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Shifted Factorial

  1. Jun 28, 2011 #1
    1. The problem statement, all variables and given/known data

    I've got to show [tex]\sum_{n=0}^{\infty} \frac{(a)_n(-1)_n}{(c)_n n!} = \frac{c-a}{c}[/tex]
    where
    [tex]\displaystyle (a)_n = \frac{\Gamma(a+n)}{\Gamma(a)} = a(a+1)...(a+n-1)[/tex]
    is the shifted factorial (Pochhammer symbol).

    3. The attempt at a solution

    I've been informed that [tex](-1)_n = 0\;\;\;\;\;\;\forall\;\;n\geq 2[/tex]
    So the sum has only 2 terms for n=0 and n=1, but what do e.g. [tex](-1)_0\,,\,(-1)_1\,,\,(a)_0\,,\,(a)_1[/tex] equal?
     
  2. jcsd
  3. Jun 28, 2011 #2

    tiny-tim

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    Hi Ted123! :smile:

    look at the definition of (a)n

    (a)0 obviously = 1 for all a (including (-1)0 = 1),

    and (a)1 = … ? :wink:
     
  4. Jun 28, 2011 #3
    So would (a)1 = a, and (-1)1 = -1 ?

    So the 2 terms of the sum give [tex]1 - \frac{a}{c} = \frac{c-a}{c}[/tex]
    Incidentally would (a)2 = a(a+1), (a)3 = a(a+1)(a+2) etc.?
     
    Last edited: Jun 28, 2011
  5. Jun 28, 2011 #4

    tiny-tim

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    yes yes yes yes and yes :smile:
     
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