Ramanujans Sum Identity

Author

Eric W. Weisstein
December 10, 2008

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

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

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

Series

a[n_]:=SeriesCoefficient[Series[(1+53x+9x^2)/(1-82x-82x^2+x^3),{x,0,n}],n]
b[n_]:=SeriesCoefficient[Series[(2-26x-12x^2)/(1-82x-82x^2+x^3),{x,0,n}],n]
c[n_]:=SeriesCoefficient[Series[(2+8x-10x^2)/(1-82x-82x^2+x^3),{x,0,n}],n]

Table[a[n],{n,0,10}]

Table[b[n],{n,0,10}]

Table[c[n],{n,0,10}]

a=Simplify[SeriesCoefficient[(1+53x+9x^2)/(1-82x-82x^2+x^3),{x,0,n}],n>0]

b=Simplify[SeriesCoefficient[(2-26x-12x^2)/(1-82x-82x^2+x^3),{x,0,n}],n>0]

c=Simplify[SeriesCoefficient[(2+8x-10x^2)/(1-82x-82x^2+x^3),{x,0,n}],n>0]

a=(1/85) (((83+9 (Sqrt[85])))^(-1-n)) (43 (((-83-9 (Sqrt[85])))^(1+n))-8 ((2^n) (101+11 (Sqrt[85]))-(1429+155 (Sqrt[85])) (((6887+747 (Sqrt[85])))^n)));
b=(2/85) (((83+9 (Sqrt[85])))^(-1-n)) (-8 (((-83-9 (Sqrt[85])))^(1+n))+7 ((2^(1+n)) (37+4 (Sqrt[85]))+(839+91 (Sqrt[85])) (((6887+747 (Sqrt[85])))^n)));
c=(1/(85 (765+83 (Sqrt[85]))))4 (((83+9 (Sqrt[85])))^(-1-n)) (3 (2^(1+n)) (53040+5753 (Sqrt[85]))-8 (((-83-9 (Sqrt[85])))^n) (63495+6887 (Sqrt[85]))+(((6887+747 (Sqrt[85])))^n) (5586795+605973 (Sqrt[85])));

Simplify[ExpandAll/@Table[(a^3)+(b^3)-(c^3),{n,20}]]

V4.2

FullSimplify[ExpandAll[(a^3)+(b^3)-(c^3)],n>0&&n∈Integers]//Timing

FullSimplify[ExpandAll[(a^3)+(b^3)-(c^3)],n∈Integers]//Timing

V5.2

FullSimplify[ExpandAll[(a^3)+(b^3)-(c^3)],n>0&&n∈Integers]//Timing

FullSimplify[ExpandAll[(a^3)+(b^3)-(c^3)],n∈Integers]//Timing

V6

FullSimplify[ExpandAll[(a^3)+(b^3)-(c^3)],n>0&&n∈Integers]//Timing

FullSimplify[ExpandAll[(a^3)+(b^3)-(c^3)],n∈Integers]//Timing

V7

FullSimplify[ExpandAll[(a^3)+(b^3)-(c^3)],n>0&&n∈Integers]//Timing

FullSimplify[(a^3)+(b^3)+(c^3),n>0&&n∈Integers]//Timing

Check Identity

First few terms

Table[a[n]^3+b[n]^3-c[n]^3-(-1)^n,{n,0,10}]

Analytic expression in Mathematica

a[n_]:=a[n]=FullSimplify[SeriesTerm[(1+53x+9x^2)/(1-82x-82x^2+x^3),{x,0,n}],n∈Integers]

a[n]

b[n_]:=b[n]=FullSimplify[SeriesTerm[(2-26x-12x^2)/(1-82x-82x^2+x^3),{x,0,n}],n∈Integers]

b[n]

c[n_]=c[n]=FullSimplify[SeriesTerm[(2+8x-10x^2)/(1-82x-82x^2+x^3),{x,0,n}],n∈Integers]

FullSimplify[ExpandAll[a[n]^3+b[n]^3-c[n]^3],n∈Integers]//Timing

a=((((83+9 (Sqrt[85])))^(-1-n)) (-172 (((-83-9 (Sqrt[85])))^n) (285769+30996 (Sqrt[85]))-(2^(5+n)) (348508+37801 (Sqrt[85]))+(2^(5-n)) (((83+9 (Sqrt[85])))^(2 n)) (4920812+533737 (Sqrt[85]))))/(85 (6887+747 (Sqrt[85])));
b=((2^(2-n)) (((83+9 (Sqrt[85])))^(-1-n)) ((2^(4+n)) (((-83-9 (Sqrt[85])))^n) (285769+30996 (Sqrt[85]))+7 (4^n) (508799+55187 (Sqrt[85]))+7 (((83+9 (Sqrt[85])))^(2 n)) (5778119+626725 (Sqrt[85]))))/(85 (6887+747 (Sqrt[85])));
c=((2^(2-n)) (((83+9 (Sqrt[85])))^(-1-n)) (-(2^(4+n)) (((-83-9 (Sqrt[85])))^n) (285769+30996 (Sqrt[85]))+3 (4^n) (954859+103569 (Sqrt[85]))+(((83+9 (Sqrt[85])))^(2 n)) (50288457+5454549 (Sqrt[85]))))/(85 (6887+747 (Sqrt[85])));

V6

FullSimplify[a^3+b^3-c^3,n∈Integers&&n>0]//Timing

V7

FullSimplify[a^3+b^3-c^3,n∈Integers&&n>0]//Timing

Hirschhorn's analytic expression

α=(83+9Sqrt[85])/2;
β=(83-9Sqrt[85])/2;

a[n_]:=((64+8Sqrt[85])(α^n)+(64-8Sqrt[85])(β^n)-43(-1)^n)/85

EIS

FormatSequence[Table[a[n],{n,0,20}],51028,
Name->"Ramanujan's a-series.",
html["RamanujanSumIdentity"],
Cross->{51029,51030},
Formula->"G.f.: f(x)=(1+53x+9x^2)/(1-82x-82x^2+x^3)."
]

FormatSequence[Table[b[n],{n,0,20}],51029,
Name->"Ramanujan's b-series.",
html["RamanujanSumIdentity"],
Cross->{51028,51030},
Formula->"G.f.: f(x)=(2-26x-12x^2)/(1-82x-82x^2+x^3)."
]

FormatSequence[Table[c[n],{n,0,20}],51030,
Name->"Ramanujan's c-series.",
html["RamanujanSumIdentity"],
Cross->{51028,51029},
Formula->"G.f.: f(x)=(2+8x-10x^2)/(1-82x-82x^2+x^3)."
]