Identification of a new integer sequence for these constants

Abstract

The alternated composition of $N=2p$ differential operators $ w_j(x)\,\partial_x^p$ of strict order $p$ on the line $\mathbb{R}\ni x$ is again a differential operator of strict order $p$; its coefficient is the constant $\mathrm{const}(p)$, depending only on the arity $N$, times the Wronskian determinant of the originally taken coefficients $w_1$, $\ldots$, $w_N$. The case $p=1$ of the Lie bracket for two vector fields fixes $\mathrm{const}(1)=1$. When $p=2$, finding $\mathrm{const}(2)=2$ is easy; we obtain $\mathrm{const}(3)=90$. The problem is to know $\mathrm{const}(p\geqslant 4)$. We express the formula of $\mathrm{const}(p)$ in terms of the sum with signs over the much smaller set of 'late-growing' permutations, thus reaching the exact values $c(p=4)= 586\,656$, $c(p=5)\approx 1.9\cdot 10^{12}$, and $c(p=6)\approx 7.9\cdot 10^{21}$; the positive integer sequence $\mathrm{const}(p)$ seems to be new.