The Theta-Functions, although arising historically from the Elliptic Functions, may be considered as in order of simplicity preceding these, and connecting themselves directly with the exponential function (e
x
or) exp.
x
; viz., they may be defined each of them as a sum of a series of exponentials, singly infinite in the case of the single functions, doubly infinite in the case of the double functions ; and so on. The number of the single functions is = 4; and the quotients of these, or say three of them each divided by the fourth, are the elliptic functions sn, cn, d n ; the number of the double functions is (4
2
= ) 16 ; and the quotients of these, or say fifteen of them each divided by the sixteenth, are the hyper-elliptic functions of two arguments depending on the square root of a sex tic function : generally the number of the
p
-tuple theta-functions is = 4
p
; and the quotients of these, or say all but one of them each divided by the remaining function, are the Abelian functions of
p
arguments depending on the irrational function
y
defined by the equation F (
x, y
) = 0 of a curve of deficiency
p
). If instead of connecting the ratios of the functions with a plane curve we consider the functions themselves as coordinates of a point in a (4
p
—1)dimensional space, then we have the single functions as the four coordinates of a point on a quadri-quadric curve (one-fold locus) in ordinary space; and the double functions as the sixteen coordinates of a point on a quadri-quadric two-fold locus in 15-dimensional space, the deficiency of this two-fold locus being of course = 2.
Eisenstein Series Identities Involving the Borweins' Cubic Theta Functions
