Airy Function Power Series Help

  • Thread starter mknut389
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  • #1
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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

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

which according to my source expands to

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

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

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

Unless I am using [itex]\Gamma[/itex] 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!
 

Answers and Replies

  • #2
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which according to my source expands to

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

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

The subscript notation is also defined there.
 
  • #3
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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
 

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