Quite similarly to this question, given the following expression: $$ P_n(x) = \sum_{0 \leq k \leq n/2} \binom{n}{k} x^k $$ Is it possible to find a closed form expression for: $$ G(x,y) = \sum_{n=0}^{+\infty} P_n(x) \frac{y^n}{n!} $$

(The difference with the former question is that here, there is an additional $k!$ in the expression of $G(x,y)$)

A few things that I seem to notice:

  • for $x=0$, we obviously have: $G(0,y) = e^y $
  • for $x$ small and $y$ large, it seems numerically quite close to $e^{(1+x)y}$

If you have any idea if this is possible to find another expression without involving power series for $G(x,y)$ I would be very grateful! Thank you !


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