Exponential polynomial

An exponential polynomial generally has both a variable x and some kind of exponential function E(x). In the complex numbers there is already a canonical exponential function, the function that maps x to e^x. In this setting the term exponential polynomial is often used to mean polynomials of the form P(x, e^x) where P ∈ C[x, y] is a polynomial in two variables.[1][2]

There is nothing particularly special about C here; exponential polynomials may also refer to such a polynomial on any exponential field or exponential ring with its exponential function taking the place of e^x above.[3] Similarly, there is no reason to have one variable, and an exponential polynomial in n variables would be of the form P(x₁, ..., x_n, e^x₁, ..., e^x_n), where P is a polynomial in 2n variables.

For formal exponential polynomials over a field K we proceed as follows.[4] Let W be a finitely generated Z-submodule of K and consider finite sums of the form

${\displaystyle \sum _{i=1}^{m}f_{i}(X)\exp(w_{i}X)\ ,}$

where the f_i are polynomials in K[X] and the exp(w_i X) are formal symbols indexed by w_i in W subject to exp(u + v) = exp(u) exp(v).

A more general framework where the term 'exponential polynomial' may be found is that of exponential functions on abelian groups. Similarly to how exponential functions on exponential fields are defined, given a topological abelian group G a homomorphism from G to the additive group of the complex numbers is called an additive function, and a homomorphism to the multiplicative group of nonzero complex numbers is called an exponential function, or simply an exponential. A product of additive functions and exponentials is called an exponential monomial, and a linear combination of these is then an exponential polynomial on G.[5][6]