what What is the solution, f$f(n)$, of the following functional equation: mf$mf(m)+nf(n)=(m+n+xmn)f(m+n+xmn)$?

whatWhat is the solution, f(n)$f(n)$, of the following functional equation: mf(m)+nf(n)=(m+n+xmn)f(m+n+xmn)

$$mf(m)+nf(n)=(m+n+xmn)f(m+n+xmn) ,$$

where f$f$ takes on integer values, m$m$ and n$n$ are integers, and x$x$ is an indeterminate? It is a fundamental step in the proof of a famous theorem of Weierstrass that a non rational-rational meromorphic function which admits an algebraic addition theorem is necessarily periodic. The equation, due to A.R. Forsyth, is "solved" by him using theaccording to his following wordsdescription: "Since the left-hand side is the sum of two functions of distinct and independent magnitudes, the form of the equation shewsshows that it can be satisfied only only if x= 0$x= 0$,so that..." I

I am unable to follow this proof that necessarily x=0$x=0$. If one can show it, then it is easy to show that the only solution of the functional equation is f(n)= a$f(n)= a$ constant.

what is the solution, f(n), of the following functional equation: mf(m)+nf(n)=(m+n+xmn)f(m+n+xmn)

what is the solution, f(n), of the following functional equation: mf(m)+nf(n)=(m+n+xmn)f(m+n+xmn) where f takes on integer values, m and n are integers, and x is an indeterminate? It is a fundamental step in the proof of a famous theorem of Weierstrass that a non rational meromorphic function which admits an algebraic addition theorem is necessarily periodic. The equation, due to A.R. Forsyth, is "solved" by him using the following words: "Since the left-hand side is the sum of two functions of distinct and independent magnitudes, the form of the equation shews that it can be satisfied only if x= 0,so that..." I am unable to follow this proof that necessarily x=0. If one can show it, then it is easy to show that the only solution of the functional equation is f(n)= a constant.

What is the solution, $f(n)$, of the following functional equation: $mf(m)+nf(n)=(m+n+xmn)f(m+n+xmn)$?

What is the solution, $f(n)$, of the following functional equation:

$$mf(m)+nf(n)=(m+n+xmn)f(m+n+xmn) ,$$

where $f$ takes on integer values, $m$ and $n$ are integers, and $x$ is an indeterminate? It is a fundamental step in the proof of a famous theorem of Weierstrass that a non-rational meromorphic function which admits an algebraic addition theorem is necessarily periodic. The equation, due to A.R. Forsyth, is "solved" by him according to his following description: "Since the left-hand side is the sum of two functions of distinct and independent magnitudes, the form of the equation shows that it can be satisfied only if $x= 0$,so that..."

I am unable to follow this proof that necessarily $x=0$. If one can show it, then it is easy to show that the only solution of the functional equation is $f(n)= a$ constant.