A ratio of gamma values

Compute $\dfrac{\Gamma\!\left(\tfrac{9}{4}\right)}{\Gamma\!\left(\tfrac{1}{4}\right)}$ .

Show hints (2)+

Use $\Gamma(z+1)=z\Gamma(z)$ to write $\Gamma(9/4)$ in terms of $\Gamma(1/4)$ .
Two steps give $\Gamma(9/4)=\tfrac54\cdot\tfrac14\,\Gamma(1/4)$ ; the ratio is $5/16$ .

Answer

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0.3125

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