order n constant coefficient differential equations and matrix exponential

Let $P$ be a degree $n>0$ monic complex polynomial in one indeterminate, let $f$ be a continuous function on the real line, let $k$ be an integer varying from 0 to $n-1$, and let $y_k$ be a complex number. The solution to the ODE

$$P(d/dt)y=f(t),y^{(k)}(0)=y_k$$(1)

is

$$y(t)=\sum y_k{g}_k(t)+{\int }_0^t{g}_{n-1}(t-x)f(x)𝑑x,$$(2)

where $g_k(t)$ is the coefficient of $z^k$ in the product of $P(z)$ by the singular part of

$$\frac{e^{tz}}{P(z)}\mathit{\hspace{1em}}.$$

Moreover, if $A$ is a complex square matrix annihilated by $P$, then

$$e^{tA}=\sum g_k(t)A^k.$$(3)

(1) into

$$Y^{\prime }-BY=f(t)v,Y(0)=Y_0$$(4)

by putting $Y_k:=y^{(k)}$, $Y_{0k}:=y_k$, and by letting $B$ be the transpose companion matrix of $P$, and $v$ the last vector of the canonical basis of ${ℂ}^n$. The solution to (4) is

$$Y(t)=e^{tB}{Y}_0+{\int }_0^{t}f(x)e^{(t-x)B}v𝑑x.$$

There is a unique $n$-tuple of functions $h_k$ such that$e^{tA}$ is the sum of the $h_k(t)A^k$ whenever $A$ is a complex square matrix annihilated by $P$. The first line of $B^k$ being the $(k+1)$-th vector of the canonical basis of ${ℂ}^n$ (for ), we obtain

$$y(t)=\sum y_k{h}_k(t)+{\int }_0^t{h}_{n-1}(t-x)f(x)𝑑x,$$

so that the proof of (2) and (3) boils down to verifying

$$h_k(t)=g_k(t).$$

a real value of $t$, let $G\in ℂ[X]$ be the sum of the $g_k(t)X^k$, form the entire function

$$\phi (z)=\frac{e^{tz}-G(z)}{P(z)}\mathit{\hspace{1em}},$$

multiply the above equality by $P(z)$, and replace $z$ by $A$.