       Re: Hypergeometric1F1 polynomial

• To: mathgroup at smc.vnet.net
• Subject: [mg91451] Re: Hypergeometric1F1 polynomial
• From: m.r at inbox.ru
• Date: Fri, 22 Aug 2008 03:12:25 -0400 (EDT)
• References: <g8je5u\$a4n\$1@smc.vnet.net>

```Alec Mihailovs wrote:
> Mathematica gives the wrong answer to the following sum,
>
> In:= Sum[Binomial[n, k]/Binomial[2 n, k]/k! (2 x)^k, {k, 0, n}]
>
> Out= 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

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:= 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/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])

Maxim Rytin
m.r at inbox.ru

```

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