= Solution
Set $r=k-1$, with $k\ge2$ for the final linear-reproduction assertion, and use the normalized knot <polynomial> $\psi_i=\omega_i/r!$. Differentiate the <Marsden identity> $r-j$ times with respect to the auxiliary variable $x$, for $0\le j\le r$. This gives
$$
\frac{r!}{j!}(x-t)^j=\sum_i\omega_i^{(r-j)}(x)N_i(t),
\qquad
\frac{(x-t)^j}{j!}=\sum_i\psi_i^{(r-j)}(x)N_i(t).
$$
The <Taylor polynomial> expansion of an arbitrary <polynomial> $p$ of degree at most $r$ about $x$ is exact:
$$
p(t)=\sum_{j=0}^r\frac{p^{(j)}(x)}{j!}(t-x)^j.
$$
Substitute the differentiated identity into each term and collect the <B-spline> coefficients. The result is
$$
p(t)=\sum_i\lambda_i(p)N_i(t),\qquad
\boxed{\lambda_i(p)=\sum_{j=0}^r(-1)^jp^{(j)}(x)\psi_i^{(r-j)}(x).}
$$
This is the <normalized Marsden dual functional>. There is no extra factorial outside the sum, because it is already included in $\psi_i$.
To prove independence of $x$ directly, differentiate the finite sum. The contribution from differentiating $\psi_i$ in term $j$ cancels the contribution from differentiating $p$ in term $j-1$. Explicitly,
$$
\frac{d}{dx}\lambda_i(p)
=\sum_{j=0}^r(-1)^jp^{(j+1)}\psi_i^{(r-j)}
+\sum_{j=0}^r(-1)^jp^{(j)}\psi_i^{(r-j+1)}=0.
$$
The two uncancelled endpoint terms vanish because $p^{(r+1)}=\psi_i^{(r+1)}=0$. Thus the coefficient functional is constant in the auxiliary variable, as required.
The leading two terms of the knot <polynomial> give
$$
\psi_i^{(r)}(x)=1,\qquad
\psi_i^{(r-1)}(x)=x-\frac1r\sum_{j=1}^rt_{i+j}=x-t_i^*.
$$
For a linear <polynomial> $p(t)=A+Bt$, only $j=0,1$ survive, whence
$$
\lambda_i(p)=p(x)-(x-t_i^*)p'(x)=A+Bt_i^*=p(t_i^*).
$$
The coefficient points $t_i^*$ are the <Greville abscissae>. Inserting these coefficients proves exact linear reproduction:
$$
\boxed{p(t)=\sum_i p(t_i^*)N_i(t)\quad\text{for every }p\in\mathcal P_1.}
$$
All identities hold on the basic knot interval $[t_k,t_{n+1}]$, including the endpoints with the standard continuous endpoint convention. Repeated knots cause no difficulty in these <polynomial> differentiations.
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