- [NC] The identity is true also for noncommutative multiplication.

- [TS] The functional equation has trigonometric sine function solutions. Note that this includes circular and hyperbolic sine functions.
- [TT] The functional equation has trigonometric tangent function solutions. Note that this includes circular and hyperbolic tangent functions.
- [JE] The functional equation has Jacobi elliptic function solutions. Note that this includes Jacobi sin am = sn, and also sc, sd, and special cases as circular and hyperbolic sine and tangent functions.
- [WS] The functional equation has Weierstrass sigma function solutions. Note that this includes Jacobi and Ramanujan theta functions and also special cases of circular and hyperbolic sine functions. See Gavrilov, arXiv:math/06031532v2 for proof of the main theorem.
- [WZ] The functional equation has reciprocal of the Weierstrass zeta function solutions and circular and hyperbolic tangent functions.
- [JZ] The functional equation has Jacobi Zeta and Epsilon function solutions and special cases as circular and hyperbolic sine and tangent functions.
- [CU] The functional equation has f(x) = k*x^3 as a solution.
The last two tags indicate special features of the identity:

- [ZS] A subset of terms sum to zero but not for the functional equation.
- [CT] A rare case where some of the terms have numeric coefficients.

{ id1_2_1_1 = +2*a -(a+a) ; }has only linear functions as solutions of the functional equation 0 = 2*f(a) - f(a+a), but this is too trivial to include in this collection. Note that

{ id2_2_1_1 = +(a-b) +(b-a) ; }, { id2_2_1_2 = +a*b -b*a ; }and other identities with only two terms I have decided are too trivial to include. Thus, all identities in the collection have at least three terms. Note that

{ id1_3_1_2 = +a*a +a*(a+a+a) -(a+a)*(a+a) ; }is a special case of id2_3_1_2a but I consider it too trivial to include, though it would be tagged [TS] and so is of interest. Other similar one variable identities such as

{ id1_3_1_4 = +a*a*a*(a+a+a+a+a) -(a+a)*(a+a)*(a+a)*(a+a+a+a) +a*(a+a+a)*(a+a+a)*(a+a+a) ; }would be tagged [WS] and its associated functional equation is the simplest one variable functional equation for the Weierstrass sigma function. Clearly, there are an unlimited number of such one variable identities, but these are reserved for, perhaps, some future collection of numerical identities. Thus, all the identities in this collection have at least two variables. You might be interested in A Collection of Algebraic Identities by Tito Piezas which is a collection of algebraic identities used for diophantine equations of special kinds like sums of powers equaling other sums of powers.

This is a directed sum of three sides of a triangle. (one vertex at origin) case n=1 of 0 = a^n - b^n - (a-b)*(a^(n-1) + ... + b^(n-1)). {} 2 vars with 3 terms highest degree 1 of total 1 [NC] { id2_3_1_1a = +a -b -(a-b) ; }

entire table as a PARI/GP program

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Michael Somos <michael.somos@gmail.com>