This is a companion question to another one that I posted here yesterday.
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. (3.8.47)), without source:
$$\sum_{n=0}^\infty \frac{(\gamma)_n}{(2n)!}H_{2n}(x)t^n = (1+t^2)^{-\gamma}\, {_1}F{_1} \left(\gamma, \frac{1}{2}, \frac{t^2}{1+t^2}x^2\right)$$
This published equation can easily be shown to be erroneous numerically. For instance, the first derivative of the RHS taken at $t=0$ is uniformly $0$, for all parameter values, while the first derivative of the LHS taken at $t=0$ is $\,\gamma (2 x^2-1)$.
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 just the original one with even powers of the variable $t$ in the LHS expansion:
$$\sum_{n=0}^\infty \frac{(\gamma)_n}{(2n)!}H_{2n}(x)t^{2n} = (1+t^2)^{-\gamma}\, {_1}F{_1} \left(\gamma, \frac{1}{2}, \frac{t^2}{1+t^2}x^2\right) $$
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.
As for the companion question, we start with 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}
(see references in the companion question). Convergence conditions are: $|u| < 1, y > 0$, where $u$ may be complex by default.
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. 18.7.19, p. 445]:
$$H_{2n}(x) = (-1)^n 2^{2n}\, n! \, L_{n}^{-\frac{1}{2}}(x^2) $$
and noting that, by Legendre's duplication formula:
$$\left(\frac{1}{2}\right)_n 2^{2n}\, n! = \frac{2^{2n}\Gamma\left(n+ \frac{1}{2}\right) \Gamma(n+1)}{\sqrt{\pi}} = (2n)!$$
straightforwardly yield the corrected version of the posted generating function:
\begin{align} &\sum_{n=0}^\infty \frac{(\gamma)_n}{(2n)!}H_{2n}(x)t^{2n} = (1+t^2)^{-\gamma}\, {_1}F{_1} \left(\gamma, \frac{1}{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 $x$.
References
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.