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bug in Legendre polinomials


Hi,

I have found a misterious bug (Version 4.0.1.0):

I have checked the identity:
Sum[Binomial[ktmp, i]^2 x^i, {i, 0, ktmp}] = (1 - x)^ktmp LegendreP[ktmp,
(x + 1)/(1 - x)]

using:

Simplify[Sum[
      Binomial[ktmp, i]^2 x^i, {i, 0, ktmp}] - (1 - x)^ktmp LegendreP[
        ktmp, (x + 1)/(1 - x)]]

which should be zero for arbitrary ktmp integers.
Mathematica gives only for ktmp<36 the correct result
for ktmp>=36 it gives a nonvanishing polinom.



I am interested also to force Mathematica to give the result of the series 
Sum[Binomial[ktmp, i]^2 x^i, {i, 0, ktmp}] 
in terms of Legendre polinomials and not as terms of Hypergeometric
functions. 

Thanks
Peter




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