Gauss's Polynomial Identity

Author

Eric W. Weisstein
September 6, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Algebra/GausssPolynomialIdentity.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/GausssPolynomialIdentity.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Identity

(∑_(n=0)^h)(((((-1))^n) (QPochhammer[(x^h), (1/x), n]))/(QPochhammer[x, x, n]))==(QPochhammer[x, (x^2), h/2])

V6

V7

(-1)^n Product[1-x^(h-k),{k,0,n-1}]/Product[1-x^k,{k,n}]

Sum[(((((-1))^n) QPochhammer[(x^h),(1/x),n])/(QPochhammer[x,x,n])),{n,0,h}]//InputForm

Product[1-(x^(2k+1)),{k,0,h/2-1}]

Union[Chop[Apply[Subtract, Table[{Sum[((-1)^n*QPochhammer[x^h, 1/x, n])/QPochhammer[x, x, n], {n, 0, h}], QPochhammer[x, x^2, h/2]}, 
      {h, 0, 20,2}], {1}] /. x -> 0.4321]]

Subtract@@@Table[{(∑_(n=0)^h)(((((-1))^n) QPochhammer[(x^h),(1/x),n])/(QPochhammer[x,x,n])),QPochhammer[x,(x^2),(h/2)]},{h,0,5,2}]//FullSimplify//FunctionExpand

Check

term[n_]:=(-1)^n Product[1-x^(h-k),{k,0,n-1}]/Product[1-x^k,{k,n}]

term[n]

terms[h_]:=Sum[term[n],{n,0,h}]

terms[3]

Table[Together[terms[h]-Product[1-x^k,{k,1,h-1,2}]],{h,2,10,2}]

Table[Together[terms[h]-Product[1-x^(2k+1),{k,0,(h-1)/2}]],{h,2,10,2}]

terms[2]

terms[4]

Table[{terms[h],Product[1-x^k,{k,1,h-1,2}]} ,{h,2,8,2}]//ColumnForm