Fine's Equation

Author

Eric W. Weisstein
August 11, 2008

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

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

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

Analytic

(∏_(n=1)^∞)(((1-(q^(2 n))) (1-(q^(3 n))) (1-(q^(8 n))) (1-(q^(12 n))))/((1-(q^n)) (1-(q^(24 n)))))==(∑_(n=1)^∞)(q^n) (Subscript[E, 1,5,7,11])(n;24)+1

V5

Product[(((1-(q^(2n)))(1-(q^(3n)))(1-(q^(8n)))(1-(q^(12n))))/((1-(q^n))(1-(q^(24n))))),{n,∞}]

V7

Product[(((1-(q^(2n)))(1-(q^(3n)))(1-(q^(8n)))(1-(q^(12n))))/((1-(q^n))(1-(q^(24n))))),{n,∞}]

FullSimplify[%]

TraditionalForm[%]

(∏_k^∞)((((x^k)+1) (1-(x^(3 k))) (1-(x^(8 k))))/((x^(12 k))+1))==(((QPochhammer2[-1, x]) (QPochhammer2[(x^3), (x^3)]) (QPochhammer2[(x^8), (x^8)]))/(QPochhammer2[-1, (x^12)]))

Product[(1+x^k)(1-x^(3k))(1-x^(8k))/(1+x^(12k)),{k,∞}]

Series[%,{x,0,50}]//Normal

1+Sum[x^k(1+x^(4k))(1+x^(6k))/(1+x^(12k)),{k,∞}]//InputForm

1+Sum[x^k(1+x^(4k))(1+x^(6k))/(1+x^(12k)),{k,∞}]//TraditionalForm

Series[1+Sum[x^k(1+x^(4k))(1+x^(6k))/(1+x^(12k)),{k,100}],{x,0,50}]//Normal

FullSimplify[((QPochhammer[-1,q] QPochhammer[(q^3),(q^3)] QPochhammer[(q^8),(q^8)])/(QPochhammer[-1,(q^12)]))-((QPochhammer[(q^2),(q^2)] QPochhammer[(q^3),(q^3)] QPochhammer[(q^8),(q^8)] QPochhammer[(q^12),(q^12)])/(QPochhammer[q,q] QPochhammer[(q^24),(q^24)]))]

Table[(((QPochhammer[-1,q])/(QPochhammer[-1,(q^12)]))-((QPochhammer[(q^2),(q^2)] QPochhammer[(q^12),(q^12)])/(QPochhammer[q,q] QPochhammer[(q^24),(q^24)])))/.q->RandomReal[WorkingPrecision->100],{100}]//Chop//Union

Series Left Side

A000377

Series[Product[(((1-(q^(2n)))(1-(q^(3n)))(1-(q^(8n)))(1-(q^(12n))))/((1-(q^n))(1-(q^(24n))))),{n,1000}],{q,0,50}]

Series[((QPochhammer[(q^2),(q^2)] QPochhammer[(q^3),(q^3)] QPochhammer[(q^8),(q^8)] QPochhammer[(q^12),(q^12)])/(QPochhammer[q,q] QPochhammer[(q^24),(q^24)])),{q,0,50}]

lhs=CoefficientList[%,q]

Series Right Side via Kronecker

rhs1=Table[Plus@@(KroneckerSymbol[-6,#]&/@Divisors[n]),{n,50}]

rhs1=Table[Plus@@(KroneckerSymbol[-6,#]&/@Divisors[n]),{n,50}]

Rest[lhs]==rhs1

Series Right Side via Divisors

FineE[n_]:=Module[{d=Divisors[n]},
Subtract@@Length/@Function[r,Select[d,MemberQ[r,Mod[#,24]]&]]/@{#,24-#}&[{1,5,7,11}]
]

rhs2=Table[FineE[n],{n,50}]

Rest[lhs]==rhs2