Power series whose coefficients are limits of coefficients of polynomial interpolations

Question

When can you reconstruct the power series of a function by taking the limits of the coefficients of the polynomials that interpolate its values at $0,1,2,\dots,j$?

More precisely:

Let $f\colon\mathbb{R}\to\mathbb{R}$. For all nonnegative integers $j$, let $p_j$ be the unique polynomial of degree $j$ with real coefficients such that $p_j(n)=f(n)$ for $n=0,\dots,j$. Let $a_{j,k}$ be the coefficient of $x^k$ in $p_j$, where $a_{j,k}=0$ if $k>j$. Under what circumstances do we have that $\lim_{j\to\infty} a_{j,k}$ exists for all $j$? In this case, let $c_k=\lim_{j\to\infty} a_{j,k}$. When $f$ is analytic, under what circumstances do we have that $f(x)=c_0+c_1 x+c_2 x^2 + c_3 x^3 +\cdots$?

Answer 1

In the paper, A note on convergence of Newton interpolating polynomials, by D. Dimitrov and J. Philipps, Journal of Computational and Applied Mathematics Volume 51, Issue 1, 30 May 1994, Pages 127-130, the following simple criterion is mentioned: $f$ is entire of exponential type less than $\log 2$, then the sequence of interpolation polynomials interpolating at non-negative integers converges to $f$.