Hermite Polynomial Integral Limit

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I'm trying to find the following limit:



$$ undersetn to inftytextrmlim
fracsqrt[3]n2^n n! sqrtpi int_sqrt2n+1^inftytextrmH^2_n(x) e^-x^2dx $$



Where H is a hermite polynomial (the physicist kind).
I'm pretty sure the limit is



$$ frac13^2/3Gamma^2(1/3) $$



This is based on an asymptotic formula using Airy functions.



Any thoughts on how to find the limit?



If it is any help, the limit can also be represented as



$$ undersetnrightarrow inftytextrmlimfracsqrt[3]n2 cdot n!fracd^n dx^n left .fractextrmerfcleft ( sqrttfrac1-x1+x sqrt2n+1right )1-x right |_x=0 $$




integration analysis limits




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    up vote
    6
    down vote

    favorite
    2












    I'm trying to find the following limit:



    $$ undersetn to inftytextrmlim
    fracsqrt[3]n2^n n! sqrtpi int_sqrt2n+1^inftytextrmH^2_n(x) e^-x^2dx $$



    Where H is a hermite polynomial (the physicist kind).
    I'm pretty sure the limit is



    $$ frac13^2/3Gamma^2(1/3) $$



    This is based on an asymptotic formula using Airy functions.



    Any thoughts on how to find the limit?



    If it is any help, the limit can also be represented as



    $$ undersetnrightarrow inftytextrmlimfracsqrt[3]n2 cdot n!fracd^n dx^n left .fractextrmerfcleft ( sqrttfrac1-x1+x sqrt2n+1right )1-x right |_x=0 $$




    integration analysis limits




    share|cite|improve this question























      up vote
      6
      down vote

      favorite
      2









      up vote
      6
      down vote

      favorite
      2






      2





      I'm trying to find the following limit:



      $$ undersetn to inftytextrmlim
      fracsqrt[3]n2^n n! sqrtpi int_sqrt2n+1^inftytextrmH^2_n(x) e^-x^2dx $$



      Where H is a hermite polynomial (the physicist kind).
      I'm pretty sure the limit is



      $$ frac13^2/3Gamma^2(1/3) $$



      This is based on an asymptotic formula using Airy functions.



      Any thoughts on how to find the limit?



      If it is any help, the limit can also be represented as



      $$ undersetnrightarrow inftytextrmlimfracsqrt[3]n2 cdot n!fracd^n dx^n left .fractextrmerfcleft ( sqrttfrac1-x1+x sqrt2n+1right )1-x right |_x=0 $$




      integration analysis limits




      share|cite|improve this question













      I'm trying to find the following limit:



      $$ undersetn to inftytextrmlim
      fracsqrt[3]n2^n n! sqrtpi int_sqrt2n+1^inftytextrmH^2_n(x) e^-x^2dx $$



      Where H is a hermite polynomial (the physicist kind).
      I'm pretty sure the limit is



      $$ frac13^2/3Gamma^2(1/3) $$



      This is based on an asymptotic formula using Airy functions.



      Any thoughts on how to find the limit?



      If it is any help, the limit can also be represented as



      $$ undersetnrightarrow inftytextrmlimfracsqrt[3]n2 cdot n!fracd^n dx^n left .fractextrmerfcleft ( sqrttfrac1-x1+x sqrt2n+1right )1-x right |_x=0 $$





      integration analysis limits





      share|cite|improve this question












      share|cite|improve this question




      share|cite|improve this question








      edited Jul 23 at 14:58
























      asked Jul 19 at 23:57









      NMK

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