Pentagonal Number Theorem

Author

Eric W. Weisstein
November 30, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/PentagonalNumberTheorem.nb.

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

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

Coefficients

V6

Series[Product[1-(x^k),{k,100}],{x,0,30}]

Series[Sum[(((-1))^k)(x^(k(3k+1)/2)),{k,-100,100}],{x,0,30}]

Series[1+Sum[(((-1))^k)((x^(k(3k-1)/2))+(x^(k(3k+1)/2))),{k,100}],{x,0,35}]

CoefficientList[%,x]

Flatten[Position[%91,_?(#≠0&)]]-1

V7

Product[1-(x^k),{k,∞}]

Sum[(((-1))^k)(x^(k(3k+1)/2)),{k,-∞,∞}]

Series[QPochhammer[x,x],{x,0,30}]

QPochhammer[.1]

With[{x=.1},NSum[(((-1))^k)(x^(k(3k+1)/2)),{k,-∞,∞}]]//Chop

1+Sum[(((-1))^k)((x^(k(3k-1)/2))+(x^(k(3k+1)/2))),{k,∞}]//InputForm

With[{x=.1},1+NSum[(((-1))^k)((x^(k(3k-1)/2))+(x^(k(3k+1)/2))),{k,∞}]]//Chop

Series[QPochhammer[x,x],{x,0,30}]

SeriesCoefficient[QPochhammer[x,x],{x,0,n}]

Generalizations

(∏_k^∞)(1-t (x^k))==(∑_(n=0)^∞)((((((-1))^n) (x^( n (n+1)/2))))/(QPochhammer[x, x, n]))(t^n) ==((QPochhammer2[t, x])/(1-t))

Product[1-(x^k)t,{k,∞}]

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

Sum[(((((-1))^n)(x^(n(n+1)/2)))/(QPochhammer[x,x,n]))(t^n),{n,0,∞}]

Series[Sum[(((((-1))^n)(x^(n(n+1)/2)))/(QPochhammer[x,x,n]))(t^n),{n,0,50}]-((QPochhammer[t,x])/(1-t)),{x,0,100}]

(∏_k^∞)(1/(1-t (x^k)))==(∑_(n=0)^∞)(((t^n) (x^n))/(QPochhammer[x, x, n]))==((1-t)/(QPochhammer2[t, x]))

Product[1/(1-(x^k)t),{k,∞}]

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

Sum[((x^n)/(QPochhammer[x,x,n]))(t^n),{n,0,∞}]

Series[Sum[((x^n)/(QPochhammer[x,x,n]))(t^n),{n,0,20}]-((1-t)/(QPochhammer[t,x])),{x,0,20}]

ser1=Series[Sum[((x^n)/(QPochhammer[x,x,n]))(t^n),{n,0,20}],{x,0,50}];

ser2=Series[((1-t)/(QPochhammer[t,x])),{x,0,20}];//Timing

ser1-ser2