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Airy Function Power Series Help

  1. Mar 28, 2012 #1
    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!
     
  2. jcsd
  3. Mar 28, 2012 #2
    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.
     
  4. Mar 28, 2012 #3
    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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