I come across the following infinite series.
$$ \sum_{n=1}^{\infty} \frac{t^n}{n!\: n^{a}}, \quad\text{for $t>0$ and $a>0$}. $$
In particular, I am interested in the case where $a=1/4$.
Thanks for any hints and references!
Anand
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Sign up to join this communityI come across the following infinite series.
$$ \sum_{n=1}^{\infty} \frac{t^n}{n!\: n^{a}}, \quad\text{for $t>0$ and $a>0$}. $$
In particular, I am interested in the case where $a=1/4$.
Thanks for any hints and references!
Anand
For natural values of a, take $\displaystyle\frac{e^t-1}t=\sum_1^\infty\dfrac{t^{n-1}}{n!}$, then apply the operator $\bigg(\displaystyle\frac1t\cdot\int\bigg)$ a times to it. For $a\not\in\mathbb N$, such as $a=\dfrac14$, welcome to the “marvelous” world of fractional calculus and Riemann-Liouville integrals.
For natural values of a, the series can also be expressed in terms of the (generalized) hypergeometric function $_{a+1}{\large F}_{a+1}\bigg(\underbrace{1,1,\ldots,1}_{a+1}~;~\underbrace{2,2,\ldots,2}_{a+1}~;~t\bigg)$.