= Solution
Put $d=k-1$ and let $e_m$ be the <elementary symmetric polynomial> of degree $m$ in the $d$ interior <spline knots> $t_{i+1},\ldots,t_{i+d}$, with $e_0=1$. Expanding the knot polynomial gives
$$
\omega_i(x)=\prod_{r=1}^d(x-t_{i+r})
=\sum_{m=0}^d(-1)^m e_m(t_{i+1},\ldots,t_{i+d})x^{d-m}.
$$
Compare the coefficient of $x^{d-m}$ in the <Marsden identity>. The left coefficient is $(-1)^m\binom dm t^m$; the right coefficient is $(-1)^m\sum_i e_m(t_{i+1},\ldots,t_{i+d})N_i(t)$. Cancelling the common sign therefore gives the <monomial B-spline coefficients>:
$$
\boxed{a_i^{(m)}=\frac{e_m(t_{i+1},\ldots,t_{i+k-1})}{\binom{k-1}{m}},\qquad 0\le m\le k-1.}
$$
Equivalently, for $m\ge1$ the numerator is $\sum_{1\le r_1<\cdots<r_m\le k-1}t_{i+r_1}\cdots t_{i+r_m}$. In particular $a_i^{(0)}=1$, so the order-zero case reproduces constants, and for $k\ge2$ the linear coefficients are $a_i^{(1)}=(t_{i+1}+\cdots+t_{i+k-1})/(k-1)$, the <Greville abscissae>. At $m=k-1$, the coefficient is the product of all interior knots. Each formula holds on the basic knot interval used in the <Marsden identity>.
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