For n in NN and x>0 we define P_n(x) = (1)/(2 pi i) oint_Sigma (Gamma(t-n))/(Gamma(t+1)^2)x^tdt where Σ is a closed contour in the t-plane that encircles the points 0,1,…,n once in the positive region.

Lisa Hardin 2022-07-16 Answered
For n N and x > 0 we define
P n ( x ) = 1 2 π i Σ Γ ( t n ) Γ ( t + 1 ) 2 x t d t
where Σ is a closed contour in the t-plane that encircles the points 0,1,…,n once in the positive region.
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Answers (1)

juicilysv
Answered 2022-07-17 Author has 17 answers
All you need to do is to multiply the integral representation by ( 1 ) n n !, so you will have the right integral representation
P n ( x ) = ( 1 ) n n ! 2 π i Σ Γ ( t n ) Γ ( t + 1 ) 2 x t d t .
For instance,
P 5 ( x ) = 1 5 x + 5 x 2 5 3 x 3 + 5 24 x 4 1 120 x 5 ,
which agrees with the Laguerre polynomial L 5 ( x )

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