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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

$$\sum_{m=0}^{n}\frac{x^{n-m}f_{m}}{m!(n-m)!}$$

j.jonah
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1 Answers1

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$$a_m=\frac{(m-1)!!\,x^{m-1-2k}}{(m-1-2k)!!}$$ $$f_{m}=\sum_{k=0}^{\frac{m-1}2} a_m=2^{\frac{m-1}{2}}\,\, e^{\frac{x^2}{2}} \,\,\Gamma \left(\frac{m+1}{2},\frac{x^2}{2}\right)$$ where appears the incomplete gamma function.

For $$F_n=\sum_{m=0}^{n}\frac{x^{n-m}\,f_{m}}{m!\,(n-m)!}$$ I am totally stuck (for a given $n$, for sure we can get the expression).

Claude Leibovici
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