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A weird expansion

  1. May 3, 2010 #1
    Dos this fractional Taylor series

    [tex] (a+x)^{-r} = \sum_{m=-\infty}^{\infty} \frac{ \Gamma (-r+1)}{\Gamma (m+\alpha+1) \Gamma(-r-n-\alpha+1)}a^{(-r-m-\alpha )}x^{m+\alpha} [/tex]

    makes sense for x < 1 or x >1 and alpha being an arbitrary real number ..for example ?? , here a and r are real numbers , the idea here is if we can define a fractional power series expansion generalizing the usuarl Laurent or Taylor series.
  2. jcsd
  3. May 5, 2010 #2
    I tried [itex]r= 1,\alpha=1/2,a=1,x=1/2[/itex].
    The series diverges since the term does not go to zero as [itex]m \to -\infty[/itex]

    P.S. You have misprint [itex]n[/itex] for [itex]m[/itex] , right?
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