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