A collection of over 600 special algebraic identities
OVERVIEW
This is a collection of special algebraic identities each of which is
a homogeneous expression in a finite number of variables with (mostly)
algebraic signs as coefficients and reduces to zero when expanded out.
Notation: "idv_t_h_d" where v is number of variables, t is the number of
terms, h is the highest degree of any factor, and d is the total degree.
The notation is not yet stable. Don't depend on it not changing in future.
The collection is sorted by the numbers in the identifier.
INFINITE FAMILIES
There is an simple infinite sequence of identities which begins with
id2_3_1_1, id4_4_1_2e, id6_5_1_3c, and so on. They're all telescoping sums.
There is an simple infinite sequence of identities which begins with
id4_3_2_2, id6_4_3_3, id8_5_4_4, and so on. They're all telescoping sums.
There is an simple infinite sequence of identities which begins with
id6_3_3_4a, id9_4_3_9, and so on. They're also telescoping sums.
TAGS
Some of the identities have "tags" associated with them. They are formed
of two upper case letters enclosed in square brackets. Two examples,
- [NC] The identity is true also for noncommutative multiplication.
- [ZS] A subset of terms sum to zero but not for the functional equation.
Each identity has an associated functional equation. In general, the
only solution of such an equation is f(x) = k*x for some constant k.
If the identity has only linear factors in its terms, then since it is
homogeneous, any constant multiple is essentially the same solution.
But, some identities have other solutions which are indicated by tags:
- [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.
For example, id2_3_1_2:
0 = +a*a -b*b -(a+b)*(a-b)
has the associated functional equation
0 = f(a)*f(a) - f(b)*f(b) + f(a+b)*f(a-b)
with trigonometric sine function solutions f(x) = sin(k*x).
Note we always assume f(-x) = -f(x) for all x to avoid sign ambiguities.
LIMITATIONS
Note that { 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. Similar other one variable equations
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 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.
THE FIRST IDENTITY
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]
d2_3_1_1a = +a -b -(a-b) ; }
entire table as a PARI/GP program
Back to my home page
Last Updated Jan 23 2020
Michael Somos <michael.somos@gmail.com>