Maclaurin series of $\exp(-ab^2 x^2+2abc x)$

Question

I have been looking for the Maclaurin series of \begin{equation} f(x) = e^{-ab^2 x^2+2abc x}, \end{equation} for $a>0$, $b\in\Bbb R$, and $c\in\Bbb R$. I need the series in the form \begin{equation} f(x) = \sum_{n=0}^{\infty}c_n x^n. \end{equation} I have been unable to find a source for this and know that I need an expression for $D^n f(x)|_{x=0}$. Seems like it may be tied to orthogonal polynomials. Is anyone aware of a source that has an expression for this series and details on its radius of convergence?

Answer 1

The result is $$\sum_{k\ge 0}\dfrac{(-ab^2)^k}{k!}x^{2k}\sum_{l\ge 0}\dfrac{(2abc)^l}{l!}x^l=\sum_{m\ge 0}c_m x^m,\,c_m:=\sum_{2k+l=m}\dfrac{(-ab^2)^k (2abc)^l}{k!l!}.$$

Answer 2

By completing the square one has: \begin{align} e^{-a b^2 x^2 + 2 a b c x} &= e^{-a ((b x)^2 - 2 (c)(b x) + c^2 - c^2} \\ &= e^{a c^2} \, e^{-a (b x -c)^2} \\ &= e^{a c^2} \, \sum_{n=0}^{\infty} \frac{(-a)^{n} \, (b x - c)^{2n}}{n!} \end{align}

Alternatively, with the use of Hermite polynomials generated by \begin{align} \sum_{n=0}^{\infty} H_{n}(x) \, \frac{t^{n}}{n!} = e^{- t^{2} + 2 x t} \end{align} one can determine that $$e^{- a b^2 x^2 + 2 a b c x} = \sum_{n=0}^{\infty} (\sqrt{a} \, b)^{n} \, H_{n}(\sqrt{a} \, c) \, \frac{x^{n}}{n!}.$$