Rational-trigonometric deformation

We discuss a rational-trigonometric deformation for two algebraic cases; for the universal enveloping algebra $U(g[u])$ of a polynomial loop algebra $g[u],$ where g is a finite-dimensional complex simple Lie algebra, and for the two-dimensional plane (x, y). In the both cases these deformations are obtained by a singular transformation (at $q=1$⁠) of the q-deformation of $U(g[u])$ and (x, y). In the first case the quantum Hopf algebra called Drinfeldian $Dqη(g)$ is a quantization of $U(g[u])$ in the direction of a classical r-matrix which is a sum of the simple rational and trigonometric r-matrices. The Drinfeldian $Dqη(g)$ contains $Uq(g)$ as a Hopf subalgebra, moreover $Uq(g[u])$ and Yangian $Yη(g)$ are its limit quantum algebras when the deformation parameters η goes to 0 and q goes to 1, respectively. Using the rational-trigonometric deformation of the plane (x, y) we introduce the $(q,η)$- and η-numbers, $(q,η)$- and η-exponentials, and $(q,η)$- and η-hypergeometric series.