Generalized Gurevič Identities

The Generalized Gurevič Identities are versions of the Gurevič Identities that do not mention any constants. They are given by:

(Au+Bnu)v(Cnv+Dnv)u=(Av+Bnv)u(Cnu+Dnu)vsuperscriptsuperscript𝐴𝑢superscriptsubscript𝐵𝑛𝑢𝑣superscriptsuperscriptsubscript𝐶𝑛𝑣superscriptsubscript𝐷𝑛𝑣𝑢superscriptsuperscript𝐴𝑣superscriptsubscript𝐵𝑛𝑣𝑢superscriptsuperscriptsubscript𝐶𝑛𝑢superscriptsubscript𝐷𝑛𝑢𝑣(A^{u}+B_{n}^{u})^{v}\cdot(C_{n}^{v}+D_{n}^{v})^{u}=(A^{v}+B_{n}^{v})^{u}\cdot% (C_{n}^{u}+D_{n}^{u})^{v}

Where n 3 𝑛 3 n\geq 3 and odd, and

A𝐴\displaystyle A =y+xabsent𝑦𝑥\displaystyle=y+x
Bnsubscript𝐵𝑛\displaystyle B_{n} =i=0n1yn(i+1)xiabsentsuperscriptsubscript𝑖0𝑛1superscript𝑦𝑛𝑖1superscript𝑥𝑖\displaystyle=\sum_{i=0}^{n-1}y^{n-(i+1)}x^{i}
Cnsubscript𝐶𝑛\displaystyle C_{n} =yn+xnabsentsuperscript𝑦𝑛superscript𝑥𝑛\displaystyle=y^{n}+x^{n}
Dnsubscript𝐷𝑛\displaystyle D_{n} =i=0n1y2(n(i+1))x2iabsentsuperscriptsubscript𝑖0𝑛1superscript𝑦2𝑛𝑖1superscript𝑥2𝑖\displaystyle=\sum_{i=0}^{n-1}y^{2(n-(i+1))}x^{2i}

When y = 1 , u = x , v = 2 x formulae-sequence 𝑦 1 formulae-sequence 𝑢 𝑥 𝑣 superscript 2 𝑥 y=1,u=x,v=2^{x} , we get the original Gurevič Identities.

References