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.