exponentialpolynomial

An exponential polynomial, also called an exponential-polynomial, is a function that can be written as a finite linear combination of terms of the form P(x) e^{λ x}, where P is a polynomial and λ is a constant (possibly complex). More generally, f(x) = ∑_{k=1}^s P_k(x) e^{λ_k x} with distinct λ_k; the coefficients of P_k may be complex, with the resulting function real-valued in suitable cases.

Such functions arise as the general solutions to linear differential equations with constant coefficients, and in the solutions to linear recurrence relations with constant coefficients in the discrete setting. If a particular exponential term corresponds to a repeated root λ of the characteristic equation, the associated polynomial factor increases in degree, leading to terms of the form x^m P(x) e^{λ x}. Thus exponential polynomials naturally appear in both continuous and discrete linear systems.

Examples include e^{2x}, x e^{3x}, and (2x^2 − 1) e^{−5x}. In the discrete case, sequences of the form ∑ P_k(n) r_k^n, where r_k are roots of the characteristic polynomial of a linear recurrence, are exponential polynomials; a repeated root multiplies the corresponding term by a power of n.

Properties include closure under addition and linear combination, and the preservation of the form under differentiation: d/dx [P(x) e^{λ x}] = (P′(x) + λ P(x)) e^{λ x}. Asymptotically, the term with the largest real part of λ (or largest |r_k| in the discrete setting) dominates growth.

See also: Exponential function, Polynomial, Linear differential equations with constant coefficients, Linear recurrences, Characteristic polynomial.