Ramanujan 6-10-8 Identity

Let $ad=bc$, then

$64[(a+b+c)^6+(b+c+d)^6-(c+d+a)^6$
$-(d+a+b)^6+(a-d)^6-(b-c)^6]$
$\times[(a+b+c)^{10}+(b+c+d)^{10}-(c+d+a)^{10}$
$-(d+a+b)^{10}+(a-d)^{10}-(b-c)^{10}]$
$=45[(a+b+c)^8+(b+c+d)^8-(c+d+a)^8$
$-(d+a+b)^8+(a-d)^8-(b-c)^8]^2.\quad$	(1)

This can also be expressed by defining

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

Then

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

and identity (1) can then be written

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

Incidentally,

$$f_2(x,y) = 0$$ (6)

$$f_4(x,y) = 0.$$ (7)

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.

Ramanujan, S. Notebooks. New York: Springer-Verlag, pp. 385-386, 1987.