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

```

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