In the acclaimed encyclopedia of special functions by Ismail and Van Assche published in 2020 the following generating function is given for Hermite polynomials (p. 77, eq. (3.8.48)), without source:
$$\sum_{n=0}^\infty \frac{(\gamma)_n}{(2n+1)!}H_{2n+1}(x)t^n = \frac{x t}{\sqrt{1 + t^2}}\, {_1}F{_1} \left(\gamma, \frac{3}{2}, \frac{t^2}{1+t^2}x^2\right)$$
This published equation can easily be shown to be erroneous numerically. For instance, the first derivative of the RHS taken at $t=0$ is $x$, for all parameter values, while the first derivative of the LHS taken at $t=0$ is $\,2\gamma\,x (\frac{2}{3} x^2-1)$.
Question: Find the correct formula, ideally preserving the first two parameters of the confluent hypergeometric function $_{1}F_{1}$ in the RHS.
Edit/Hint
The correct formula is:
$$\sum_{n=0}^\infty \frac{(\gamma)_n}{(2n+1)!}H_{2n+1}(x)t^{2n} = 2x(1+t^2)^{-\gamma}\, {_1}F{_1} \left(\gamma, \frac{3}{2}, \frac{t^2}{1+t^2}x^2\right)$$
I have a tentative proof, which I will make into an answer within two weeks, hoping that the community will find a better one in the meantime.
Reference
Ismail, Mourad E. H. and Van Assche, W., Encyclopedia of Special Functions: The Askey-Bateman Project, Volume I - Univariate Orthogonal Polynomials, Cambridge University Press, 2020.
Let us start from the following generating function of the generalized Laguerre polynomials:
\begin{equation} (1-u)^{-\gamma} {_1}F{_1}\left(\gamma, 1 + \alpha, \frac{ y\, u}{u-1}\right) = \sum_{n=0}^\infty \frac{(\gamma)_n}{(1+\alpha)_n} L_n^\alpha (y) u^n \tag{1} \end{equation}
Equation $(1)$ was actually first published in 1937 by Erdelyi [1937, Eq. (5,7)] then reestablished on independent grounds by F. Brafman [1951, p. 948, Eq. (28)], as a direct consequence of a general formula first obtained in 1943 by T.W. Chaundy [1943, p. 62], expanding generalized hypergeometric functions in series of associated polynomials.
Convergence conditions given in Erdelyi et al. [1953, Vol.~1, p.~276, 5.2 Eq.(6)] are: $|u| < 1, y > 0$, where $u$ may be complex by default. In Erdelyi's original publication, the condition on $y$ is more general, it is stated as $|\arg(y)| < \frac{3\pi}{4}$.
Setting:
$$\alpha = \frac{1}{2},\; y = x^2,\; u=-t^2 \quad \text{ with } \quad |t| < 1$$
in $(1)$, then recalling that Laguerre and Hermite polynomials are related by the well-known equation [NIST 2010, eq. 18.7.20, p. 445]:
$$H_{2n+1}(x) = (-1)^n 2^{2n+1}\, n! \, x\, L_{n}^{\frac{1}{2}}(x^2) $$
and noting that, by Legendre's duplication formula:
$$\left(\frac{3}{2}\right)_n 2^{2n+1}\, n! = \frac{2^{2n+2}\Gamma\left(n+ \frac{3}{2}\right) \Gamma(n+1)}{\sqrt{\pi}} = 2\,(2n+1)!$$
straightforwardly yield the corrected version of the posted generating function:
\begin{align} &\sum_{n=0}^\infty \frac{(\gamma)_n}{(2n+1)!}H_{2n+1}(x)t^{2n} = 2x(1+t^2)^{-\gamma}\, {_1}F{_1} \left(\gamma, \frac{3}{2}, \frac{t^2}{1+t^2}x^2\right) \end{align}
which is valid for all $x$ real and $t$ possibly complex, $|t|<1$, hence in domains compatible with iterated derivation of the RHS with respect to the real variable $t$ in the neighbourhood of zero for all real values of $x$.
(Edit)
The fact that $u=t^2$ can be complex (with $|u| < 1$) leads to a very short proof of this posted expansion of Dawson's function in a series of Hermite polynomials.
References
Brafman 1951, Generating functions of Jacobi and related polynomials, Proceedings of the American Mathematical Society, 2(6), pp. 942-949, American Mathematical Society.
Chaundy 1943, An extension of hypergeometric functions (I), The Quarterly Journal of Mathematics, os-14(1), pp. 55-78.
Erdélyi 1937, Funktionalrelationen mit konfluenten hypergeometrischen Funktionen, Mathematische Zeitschrift, 42(1), pp. 641-670
NIST 2010, Handbook of Mathematical Functions Hardback and CD-ROM Olver, F.W.J. and National Institute of Standards and Technology (U.S.) and Lozier, D.W. and Boisvert, R.F. and Clark, C.W.