Re: Hypergeometric1F1 polynomial
- To: mathgroup at smc.vnet.net
- Subject: [mg91469] Re: Hypergeometric1F1 polynomial
- From: "Jean-Marc Gulliet" <jeanmarc.gulliet at gmail.com>
- Date: Sat, 23 Aug 2008 01:40:13 -0400 (EDT)
- References: <g8je5u$a4n$1@smc.vnet.net> <48ADCC77.9070400@gmail.com>
On Fri, Aug 22, 2008 at 7:13 AM, Alec Mihailovs <alec at mihailovs.com> wrote:
>> The polynomial form you are expecting (see In[1]) can be obtained by
>> taking the series expansion about x == 0 to the order n (see In[2]).
>
> Well, I can obtain it even without series expansion. For example, as
>
> In[3]:= Hypergeometric1F1[-1, -2, 2 x]
>
> Out[3]= 1 + x
>
> The problem is that the answers given by Mathematica to the Sum problem, are
> not the same - they are not polynomials, with the series expansion, or
> without.
Hum, with series expansion they are (at least on my system). For instance,
In[1]:= s = Sum[Binomial[n, k]/Binomial[2 n, k]/k! (2 x)^k, {k, 0, n}]
Out[1]=
-(1/2) - n x 1/2 + n 1 1
2 E x BesselI[- (-1 - 2 n), x] Gamma[- - n]
2 2
In[2]:= Table[Series[s, {x, 0, n}] // Normal, {n, 0, 5}] // TableForm
Out[2]//TableForm=
1
1 + x
2
x
1 + x + --
3
2 3
2 x x
1 + x + ---- + --
5 15
2 3 4
3 x 2 x x
1 + x + ---- + ---- + ---
7 21 105
2 3 4 5
4 x x x x
1 + x + ---- + -- + -- + ---
9 9 63 945
In[3]:= FullSimplify[s]
Table[Series[%, {x, 0, n}] // Normal, {n, 0, 5}] // TableForm
Out[3]=
2
x 1 x
E Hypergeometric0F1[- - n, --]
2 4
Out[4]//TableForm=
1
1 + x
2
x
1 + x + --
3
2 3
2 x x
1 + x + ---- + --
5 15
2 3 4
3 x 2 x x
1 + x + ---- + ---- + ---
7 21 105
2 3 4 5
4 x x x x
1 + x + ---- + -- + -- + ---
9 9 63 945
In[5]:= Table[Hypergeometric1F1[-n, -2 n, 2 x], {n, 0, 5}] // TableForm
Out[5]//TableForm=
1
1 + x
2
x
1 + x + --
3
2 3
2 x x
1 + x + ---- + --
5 15
2 3 4
3 x 2 x x
1 + x + ---- + ---- + ---
7 21 105
2 3 4 5
4 x x x x
1 + x + ---- + -- + -- + ---
9 9 63 945
In[7]:= Table[FullSimplify[s] == Hypergeometric1F1[-n, -2 n, 2 x], {n,
0, 5}, {x, 1,
5}]
Out[7]= {{False, False, False, False, False}, {False, False, False, False,
False}, {False, False, False, False, False}, {False, False, False, False,
False}, {False, False, False, False, False}, {False, False, False, False,
False}}
In[9]:= $Version
Out[9]= "6.0 for Mac OS X x86 (64-bit) (May 21, 2008)"
Best regards,
--
Jean-Marc