Complete systems of recursive integrals and Taylor series for solutions of Sturm-Liouville equations

TL;DR

This paper introduces a recursive integral system generalizing power functions, explores its completeness and relation to Taylor expansions, and provides explicit formulas for derivatives of solutions to Sturm-Liouville equations.

Contribution

It develops a new recursive integral system related to transmutation operators and generalizes Taylor expansions for Sturm-Liouville solutions with explicit derivative formulas.

Findings

The recursive integral system is complete in various functional spaces.

A generalized Taylor theorem with explicit remainder is proven.

Explicit formulas for derivatives of Sturm-Liouville solutions are derived.

Abstract

Consider an arbitrary complex-valued, twice continuously differentiable, nonvanishing function $\phi$ defined on a finite segment $[a,b]\subset \mathbb{R}$. Let us introduce an infinite system of functions constructed in the following way. Each subsequent function is a primitive of the preceding one multiplied or divided by $\phi$ alternately. The obtained system of functions is a generalization of the system of powers ${(x-x_{0}%)^{k}}_{k=0}^{\infty}$. We study its completeness as well as the completeness of its subsets in different functional spaces. This system of recursive integrals results to be closely related to so-called $L$-bases arising in the theory of transmutation operators for linear ordinary differential equations. Besides the results on the completeness of the system of recursive integrals we show a deep analogy between the expansions in terms of the recursive integrals and Taylor expansions. We prove a generalization of the Taylor theorem with the Lagrange form of the remainder term and find an explicit formula for transforming a generalized Taylor expansion of a function in terms of the recursive integrals into a usual Taylor expansion. As a direct corollary of the formula we obtain the following new result concerning solutions of the Sturm-Liouville equation. Given a regular nonvanishing complex valued solution $y_{0}$ of the equation $y^{\prime\prime}+q(x)y=0$, $x\in(a,b)$, assume that it is $n$ times differentiable at a point $x_{0}% \in\lbrack a,b]$. We present explicit formulas for calculating the first $n$ derivatives at $x_{0}$ for any solution of the equation $u^{\prime\prime}+q(x)u=\lambda u$. That is, an explicit map transforming the Taylor expansion of $y_{0}$ into the Taylor expansion of $u$ is constructed.