Power-type initial conditions and diagonal-ratio asymptotics

Determine whether, for the recurrence with boundary initial conditions \(f_{n,0}=f_{0,n}=n^n\), the diagonal ratio satisfies \(f_{n+1,n+1}/f_{n,n}\approx e\,(n+1/2)\) as \(n\to\infty\).

Background

The paper also considers boundary data with superexponential growth, specifically $f_{n,0}=f_{0,n}=n^n$. As with factorial boundary values, the corresponding generating function has radius of convergence zero, placing the problem outside the principal asymptotic framework developed for geometric boundary conditions. The authors formulate a conjectural approximation for the consecutive-diagonal ratio and note that the diagonal sequence remains P-recursive in this case.