Functions with nonzero finite difference (Q1813410)

Let \(F(z)\) be an analytic function in the domain \(\Pi=\{\text{Re} z\geq 0\}\), \(\overline \Pi=\{\text{Re} z\geq 0\}\backslash\{0\}\), \(z\in \Pi\), \(\zeta_ 1,\dots,\zeta_ n\in\overline \Pi\). The \(n\)-th finite difference \(\Delta_ n[F(z);\zeta_ 1,\dots,\zeta_ n]\) is defined by the relations \[ \Delta_ 1[F(z);\zeta_ 1]=F(z+\zeta_ 1)-F(z), \] \[ \Delta_ n[F(z);\zeta_ 1,\dots,\zeta_ n]=\Delta_ 1[\Delta_{n- 1}[F(z);\zeta_ 1,\dots, \zeta_{n-1}];\zeta_ n]. \] The properties of \(n\)-th finite differences are studied and, in particular, \(n\)-th finite differences of elementary functions such as polynomials, exponential functions, sine- and cosine-functions are evaluated. The class \(Q_ n(\Pi)\) of analytic functions \(F(z)\) in \(\Pi\) the \(n\)-th finite differences \(\Delta_ n[F(z);\zeta_ 1,\dots,\zeta_ n]\) of which do not vanish for any \(z\in\Pi\), \(\zeta_ 1,\dots,\zeta_ n\in\overline\Pi\) is introduced. The properties of functions \(F(z)\in Q_ n(\Pi)\) are investigated and the conditions for a function \(F(z)\) to be an entire function are given.