Continued fractions and an annelidic PDE (Q794218)

One constructs the general solution of the P.D.E. \[ (1)\quad \partial f_ n(x_ 1,...,x_ n)/\partial x_ j=f_{j-1}(x_ 1,...,x_{j- 1})f_{n-j}(x_{j+1},...,x_ n) \] where \(f_ 0=1\), \(f_ 1(x_ 1),...\); \(f_ n(x_ 1,...;x_ n)\), and 1\(\leq j\leq n\). In particular (1) is satisfied by a given determinant which is considered in the study of continued fractions and has a curious analogy to the following property: if one cuts an earthworm in two, each half regenerates the missing part and becomes a new earthworm. The general solution of (1) is given by \(\Sigma c(I)x_{i_ 1}...x_{i_ k}\), where the sum is taken over all \(2^ n\) subsets \(I=\{i_ 1,...,i_ k\} \subseteq \{1,...,n\}\) and \(c(I)=a(i_ 1- 1)a(i_ 2-i_ 1-1)...a(n-i_ k)\), \(a(n)=f_ n(0,...,0)\).