一类多元线性函数方程(Ⅰ)

设函数f(x1，x2，…，xn)对xn有连续二阶偏导数，我们寻求函数方程n↑∑i=1(-1)^i-1f(x1，…，xi xi 1，…，xi 1) f(x1，…，xi-xi-x(i 1)，…，x(n 1))] (-1)^n2f(x1，x2，…，xn)=0的一般解．首先，给出了方程n↑∑i=l(-1)^i-1F(x1，…，xi x(i 1)，…，x(n 1)) F(x1，…，xi-x(i 1)，…，x(n 1)]=0的一般解，其次，上述第1式对x(n 1)两次微分，并简化得到形如第2式的方程．第1个函数方程的一般解为f(x1，x2，…，xn)=(n-1)↑∑i=1(-1)^i-1A(x1，…，xi x(i 1)，…，xn) A(x1，…，xi-x(i 1)),…,xn)] (-1)^n-1 2A(xi,x2,…,x(n-1).其中A(x1，x2，…，x(n-1))是对x(n-1)具有连续二阶导数的任意函数。

On a Class of Linear Functional Equations for Functions of Several Variables (Ⅰ)

By letting the function f(x1,x2,…,xn) have continuous partial derivatives of second order with respect to xn,the functional equation ni=1(-1)i-1f(x1,…,xi+xi+1,…,xn+1)+f(x1,…,xi-xi+1,…,xn+1)]+(-1)n2f(x1,x2,…,xn)=0 is considered.First,the general solution of the equation ni=1(-1)i-1F(x1,…,xi+xi+1,…,xn+1)+F(x1,…,xi-xi+1,…,xn+1)]=0 was presented.Then,the first functional equation was twice differentiated with respect to xn+1 and reduced to an equation of the aforementioned type.It is found that the general solution of the first functional equation is f(x1,x2,…,xn)=n-1i=1(-1)i-1A(x1,…,xi+xi+1,…,xn)+A(x1,…,xi-xi+1,…,xn)]+(-1)n-12A(x1,x2,…,xn-1).Where A(x1,x2,…xn-1) is an arbitrary twice continuous differentiable with respect to xn-1.