Laguerre expansion with the Laguerre polynomials
$$L_n(x)=\sum_{k=0}^n (-1)^k \binom{n}{k}\frac{x^{k}}{k!}= \triangledown_{k=0}^{n} \frac{x^k}{k!} $$
formally gives
$$\frac{x^{s-1}}{(s-1)!}= \sum_{n \geq 0} L_n(x) \int_0^\infty e^{-t} L_n(t) \frac{t^{s-1}}{(s-1)!}dt $$
$$= \sum_{n \geq 0}(-1)^n \binom{s-1}{n} L_n(x)= \triangledown_{n=0}^{s-1}L_n(x) =\triangledown_{n=0}^{s-1}\triangledown_{k=0}^{n} \frac{x^k}{k!} $$
$$= \int_0^\infty e^{-t}[\sum_{n \geq 0} L_n(x) L_n(t)] \frac{t^{s-1}}{(s-1)!}dt= \int_0^\infty \delta(x-t) \frac{t^{s-1}}{(s-1)!}dt .$$
Norlund in his book Lecons sur les Series d'Interpolation (1926, p. 165) established that
$$\frac{x^{s-1}}{(s-1)!}= \triangledown_{n=0}^{s-1}\triangledown_{k=0}^{n} \frac{x^k}{k!} $$
holds for $x >0$ and $Re(s) > 1/4$.
Wigert had shown earlier in "Contributions à la théorie des polynômes d'Abel-Laguerre" [Arkiv for Matematik, Astronomi och Fysik (Stockholm), t. XV, 1921, no. 25] on pp. 16-7, (frames 595-6)] that the Newton series identity holds for $s > 1$. Furthermore, in "[Sur une transformation de la série $\sum_{n} a_n \binom{s}{n}$][2]" [Arkiv Mat. Astr. och Fysik (Stockholm), t. VII, no. 26, 1911, p. 1 (frame 555)], he claims that the Newton series
$$\triangledown_{n = 0}^s a_n$$
and the Dirichlet series
$$\sum_{n \geq 1} (-1)^n \frac{a_n}{n^{s+1}}$$
have the same domain of convergence.
Question:
Is the Newton series identity
$$\frac{x^{s-1}}{(s-1)!}= \triangledown_{n=0}^{s-1}\triangledown_{k=0}^{n} \frac{x^k}{k!} $$
valid for $0 < Re(s) \leq 1/4$?