Cfrgtkky

I am interested in the calculation of the general $N$th derivative w.r.t. a parameter $lambda$ of a quantum mechanical exponential operator with the following structure:
begin{equation*}
frac{mathrm d^N}{mathrm d lambda^N} e^{-beta hat H(lambda)}
end{equation*}
In the case $n=1$ this reads as
begin{equation*}
sum_{n=0}^{+infty}sum_{m=0}^{n-1}frac{1}{n!}[hat H(lambda)]^{m}frac{mathrm dhat H(lambda)}{mathrm d lambda}[hat H(lambda)]^{n-m-1}
end{equation*}
Is there a compact fashion to rearrange the expression above for the general $N$th derivative?

Hint: Use repeatedly the identity for the 1st derivative
$$ frac{d}{dlambda}e^{that{A}} ~=~ int_0^t!dt_1~ e^{(t-t_1)hat{A}}frac{dhat{A}}{dlambda}e^{t_1hat{A}}~=~ iint_{mathbb{R}^2_+}!dt_1~dt_2~delta(t!-!t_1!-!t_2)~ e^{t_2hat{A}}frac{dhat{A}}{dlambda}e^{t_1hat{A}} , qquad t~in~mathbb{R}_+ .tag{1} $$
For a proof of eq. (1) see e.g. my Phys.SE answer here. Then the 2nd derivative becomes
$$begin{align} frac{1}{2!}frac{d^2}{dlambda^2}e^{that{A}}
&~=~iiint_{mathbb{R}^3_+}!dt_1~dt_2~dt_3~delta(t!-!t_1!-!t_2!-!t_3)~ e^{t_3hat{A}}frac{dhat{A}}{dlambda}e^{t_2hat{A}}frac{dhat{A}}{dlambda}e^{t_1hat{A}} cr
&~+~ iint_{mathbb{R}^2_+}!dt_1~dt_2~delta(t!-!t_1!-!t_2)~ e^{t_2hat{A}}frac{1}{2!}frac{d^2hat{A}}{dlambda^2}e^{t_1hat{A}} , qquad t~in~mathbb{R}_+ ,end{align}tag{2} $$
the 3rd derivative becomes
$$begin{align} frac{1}{3!}frac{d^3}{dlambda^3}e^{that{A}}
&~=~iiiint_{mathbb{R}^4_+}!dt_1~dt_2~dt_3~dt_4~delta(t!-!t_1!-!t_2!-!t_3!-!t_4)~ e^{t_4hat{A}}frac{dhat{A}}{dlambda}e^{t_3hat{A}}frac{dhat{A}}{dlambda}e^{t_2hat{A}}frac{dhat{A}}{dlambda}e^{t_1hat{A}} cr
&~+~iiint_{mathbb{R}^3_+}!dt_1~dt_2~dt_3~delta(t!-!t_1!-!t_2!-!t_3)~ e^{t_3hat{A}}frac{dhat{A}}{dlambda}e^{t_2hat{A}}frac{1}{2!}frac{d^2hat{A}}{dlambda^2}e^{t_1hat{A}} cr
&~+~iiint_{mathbb{R}^3_+}!dt_1~dt_2~dt_3~delta(t!-!t_1!-!t_2!-!t_3)~ e^{t_3hat{A}}frac{1}{2!}frac{d^2hat{A}}{dlambda^2}e^{t_2hat{A}}frac{dhat{A}}{dlambda}e^{t_1hat{A}} cr
&~+~ iint_{mathbb{R}^2_+}!dt_1~dt_2~delta(t!-!t_1!-!t_2)~ e^{t_2hat{A}}frac{1}{3!}frac{d^3hat{A}}{dlambda^3}e^{t_1hat{A}} , qquad t~in~mathbb{R}_+ ,end{align}tag{3} $$
and so forth. The $N$th derivative $$frac{1}{N!}frac{d^N}{dlambda^N}e^{that{A}}~=~sumstackrel{t_{n+1}}{rule{1cm}{.5mm}}fbox{$k_n$}stackrel{t_n}{rule{1cm}{.5mm}}cdots stackrel{t_3}{rule{1cm}{.5mm}}fbox{$k_2$}stackrel{t_2}{rule{1cm}{.5mm}}fbox{$k_1$}stackrel{t_1}{rule{1cm}{.5mm}}tag{N}$$
is a sum of possible Feynman diagrams with Schwinger propagators $$stackrel{t_i}{rule{1cm}{.5mm}}~=~e^{t_ihat{A}}, qquad t_i~in~mathbb{R}_+,tag{P}$$
and 2-vertices $$fbox{$k_i$}~=~frac{1}{k_i!}frac{d^{k_i}hat{A}}{dlambda^{k_i}}, qquad k_i~in~mathbb{N}.tag{V}$$ Each diagram in the sum has weight 1.