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Re: Hypergeometric1F1 polynomial
*To*: mathgroup at smc.vnet.net
*Subject*: [mg91492] Re: [mg91437] Hypergeometric1F1 polynomial
*From*: Devendra Kapadia <dkapadia at wolfram.com>
*Date*: Sat, 23 Aug 2008 01:44:25 -0400 (EDT)
*References*: <200808210956.FAA10365@smc.vnet.net>
On Thu, 21 Aug 2008, Alec Mihailovs wrote:
> 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
>
Hello Alec,
Thank you for reporting the problem with the answer returned by Sum in the
above example. As noted by you, this sum can be evaluated in terms of
Hypergeometric1F1.
A partial workaround for the problem is to introduce a parameter 'a'
as shown in In[2] below, to obtain a Hypergeometric1F1 function.
The incorrect behavior in your example is caused by
auto-simplification of this hypergeometric function (see In[4] below).
=========================
In[2]:= Sum[(Binomial[n, k]/Binomial[a*n, k]/k!)*(2*x)^k, {k, 0, n}]
Out[2]= Hypergeometric1F1[-n, -(a n), 2 x]
In[3]:= Table[% /. {a -> 2}, {n, 0, 3}] // InputForm
Out[3]//InputForm= {1, 1 + x, 1 + x + x^2/3, 1 + x + (2*x^2)/5 + x^3/15}
In[4]:= %% /. {a -> 2} // InputForm
Out[4]//InputForm=
2^(-1/2 - n)*E^x*x^(1/2 + n)*BesselI[(-1 - 2*n)/2, x]*Gamma[1/2 - n]
==========================
I apologize for the confusion caused by this problem.
Sincerely,
Devendra Kapadia,
Wolfram Research, Inc.
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