On the general solution of a nonsymmetric partial difference functional equation analogous to the wave equation

Summary

We give the general solution of the nonsymmetric partial difference functional equationf(x + t,y) + f(x − t,y) − 2f(x,y)/t ² =f(x,y + s) + f(x,y − s) − 2f(x,y)/s ² (N) analogous to the well-known wave equation (∂ ²/∂x ² − ∂ ²/∂y ²)f(x,y) = 0 with the aid of generalized polynomials when no regularity assumptions are imposed onf. The result is as follows.

Theorem.Let R be the set of all real numbers. A function f: R × R → R satisfies the functional equation (N)for all x, y ∈ R, s, t ∈ R\{0}, and s ≠ t if and only if there exist

• (i) additive functions A, B: R → R

• (ii) a function C: R × R → R which is additive in each variable, and

• (iii) polynomials

P ₁ (x) = a ₁ +a ₃ x ²/2 +a ₅ x ³/6

P ₂ (y) = a ₇ +a ₃ y ²/2 +a ₄ y ³/6

P ₃ (x, y) = a ₅ xy ²/2 +a ₄ yx ²/2 +a ₆ xy ³/6 +a ₆ yx ³/6,where a ₁,a ₃,a ₄,a ₅,a ₆,and a ₇ are constants, such that f(x,y) = A(x) + B(y) + C(x,y) + P ₁ (x) + P ₂ (y) + P ₃ (x,y) for all x,y ∈ R.