Re: Re: a curious answer

*To*: mathgroup at smc.vnet.net*Subject*: [mg69188] Re: [mg69149] Re: [mg69055] a curious answer*From*: "Chris Chiasson" <chris at chiasson.name>*Date*: Fri, 1 Sep 2006 06:41:29 -0400 (EDT)*References*: <200608290725.DAA28971@smc.vnet.net> <200608310839.EAA19560@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

After seeing Devendra's response, I realize that I am blind. I thought the input didn't have an x in it. On 8/31/06, Devendra Kapadia <dkapadia at wolfram.com> wrote: > 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 > > expected no answer-not gibberish. > > > > 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, > > Devendra Kapadia. > Wolfram Research, Inc. > > -- http://chris.chiasson.name/