Let K be either the field of
complex numbers C or the field of real numbers R. Let n be a fixed integer > 2, and
𝜃 denote the number exp(2πi∕n). Let f,f_{j} : C → K for j = 0,⋯,n. Define Λ_{n} and Ω_{n}
by
Λ_{n}(x,y)  = n^{−1}∑
_{j=0}^{n−1}f(x + 𝜃^{j}y)− f(x),  
 Ω_{n}(x,y)  = n^{−1}∑
_{j=0}^{n−1}f_{
j}(x + 𝜃^{j}y)− f_{n}(x),   
for all x,y ∈ C. Our main result is the following. If (n + 1) unknown functions
f_{j} : C → K for j = 0,1,⋯,n satisfy the quasi mean value property Ω_{n}(x,y) = 0 for
all x,y ∈ C, then (n + 1) unknown functions f_{j} satisfy the difference functional
equation Δ_{u}^{n}f_{j}(x) = 0 for all u,x ∈ C and for each j = 0,1,⋯,n, where the usual
difference operator Δ_{u} is defined by Δ_{u}f(x) = f(x + u) −f(x). By using this result
we prove somewhat stronger results than the theorem of S. KakutaniM.
Nagumo (Zenkoku, Sūgaku Danwakai, 66 (1935), 10–12) and J. L. Walsh
(Bull. Amer. Math. Soc., 42 (1936), 923–930) for the mean value property
Λ_{n}(x,y) = 0 of harmonic and complex polynomials.
