Let 
,
then Hirschhorn's 3-7-5 identity, inspired by the Ramanujan
6-10-8 identity, is given by
![25[(b+c+d)^3+(a-d)^3-(a+b+c)^3-(c+d+a)^3-(b-c)^3+(d+a+b)^3][(b+c+d)^7+(a-d)^7-(a+b+c)^7-(c+d+a)^7-(b-c)^7+(d+a+b)^7]
=21[(b+c+d)^5+(a-d)^5-(a+b+c)^5-(c+d+a)^5-(b-c)^5+(d+a+b)^5]^2.](/images/equations/Hirschhorn3-7-5Identity/NumberedEquation1.svg) |
(1)
|
Another version of this identity can be given using linear forms. Let 
, then,
![25{[ax+(b+c)y]^3+[bx-(a+c)y]^3-[cx-(a-b)y]^3-[ax-(b+c)y]^3-[bx+(a+c)y]^3+[cx+(a-b)y]^3}{[ax+(b+c)y]^7+[bx-(a+c)y]^7-[cx-(a-b)y]^7-[ax-(b+c)y]^7-[bx+(a+c)y]^7+[cx+(a-b)y]^7}
=21{[ax+(b+c)y]^5+[bx-(a+c)y]^5-[cx-(a-b)y]^5-[ax-(b+c)y]^5-[bx+(a+c)y]^5+[cx+(a-b)y]^5}^2.](/images/equations/Hirschhorn3-7-5Identity/NumberedEquation2.svg) |
(2)
|
The situation can be understood better considering that
![25[p^3+q^3-(p+q)^3-r^3-s^3+(r+s)^3]
×[p^7+q^7-(p+q)^7-r^7-s^7+(r+s)^7]-21[p^5+q^5-(p+q)^5-r^5-s^5+(r+s)^5]^2
=-525pq(p+q)rs(r+s)(p^2+pq+q^2-r^2-rs-s^2)^2,](/images/equations/Hirschhorn3-7-5Identity/NumberedEquation3.svg) |
(3)
|
and hence is reduced to finding expressions 
such that
 |
(4)
|
which is satisfied by the two given versions.
See also
Ramanujan 6-10-8 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, 1994.Hirschhorn,
M. "Two Or Three Identities of Ramanujan." Amer. Math. Monthly 105,
52-55, 1998.Piezas, T. "Ramanujan and the Quartic Equation 
."
http://www.geocities.com/titus_piezas/RamQuad.pdf.Referenced
on Wolfram|Alpha
Hirschhorn 3-7-5 Identity
Cite this as:
Piezas, Tito III. "Hirschhorn 3-7-5 Identity." From MathWorld--A Wolfram Web Resource, created by Eric
W. Weisstein. https://mathworld.wolfram.com/Hirschhorn3-7-5Identity.html
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