This paper is inspired by a class of infinite order differential operators arising in quantum mechanics. They turned out to be an important tool in the investigation of evolution of superoscillations with respect to quantum fields equations. Infinite order differential operators act naturally on spaces of holomorphic functions or on hyperfunctions. Recently, infinite order differential operators have been considered and characterized on the spaces of entire monogenic functions, i.e. functions that are in the kernel of the Dirac operators. The focus of this paper is the characterization of infinite order differential operators that act continuously on a different class of hyperholomorphic functions, called slice hyperholomorphic functions with values in a Clifford algebra or also slice monogenic functions. This function theory has a very reach associated spectral theory and both the function theory and the operator theory in this setting are subjected to intensive investigations. Here we introduce the concept of proximate order and establish some fundamental properties of entire slice monogenic functions that are crucial for the characterization of infinite order differential operators acting on entire slice monogenic functions.

We prove that any subset of $\overline {\mathbb {Q}}^m$ (closed under complex conjugation and which contains the origin) is the exceptional set of uncountably many transcendental entire functions over $\mathbb {C}^m$ with rational coefficients. This result solves a several variables version of a question posed by Mahler for transcendental entire functions [Lectures on Transcendental Numbers, Lecture Notes in Mathematics, 546 (Springer-Verlag, Berlin, 1976)].

The aim of this paper is twofold. The first aim is to describe the entire solutions of the partial differential equation (PDE) $u_{z_1}^2+2Bu_{z_1}u_{z_2}+u_{z_2}^2=e^g$, where B is a constant and g is a polynomial or an entire function in $\mathbb {C}^2$. The second aim is to consider the entire solutions of another PDE, which is a generalization of the well-known PDE of tubular surfaces.

This paper gives an equivalent form of Picard’s theorem via entire solutions of the functional equation ${{f}^{2}}\,+\,{{g}^{2}}\,=\,1$ and then its improvements and applications to certain nonlinear (ordinary and partial) differential equations.

The paper describes entire solutions to the eiconal type non-linear partial differential equations, which include the eiconal equations ${{({{X}_{1}}(u))}^{2}}\,+\,{{({{X}_{2}}(u))}^{2}}\,=\,1$ as special cases, where ${{X}_{1}}\,=\,{{p}_{1}}\partial /\partial {{z}_{1}}\,+\,{{p}_{2}}\partial /\partial {{z}_{2}},\,{{X}_{2}}\,=\,{{p}_{3}}\partial /\partial {{z}_{1}}\,+\,{{p}_{4}}\partial /\partial {{z}_{2}}$ are linearly independent operators with ${{p}_{j}}$ being arbitrary polynomials in ${{\mathbf{C}}^{2}}$.

Let $f\,:\,{{\mathbb{R}}^{n}}\,\to \,\mathbb{R}$ be ${{C}^{\infty }}$ and let $h\,:\,{{\mathbb{R}}^{n}}\,\to \,\mathbb{R}$ be positive and continuous. For any unbounded nondecreasing sequence $\{{{c}_{k}}\}$ of nonnegative real numbers and for any sequence without accumulation points $\{{{x}_{m}}\}$ in ${{\mathbb{R}}^{n}}$, there exists an entire function $g\,:\,{{\mathbb{C}}^{n}}\,\to \,\mathbb{C}$ taking real values on ${{\mathbb{R}}^{n}}$ such that

$$\left| {{g}^{\left( \alpha \right)}}\left( x \right)-{{f}^{\left( \alpha \right)}}\left( x \right) \right|\text{ }<h\left( x \right),\left| x \right|\ge {{c}_{k}},\left| \alpha \right|\le k,k=0,1,2,...,$$

$${{g}^{\left( \alpha \right)}}\left( {{x}_{m}} \right)\,=\,{{f}^{\left( \alpha \right)}}\left( {{x}_{m}} \right),\,\,\,\left| {{x}_{m}} \right|\,\ge \,{{c}_{k}},\,\left| \alpha \right|\,\le \,k,\,m,\,k\,=\,0,\,1,\,2,\,.\,.\,.\,.$$

This is a version for functions of several variables of the case $n\,=\,1$ due to $L$. Hoischen.

The paper is concerned with description of entire solutions of the partial differential equations where m ≥ 2, n ≥ 2 are integers and g is a polynomial or an entire function in C2. Descriptions are given and complemented by various examples.

The paper gives a good approximation of a Ck function on Rn and its derivatives by the restriction of an entire function on Cn and its derivatives respectively.