A closed-form for higher-order derivatives of the Dawson integral?

Question

We have as the Dawson integral $$ \mathcal D(x):=e^{-x^2}\int_0^xe^{t^2}\,\mathrm dt. $$ I am interested in an expression for $\mathcal D^{(n)}(x)$ that does not involve sums. For example, we could use the relation $$ \mathcal D(x)=\tfrac{1}{2}\sqrt\pi e^{-x^2}\operatorname{erfi}(x) $$ to write $$ \mathcal D^{(n)}(x)=\tfrac{1}{2}\sqrt\pi \sum_{k=0}^n\binom{n}{k} (\partial_x^ke^{-x^2})(\partial_x^{n-k}\operatorname{erfi}(x)) $$ and each of the remaining higher-order derivatives have closed-forms in terms of hypergeometric functions and hermite polynomials. But can we write $\mathcal D^{(n)}(x)$ without sums? My thought was to find a hypergeometric form for $\mathcal D(x)$ and then apply the well-known formulae for derivatives of hypergeometric functions to obtain the final result; however, I have not been able to find such a form of the Dawson integral. Thoughts?

Answer 1

@ClaudeLeibovici Provided reference to the paper The derivatives of Dawson's function (Barakat,1971) found here. The paper gives the form $$ \mathcal D^{(n)}(x)=(-1)^n(H_n(x)\mathcal D(x)-G_{n-1}(x)), $$ where $H_n(x)=(-1)^ne^{x^2}\partial_x^ne^{-x^2}$ is the $n$th degree Hermite polynomial and $G_n(t)$ is an $n$th degree polynomial satisfying $$ G_n(x)=2xG_{n-1}(x)-2nG_{n-2}(x) $$ with $G_0(x)=1$ and $G_1(x)=2x.$