TITLE: Expressing inductive function as power series in summation
QUESTION [1 upvotes]: Consider the following power series of $f$:
$$t^2+\frac{t^5}{5}+\frac{t^8}{5\cdot 8}+\frac{t^{11}}{5\cdot 8\cdot 11}+\dots$$
The best summation notation expression I could come up with is this:
$$\sum\limits_{n=1}^\infty \frac{t^{3(n-1)+2}}{[3(n-1)+2]\cdot [3(n-2)+2]\cdot\dots\cdot 5}$$
Is there a more precise way to express this series?

REPLY [1 votes]: $$f(t)=\sum_{n\geq 0}\frac{t^{3n+2}}{\prod_{k=1}^{n}(3k+2)}=\sum_{n\geq 0}\frac{t^{3n+2}\Gamma\left(\tfrac{5}{3}\right)}{3^n \Gamma\left(n+\tfrac{5}{3}\right)} $$
is a hypergeometric function, $t^2\cdot{}_1 F_1\left(1;\frac{5}{3};\frac{t^3}{3}\right)$, which for any $t>0$, due to Euler's Beta function, can be written in terms of the incomplete Gamma function:
$$ f(t) = \frac{1}{\sqrt[3]{3}}\,e^{\frac{t^3}{3}} \left[3\, \Gamma\left(\tfrac{5}{3}\right)-2 \,\Gamma\left(\tfrac{2}{3},\tfrac{t^3}{3}\right)\right]. $$
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