• To: mathgroup at smc.vnet.net
• Subject: [mg69149] Re: [mg69055] a curious answer
• Date: Thu, 31 Aug 2006 04:39:03 -0400 (EDT)
• References: <200608290725.DAA28971@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```On Tue, 29 Aug 2006, rick wrote:

> Hi,
>
> Can anyone explain these answers (Out[5] and Out[6])?
>
> Line 1 defines a polynomial in x that depends on n;
> line 2 tests the definition when n= 4;
> line 3 lists the coefficients of that polynomial and
> line 4 checks the list when n= 4; lines 5 and 6 ask for a closed form
> for the polynomial and coefficients (which is probably not possible). I
>
> In[1]:=
> k[n_]:=Expand[Product[(j*x+n-j),{j,1,n-1}]]
>
> In[2]:=
> k[4]
>
> Out[2]=
> \!\(6 + 26\ x + 26\ x\^2 + 6\ x\^3\)
>
> In[3]:=
> cL[s_]:=CoefficientList[k[s],x]
>
>
> In[4]:=
> cL[4]
>
> Out[4]=
> {6,26,26,6}
>
> In[5]:=
> cL[n]
>
> Out[5]=
> \!\({\(-\((\(-1\))\)\^n\)\ n\^\(\(-1\) + n\)\ \(\((\(-1\) + n)\)!\)}\)
>
> In[6]:=
> k[n]
>
> Out[6]=
> \!\(\(-\((\(-1\))\)\^n\)\ n\^\(\(-1\) + n\)\ \(\((\(-1\) + n)\)!\)\)
>
>
> Thanks,
>

Hello Rick,

Thank you for reporting the problem with the above finite product.

In this example, Product fails to identify the coefficient of 'j'
in the first argument (j*x+n-j) correctly, and returns an answer
independent of 'x'.

A workaround for this problem is to use Collect in the definition
of k[n]. This seems to work well and gives a closed form for the
product in terms of Pochhammer (Out[7] below). The message from
CoefficientList is given to indicate that the result from Product
depends on 'x' but is not a polynomial in 'x' for symbolic 'n'.

===================================

In[1]:= \$Version

Out[1]= 5.2 for Linux (June 27, 2005)

In[2]:= k[n_] := Expand[Product[Collect[(j*x + n - j), j], {j, 1, n - 1}]]

In[3]:= k[4]

2      3
Out[3]= 6 + 26 x + 26 x  + 6 x

In[4]:= cL[s_] := CoefficientList[k[s], x]

In[5]:= cL[4]

Out[5]= {6, 26, 26, 6}

In[6]:= cL[n]

-1 + n                  n
General::poly: (-1 + x)       Pochhammer[1 + ------, -1 + n]
-1 + x
is not a polynomial.

n                  n
Out[6]= {(-1 + x)  Pochhammer[1 + ------, -1 + n]}
-1 + x

In[7]:= k[n]

-1 + n                  n
Out[7]= (-1 + x)       Pochhammer[1 + ------, -1 + n]
-1 + x

In[8]:= Expand[Together[% /. {n -> 4}]]

2      3
Out[8]= 6 + 26 x + 26 x  + 6 x

=================================

I apologize for the inconvenience caused by this problem.

Sincerely,