For n ∈ N and x &gt; 0 we define P n ( x ) = 1 2 π i ∮ Σ Γ ( t − n ) Γ ( t + 1 ) 2 x t d twhere 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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For   n  ∈      N   and   x  &amp;gt;  0 we define      P    n    (  x  )  =      1          2      π      i            ∮    Σ              Γ      (      t      −      n      )              Γ      (      t      +      1              )        2                  x    t    d  twhere Σ is a closed contour in the t-plane that encircles the points 0,1,…,n once in the positive region.

juicilysv · Accepted Answer

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  )

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.

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