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Math Help - Gamma and factorial

  1. #1
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    Gamma and factorial

    Is this expression true?
     (n+s)! \Gamma (s+1)= s! \Gamma (n+s+1)
    i was looking at the proof of  J_{-n} (x)= (-1)^n J_n (x) and i deduced that expression must be true for the proof provided to be sound, the proof was not from a rigourous source so it may have cut corners & have botched up somewhere.
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  2. #2
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    Quote Originally Posted by phycdude View Post
    Is this expression true?
     (n+s)! \Gamma (s+1)= s! \Gamma (n+s+1) .
    For this to make sense requires that s and n be natural numbers. In which case the gammas may be replaced by factorials and both sides are equal to s!(n+s)!

    CB
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  3. #3
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    Quote Originally Posted by phycdude View Post
    Is this expression true?
     (n+s)! \Gamma (s+1)= s! \Gamma (n+s+1)
    What makes sense if s is non-integer and n is positive integer is the following: \Gamma(n+s+1)=(n+s)\Gamma(n+s)=(n+s)(n+s-1)\Gamma(n+s-1) =\cdots=(n+s)\cdots (s+1)\Gamma(s+1). On the other hand, it is not uncommon to define s!=\Gamma(s+1) for non-integer s, in which case your expression is trivially true.
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  4. #4
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    Quote Originally Posted by Laurent View Post
    What makes sense if s is non-integer and n is positive integer is the following: \Gamma(n+s+1)=(n+s)\Gamma(n+s)=(n+s)(n+s-1)\Gamma(n+s-1) =\cdots=(n+s)\cdots (s+1)\Gamma(s+1). On the other hand, it is not uncommon to define s!=\Gamma(s+1) for non-integer s, in which case your expression is trivially true.
    thanks , makes lots of sense now
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