What Does the Subscript in the Airy Function Power Series Notation Mean?

[mknut389]

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!

 

[zmhl0910]

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.

 

[mknut389]

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