Carlson's Theorem

If $f(z)$ is regular and of the form ${\mathcal O}(e^{k\vert z\vert})$ where $k<\pi$, for $\Re[z]\geq 0$, and if $f(z)=0$ for $z=0$, 1, ..., then $f(z)$ is identically zero.

See also Generalized Hypergeometric Function

References

Bailey, W. N. ``Carlson's Theorem.'' §5.3 in Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, pp. 36-40, 1935.