Re: Hypergeometric1F1 polynomial
- To: mathgroup at smc.vnet.net
- Subject: [mg91476] Re: Hypergeometric1F1 polynomial
- From: "Alec Mihailovs" <alec at mihailovs.com>
- Date: Sat, 23 Aug 2008 01:41:29 -0400 (EDT)
- References: <g8je5u$a4n$1@smc.vnet.net> <g8lpat$ijd$1@smc.vnet.net>
Maxim Rytin wrote
> Here's one way to obtain the correct answer. The intermediate steps
> are only formally correct but in the end the singularities cancel out:
>
> In[1]:= Sum[
> Binomial[n, k] (2 x)^k/(Binomial[2 n, k] k!) // FunctionExpand //
> # /. Gamma[a_ - k] :> (-1)^k Pi Csc[a Pi]/Gamma[1 - a + k]&,
> {k, 0, n}] //
> FullSimplify // Simplify[#, Element[n, Integers]]&
>
> Out[1]= 1/Gamma[1/2 + n] 2^-n ((-2)^n E^x Pi
> Hypergeometric0F1Regularized[1/2 - n, x^2/4] + Sqrt[Pi] (-x)^(1 + n)
> HypergeometricPFQRegularized[{1, 1}, {1 - n, 2 + n}, 2 x])
That seems to be correct answer. It is an interesting way of obtaining it.
Is there a way to simplify it to one of 2 other forms of the correct answer
that I mentioned in the original post?
Alec