I need to find the $m^\text{th}$ term for the following expression:
$$ \left.\frac{\partial^m}{\partial t^m}e^{a t^2}\right|_{t=0}$$ I computed first few terms and used mathematica "FindSequenceFunction", which yielded the $m^\text{th}$ term as:
$$ \frac{2^{m-1} \left(1+(-1)^{m}\right) a^{\frac{m}{2}} \Gamma \left(\frac{m+1}{2}\right)}{\sqrt{\pi }}$$
Question 1: Instead of following the above approach, I want to follow a direct approach:
res = D[Exp[a t^2], {t, m}]
res1 = res /. {t -> 0}
FullSimplify[%]
The corresponding three outputs are: $$\sqrt{\pi } 2^m t^{-m} \, _2\tilde{F}_2\left(\frac{1}{2},1;\frac{1}{2} (-m-1)+1,1-\frac{m}{2};a t^2\right)$$
$$\frac{\sqrt{\pi } 0^{-m}}{\Gamma \left(\frac{1-m}{2}\right) \Gamma \left(\frac{2-m}{2}\right)}$$ $$\frac{0^{-m}}{\Gamma (1-m)}$$
Since $a$ is not present in the answer, hence it is not correct.
Question 2: Hermite polynomial generating function:
resh = D[Exp[x t - t^2/2], {t, m}]
resh1 = resh /. {t -> 0}
FullSimplify[%]
Any suggestion on how to proceed further in this regard will be appreciated.