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Ramanujan 6-10-8 Identity

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Let ad=bc, then

64[(a+b+c)^6+(b+c+d)^6-(c+d+a)^6-(d+a+b)^6+(a-d)^6-(b-c)^6][(a+b+c)^(10)+(b+c+d)^(10)-(c+d+a)^(10)-(d+a+b)^(10)+(a-d)^(10)-(b-c)^(10)] =45[(a+b+c)^8+(b+c+d)^8-(c+d+a)^8-(d+a+b)^8+(a-d)^8-(b-c)^8]^2.
(1)

This can also be expressed by defining

F_(2m)(a,b,c,d)=(a+b+c)^(2m)+(b+c+d)^(2m)-(c+d+a)^(2m)-(d+a+b)^(2m)+(a-d)^(2m)-(b-c)^(2m)
(2)
f_(2m)(x,y)=(1+x+y)^(2m)+(x+y+xy)^(2m)-(y+xy+1)^(2m)-(xy+1+x)^(2m)+(1-xy)^(2m)-(x-y)^(2m).
(3)

Then

F_(2m)(a,b,c,d)=a^(2m)f_(2m)(x,y),
(4)

and identity (1) can then be written

64f_6(x,y)f_(10)(x,y)=45f_8^2(x,y).
(5)

Incidentally,

f_2(x,y)=0
(6)
f_4(x,y)=0.
(7)

Another version of the identity can be given in terms of linear forms. Let c=a+b, then,

64{[ax+(b+c)y]^6+[bx-(a+c)y]^6+[cx-(a-b)y]^6 -[ax-(b+c)y]^6-[bx+(a+c)y]^6-[cx+(a-b)y]^6]} ×{[ax+(b+c)y]^(10)+[bx-(a+c)y]^(10)+[cx-(a-b)y]^(10) -[ax-(b+c)y]^(10)-[bx+(a+c)y]^(10)-[cx+(a-b)y]^(10)} =45{[ax+(b+c)y]^8+[bx-(a+c)y]^8+[cx-(a-b)y]^8 -[ax-(b+c)y]^8-[bx+(a+c)y]^8-[cx+(a-b)y]^8}^2.
(8)

This can be understood better considering that

64[p^6+q^6+(p+q)^6-r^6-s^6-(r+s)^6] ×[p^(10)+q^(10)+(p+q)^(10)-r^(10)-s^(10)-(r+s)^(10)] -45[p^8+q^8+(p+q)^8-r^8-s^8-(r+s)^8]^2 =(p^2+pq+q^2-r^2-rs-s^2)P(z),
(9)

where P(z) is a homogeneous polynomial of degree 14. The situation is then reduced to finding expressions (p,q,r,s) such that

p^2+pq+q^2=r^2+rs+s^2
(10)

(Piezas).

See also

Hirschhorn 3-7-5 Identity, Eisenstein Integer

This entry contributed by Tito Piezas III (author's link)

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References

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 3 and 102-106, 1994.Berndt, B. C. and Bhargava, S. "A Remarkable Identity Found in Ramanujan's Third Notebook." Glasgow Math. J. 34, 341-345, 1992.Berndt, B. C. and Bhargava, S. "Ramanujan--For Lowbrows." Amer. Math. Monthly 100, 644-656, 1993.Bhargava, S. "On a Family of Ramanujan's Formulas for Sums of Fourth Powers." Ganita 43, 63-67, 1992.Hirschhorn, M. D. "Two or Three Identities of Ramanujan." Amer. Math. Monthly 105, 52-55, 1998.Nanjundiah, T. S. "A Note on an Identity of Ramanujan." Amer. Math. Monthly 100, 485-487, 1993.Piezas, T. "Ramanujan and the Quartic Equation 2^4+2^4+3^4+4^4+4^4=5^4." http://www.geocities.com/titus_piezas/RamQuad.pdf.Ramanujan, S. Notebooks. New York: Springer-Verlag, pp. 385-386, 1987.

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Ramanujan 6-10-8 Identity

Cite this as:

Piezas, Tito III. "Ramanujan 6-10-8 Identity." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Ramanujan6-10-8Identity.html

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