Mathematica 9 is now available
Services & Resources / Wolfram Forums / MathGroup Archive
-----

MathGroup Archive 2008

[Date Index] [Thread Index] [Author Index]

Search the Archive

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.


  • Prev by Date: Re: Help to remove equivalent (redundant) solutions from FindRoot[]
  • Next by Date: Re: Multifile search
  • Previous by thread: Hypergeometric1F1 polynomial
  • Next by thread: Re: Hypergeometric1F1 polynomial