Gamma Funk'tion
Evaluate
Γ ( 3 1 0 ) Γ ( 3 1 6 ) ,
where Γ ( z ) denotes the gamma function.
If the value of the above expression can be expressed in the form of b a , where a and b are coprime positive integers, find a + b .
The answer is 139.
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3 solutions
Please do change the subject of the problem to Calculus.
Γ ( z ) = ( z − 1 ) Γ ( z − 1 ) = ( z − 1 ) ( z − 2 ) Γ ( z − 2 ) so Γ ( z − 2 ) Γ ( z ) = ( z − 1 ) ( z − 2 )
For z = 3 1 6 this ratio is ( 3 1 6 − 1 ) ( 3 1 6 − 2 ) = 9 1 3 0 so that the sum we seek is 1 3 9
Moderator note:
What can we do if the difference of the variables was not an integer? In such a case, must we evaluate the Gamma function?
Γ ( 3 1 0 ) Γ ( 3 1 6 ) = Γ ( 3 1 0 ) 3 1 3 × Γ ( 3 1 3 ) = Γ ( 3 1 0 ) 9 1 0 × 9 1 3 × Γ ( 3 1 0 ) = 9 1 3 0
1 3 0 + 9 = 1 3 9
We use here the recursive formula, Γ ( z ) = ( z − 1 ) Γ ( z − 1 ) . Now we have, Γ 3 1 0 Γ 3 1 6 = Γ 3 1 0 ( 3 1 6 − 1 ) Γ ( 3 1 6 − 1 ) = 3 1 3 Γ 3 1 0 Γ 3 1 3 = 3 1 3 Γ 3 1 0 ( 3 1 3 − 1 ) Γ ( 3 1 3 − 1 ) = 3 1 3 ⋅ 3 1 0 ⋅ Γ 3 1 0 Γ 3 1 0 = 9 1 3 0 .