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Let $a,b,c:\mathbb{R}\rightarrow\mathbb{R}$ be differentiable functions, with $a(t)\rightarrow -\infty$ and $c(t)\rightarrow -\infty$ as $t\rightarrow -\infty$, and with $a(t)\rightarrow\infty$ and $c(t)\rightarrow\infty$ as $t\rightarrow\infty$.

Let $\phi:\mathbb{R}\rightarrow\mathbb{R}^+$ be the density of a standard normal random variable, meaning that for all $z\in\mathbb{R}$: $$\phi(z)=\frac{1}{\sqrt{2\pi}} \exp{\left(-\frac{z^2}{2}\right)}.$$

For $j\in\mathbb{N}$, let $z_j:\mathbb{R}\rightarrow\mathbb{R}$ with: $$z_j(t)=\int_{-\infty}^t{\exp{\left(a(\tau)-a(t)\right)}\,\left(b(\tau)+c(t)-c(\tau)\right)^j\,\phi\left(b(\tau)+c(t)-c(\tau)\right)\,d\tau},$$ for all $t\in\mathbb{R}$, where you may assume the integral exists and is finite for all $j\in\mathbb{N}$ and $t\in\mathbb{R}$.

Finally, let $z_{-1}:\mathbb{R}\rightarrow\mathbb{R}$.

Then from differentiating $z_j(t)$ (in $t$), we have that for $j\in\mathbb{N}$ and $t\in\mathbb{R}$: $$z_j'(t)=\left(b(t)\right)^j\,\phi\left(b(t)\right)-a'(t)z_j(t)+c'(t)\left(jz_{j-1}(t)-z_{j+1}(t)\right),$$ where, as usual, dashes denote first derivatives.

I want to derive a differential equation for $z_0(t)$ that does not depend on $z_1(t),z_2(t),\dots$. Is there a way to solve the recurrence equation (holding $t$ fixed) here to leave a differential equation? Or is there some other direct way to get a differential equation for $z_0(t)$?

The solution should be somehow related to the recurrence relation for $z_j$ ($j\in\mathbb{N}$): $$0=\beta^j\varphi-\alpha z_j+\gamma\left(jz_{j-1}-z_{j+1}\right).$$ When $\phi=0$ or $\beta=0$, and $\alpha=-\gamma$, this gives: $$z_{j+1}=z_j+jz_{j-1},$$ which is the sequence here and here (the number of involutions on a finite set) for appropriate initial conditions.

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