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Hypergeometric1F1 polynomial

  • To: mathgroup at smc.vnet.net
  • Subject: [mg91437] Hypergeometric1F1 polynomial
  • From: "Alec Mihailovs" <alec at mihailovs.com>
  • Date: Thu, 21 Aug 2008 05:56:55 -0400 (EDT)

Mathematica gives the wrong answer to the following sum,

In[1]:= Sum[Binomial[n, k]/Binomial[2 n, k]/k! (2 x)^k, {k, 0, n}]

Out[1]= 2^(-(1/2) - n) E^x x^(1/2 + n)
  BesselI[1/2 (-1 - 2 n), x] Gamma[1/2 - n]

The correct answer is 1 for n=0 and Hypergeometric1F1[-n, -2 n, 2 x] for 
integer n>0, which would be equal to the expression given by Mathematica if 
n was not a positive integer.

Another form of the correct answer is

(2 x)^(n+1/2) E^x BesselK[n+1/2,x] n!/(2 n)!/Sqrt[Pi]

Is there a way to apply some assumptions to get the correct answer?

Alec


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