On the arithmetic nature of hypertranscendental functions at complex points

Abstract

Most well-known transcendental functions usually take transcendental values at algebraic points belonging to their domains, the algebraic exceptions forming the so-called exceptional set. For instance, the exceptional set of the function e z − 2 is the set { 2 } , as follows from the Hermite–Lindemann theorem. In this paper, we shall use interpolation formulae to prove that any subset of Q ¯ is the exceptional set of uncountably many hypertranscendental entire functions with order of growth as small as we wish. Moreover these functions are algebraically independent over C .