I've got a polynomial (which comes from solutions of the heat conduction PDE) which seems so simple I'm wondering if anyone recognizes it


$$f\_{m}=x^{m-1}
+(m-1)x^{m-3}+(m-1)(m-3)x^{m-5}
+(m-1)(m-3)(m-5)x^{m-7}
+\cdots$$


where the sum terminates when the exponent becomes negative
This can be written with double factorials where we sum over k


$$f\_{m}=\sum\_{0\le k\le (m-1)/2} \frac{(m-1)!!x^{m-1-2k}}{(m-1-2k)!!}$$


it seems so simple, I was wondering if anyone recognized it.


What I need to calculate down the road is the polynomial


$$\sum\_{m=0}^{n}\frac{x^{n-m}f\_{m}}{m!(n-m)!}$$



do you recognize this polynomial with double factorials?
For $m$ an odd integer there is a closed-form expression in terms of the incomplete Gamma function$\Gamma(a,z)$,
$$f\_{m}=\sum\_{0\le k\le (m-1)/2} \frac{(m-1)!!x^{m-1-2k}}{(m-1-2k)!!}=2^{\frac{m-1}{2}} e^{\frac{x^2}{2}} \Gamma \left(\frac{m+1}{2},\frac{x^2}{2}\right).$$


I do not have a closed-form expression for $m$ an even integer (and the sum running up to $m/2-1$).


