I need to apply the operator
$$\exp\left( \alpha \frac{\partial^2}{\partial q^2}\right) \tag{1} \label{1}$$
To the function
$$M(x) N(y +C_{1}p)\mathcal{F}[f(q)](p) \tag{2} \label{2}$$
where $M(x)$ and $N(y + C_{1}p)$ are arbitrary functions, $C_{1}$ is a real constant and $$\mathcal{F}[f(q)](p)=\frac{1}{\sqrt{2\pi}} \int_{\infty}^{\infty}f(q)e^{-iqp}dq \tag{3} \label{D.1} $$
is the Fourier transform of the arbitrary function $f(q)$; is important to note that $q$ and $p$ is a conjugate pair (analogous to the position and moment variables of quantum mechanics), such that, in a quantum mechanical sense, the $y$ and $p$ variables are entangled. Specifically I need to know if I am correctly applying the operator $(\ref{1})$ to the fuction $(\ref{2})$, my process is as follows:
First, I write the function $(\ref{2})$ with the explicit Fourier transform and I expand the operator ($[\ref{1}]$) as an infinite sum of derivatives
$$\sum_{n=0}^{\infty} \frac{1}{n!}\alpha^n \frac{\partial^{2n}}{\partial q^{2n}} M(x) N(y +C_{1}p)\mathcal{F}[f(q)](p) $$ $$=\frac{1}{\sqrt{2\pi}} M(x)N(y+C_{1}p) \quad \sum_{n=0}^{\infty} \frac{1}{n!}\alpha^n \frac{\partial^{2n}}{\partial q^{2n}} \int_{-\infty}^{\infty}f(q)e^{-iqp}dq \tag{4}\label{4} $$
Second, I introduce the derivative inside the integral $$=\frac{1}{\sqrt{2\pi}} M(x)N(y+C_{1}p) \quad \sum_{n=0}^{\infty} \frac{1}{n!}\alpha^n \int_{-\infty}^{\infty} \frac{\partial^{2n}}{\partial q^{2n}} f(q)e^{-iqp}dq \tag{5}\label{5} $$
and then I use the derivative theorem of the Fourier transform:
$$\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \frac{\partial^{2n}}{\partial q^{2n}} f(q)e^{-iqp}dq =\frac{1}{\sqrt{2\pi}} (ip)^{2n} \int_{-\infty}^{\infty} f(q)e^{-iqp}dq \tag{D.2}$$
such that the line (\ref{5}) is
$$\frac{1}{\sqrt{2\pi}} M(x)N(y+C_{1}p) \quad \sum_{n=0}^{\infty} \frac{1}{n!}\alpha^n (ip)^{2n} \int_{-\infty}^{\infty} f(q)e^{-iqp}dq $$
$$= M(x)N(y+C_{1}p)e^{-\alpha p^2} \quad \mathcal{F}[f(q)](p) \tag{7}\label{7} $$
where I have reversed the infinite sum, then, the last line is my final result.
So, my question is: Am I correctly applying operator ($\ref{1}$) to the function ($\ref{2}$)?