Proof/disproof of conjecture involving noncentral Stirling numbers of the second kind I have a conjecture I am looking at involving the noncentral Stirling numbers of the second kind (for explanation of these numbers, see e.g., Koutras 1982).  I'm having some difficulty proving it.  I've tried expanding the noncentral numbers to the central Stirling numbers but this is just making a mess.  Can anyone prove/disprove this?

Conjecture: For large $n$ we have the asymptotic equivalence:$$\frac{S_a(n,k)}{S_b(n,k)} \sim \bigg( \frac{k+a}{k+b} \bigg)^{n-k}.$$
(If necessary, you can add the conditions $k \geqslant 0$, $a>0$ and $b>0$ since that is the case I am interested in.)

 A: According to formula (2.6) of the link provided by the OP,
$$ \tag{1}S_a(n,k)=\frac{1}{k!}\sum_{m=0}^k(-1)^{m-k}\binom{k}{m}(m-a)^n $$
According to formula (2.3) of reference [1]
$$ \tag{2} B_{n-k}^{-k}(-a) = \frac{(n-k)!}{n!} \sum_{m=0}^k (-1)^{m-k}\binom{k}{m}(m-a)^n $$
where $B_n^k(x)$ are the generalized Bernoulli polynomials, and are implemented in Mathematica as NorlundB[n,k,a].  Thus it is easy to conclude
$$S_a(n,k) = \binom{n}{k}B_{n-k}^{-k}(-a) $$
Reference [1] states asymptotic expansions for various regimes of parameter variation. The OP did not specify how $a$ and $k$ vary with respect to $n,$ so I assume they are fixed and $a<k.$(and rather small for numerical calculation.)  Then formula (2.8) applies, and by cancelling factors without $a$ or $b$ dependence one easily obtains
$$ \tag{3} \frac{B_{n-k}^{-k}(-a)}{B_{n-k}^{-k}(-b)} \sim \frac{\big( 1- \exp{(-\frac{n}{k-a})}\big)^k\big(k-a\big)^n} {\big( 1- \exp{(-\frac{n}{k-b})}\big)^k \big(k-b\big)^n} $$
The only way to get something that looks like the OP's conjecture is to assume the argument of the exponential is small, but this is impossible for fixed $a$ and $k.$  (Furthermore, the signs on $a$ and $b$ differ from the proposed asymptotics.) For a numerical example, let's calculate the left-hand side of (3) versus the RHS with n=50, a=4/10, b=7/10, and k=4.  Then the LHS = 77.522527 and the RHS = 77.522342 for about a 0.000024% error.
$[1]$ Large degree asymptotics of generalized Bernoulli and Euler polynomials, J. Lopez, N. Temme in J. Math Analysis & Applic. 363 (2010) 197-208.
