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

**Follow-Ups**:**Re: Hypergeometric1F1 polynomial***From:*Devendra Kapadia <dkapadia@wolfram.com>