Question
Let $\Gamma(p)$ denote the gamma function. Show that$$\Gamma(p+1)[(p+1)(p+2) \cdots(p+k)]=\Gamma(p+k+1)$$
Step 1
Step 1: We know that the gamma function is defined as $\Gamma(p) = (p-1)!$ for any positive integer $p$. Show more…
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The Gamma Function The gamma function is defined by the formula $$ \Gamma(x)=\int_{0}^{+\infty} t^{x-1} e^{-t} d t $$ a. Find $\Gamma(1)$ and $\Gamma(2)$. b. Use integration by parts to show that for every positive integer $n, \Gamma(n+1)=n \Gamma(n)$. c. Deduce that $\Gamma(n)=(n-1) ![=(n-1)(n-2) \cdots 2 \cdot 1]$ for every positive integer $n$.
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