# Airy Function Power Series Help

## Main Question or Discussion Point

I am currently working on a solution to an differential equation of the form I(x)-xI(x)=0.

The solution is the airyai and airybi functions, and I have found the power series equations for these.

I am using two different mathematical programs to evaluate the solution, and each are giving me different answers, and I am attempting to verify which is correct.

My issue is there is a notation in the power series that I am unfamiliar with, and with all my searching I cannot find a explanation, so I am turning to this forum to see if anyone here could help.

The power series for the airyai function is

$\frac{1}{(3)^{2/3}\Gamma(\frac{2}{3})}\sum\frac{1}{(\frac{2}{3})_{k}k!}(\frac{z^{3}}{9})^{k}$-$\frac{1}{(3)^{1/3}\Gamma(\frac{1}{3})}\sum\frac{1}{(\frac{4}{3})_{k}k!}(\frac{z^{3}}{9})^{k}$

which according to my source expands to

$\frac{1}{(3)^{2/3}\Gamma(\frac{2}{3})}(1+\frac{z^{3}}{6}+\frac{z^{6}}{180}+...)$-$\frac{1}{(3)^{1/3}\Gamma(\frac{1}{3})}(1+\frac{z^{3}}{12}+\frac{z^{6}}{504}+...)$

My notation question is what does the subscript on the fraction in both summations mean
i.e. $(\frac{2}{3})_{k}$ and $(\frac{4}{3})_{k}$

Through my searching I came across one topic that stated it was a special type of factorial:
$x_{n}=\frac{x!}{(x-n)!}$
which since have fractions would be
$x_{n}=\frac{\Gamma(x+1)}{\Gamma(x+1-n)!}$

Unless I am using $\Gamma$ incorrectly, when using this within the summation, it does not provide me with the values shown in the expansion.

For the life of me I can find no explanation as to what the subscript may mean. Please help!

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which according to my source expands to

$\frac{1}{(3)^{2/3}\Gamma(\frac{2}{3})}(1+\frac{z^{3}}{6}+\frac{z^{6}}{180}+...)$-$\frac{1}{(3)^{1/3}\Gamma(\frac{1}{3})}(1+\frac{z^{3}}{12}+\frac{z^{6}}{504}+...)$
I'm certain your power series expansion is incorrect. There should be terms $z^{3k+1}$ for $k\in\mathbb{Z}_{\ge 0}$.

The correct power series expansion is provided in page 446 of Abramowitz and Stegun.

The subscript notation is also defined there.

Thank you. That link lead me to another chapter in the book which gave me the explanation, as well as some more info on the airy function