%for a more readable version of this page.
%
Since%
$$
\dvd{t_0,\ldots,t_k}f = \int M(\cdot|t_0,\ldots,t_k) D^kf/k!,
$$%
with $M$ the B-spline normalized to have integral 1, some integrals of the form%
$$
\int M(\cdot|t_0,\ldots,t_k) g
$$%
can (and certainly have been) computed from knowledge of divided differences,
as long as it is easy to obtain $D^{-k}g$ and compute its divided difference.
See pages004 %Moments and Fourier Transform of a
B-spline
for moments or the Fourier transform of a B-spline.
%
In particular, with%
$$
N(y|t_j,\ldots,t_{j+h}) = (t_{j+h}-t_j)\dvd{t_j,\ldots,t_{j+h}}(\cdot-y)^{h-1}
$$%
the B-splines (of order $h$) normalized to sum to 1,
we compute%
$$
\eqalign{\int M(\cdot| t_i,\ldots, t_{i+k}) N(\cdot| t_j,\ldots,t_{j+h})&\cr
&\kern-4truecm =(-1)^k{k+h-1\choose k}(t_{j+h}-t_j)
\dvd{t_i,\ldots,t_{i+k}}_x\dvd{t_j,\ldots,t_{j+h}}_y (x-y)_+^{k+h-1}.\cr}
$$%
The special case $h=k$ appears already, in slightly different normalization,
on p.~662 in
%
%JeromeSchumaker68 % larry %\refJ Jerome, J., Schumaker, L. L.; A note on obtaining natural spline functions by the abstract approach of Atteia and Laurent; SIAM J. Numer. Anal.; 5; 1968; 657--663; \bigskip\rightline{last updated \updated} % % \bye