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I've just stumbled upon this $$ Q_{m}(x)=\sum _{n=0}^{\infty } \frac{ (-1)^n}{n+m}\frac{x^{n}}{n!} $$ and i'd like to know if it has a closed form.

Note that $m \geq1 $ is an integer.

Thanks.

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    $\begingroup$ Given the very fast decrease of the coefficients, I doubt it. $\endgroup$
    – user65203
    Commented Apr 21, 2018 at 19:21
  • $\begingroup$ This looks like we need the coefficient of $x^{-m}$ in the product $e^{-x}\sum_{k=0}^{k=\infty} \frac{x^{-(m+k)}}{m+k}$ $\endgroup$
    – sku
    Commented Apr 21, 2018 at 21:26
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    $\begingroup$ Note that $$(x^mQ_m(x))'=x^{m-1}e^{-x}$$ and $Q_m(0)=\frac1m$ hence $$Q_m(x)=x^{-m}\int_0^xt^{m-1}e^{-t}dt$$ $\endgroup$
    – Did
    Commented Apr 21, 2018 at 22:24
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    $\begingroup$ @Did, which is essentially lower incomplete gamma function $Q_m(x)=x^{-m}\gamma(m,x)$ en.wikipedia.org/wiki/Incomplete_gamma_function $\endgroup$
    – Yuriy S
    Commented Apr 24, 2018 at 12:05
  • $\begingroup$ @YuriyS Which is mostly giving a name to a formula. $\endgroup$
    – Did
    Commented May 5, 2018 at 14:50

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