Factorial initial conditions and diagonal-ratio asymptotics

Determine whether, for the recurrence with parameters \(\alpha=\beta=\gamma=1\) and boundary initial conditions \(f_{n,0}=f_{0,n}=n!\), the diagonal ratio satisfies \(f_{n+1,n+1}/f_{n,n}\approx n+1\) as \(n\to\infty\).

Background

The paper proposes extending the weighted Delannoy recurrence to boundary data more general than geometric sequences. When the boundary values grow factorially, the associated generating function has radius of convergence zero, so the asymptotic behavior differs from the nonnegative geometric-boundary cases treated earlier. For the specific choice α=β=γ=1\alpha=\beta=\gamma=1 and fn,0=f0,n=n!f_{n,0}=f_{0,n}=n!, the authors report a conjectural first-order approximation for the ratio of consecutive diagonal terms. The sequence fn,nf_{n,n} is nevertheless stated to be P-recursive in this example.

References

This happens, for instance, if \alpha=\beta=\gamma=1 and f_{n,0}=f_{0,n}=n!. In this example OEIS A346374 we conjecture that \frac{f_{n+1,n+1}}{f_{n,n}} \approx n+1 (where a_n \approx b_n means, as usual, that a_n-b_n \to 0 when n \to \infty).

— A class of weighted Delannoy numbers  (2501.09726 - Grau et al., 16 Jan 2025) in Section 5, “Further work”