Elliptic solutions of the Yang-Baxter equation and modular hypergeometric functions

Various results in algebra, analysis, and geometry can be generalized by replacing the ordinary numbers (integer, real or complex) by their trigonometric analogues. For x ∈ ℂ, the trigonometric number [x] h ∈ ℂ is defined by $$[x]_h = \frac{{\sin (\pi hx)}}{{\sin (\pi h)}}$$ (0.a) where h ∈ ℂ\ℤ is a fixed parameter. It is clear that \(\mathop {\lim }\limits_{h \to 0} [x]_h = x\) thus, [x] h may be viewed as a one-parameter deformation of x. The trigonometric numbers are not additive: generally speaking [x+y] h ≠ [x] h +[y] h . However, they satisfy a kind of additivity of “second order”: for any x, y, z ∈ ℂ, $$\left[ {x + z} \right]h\left[ {x - z} \right]h = \left[ {x + y} \right]h\left[ {x - y} \right]h\left[ {y + z} \right]h\left[ {y - z} \right]h.$$ (0.b) Many identities between ordinary numbers can be proved using only the additivity of second order and therefore allow a trigonometric deformation.