The power series 1+zΓ(1+i)+z2Γ(1+2i)+z3Γ(1+3i)+Γ has a natural boundary!

摘要

The lacunary series are the most classic examples among all the power series whose circle of convergence constitutes a natural boundary (Dienes, 1931 [4, §93-94, pp. 372-383], Titchmarsh, 1939 [8, §7.43, p. 223], ...). In this Note, we study a family of non-lacunary power series whose coefficients are given by means of values of the Gamma function over vertical line. We explain how to transform these series into lacunary Dirichlet series, which allows us to conclude the existence of their natural boundary. Our results, which illustrate in what manner the Gamma function may have an unpredictable behaviour on any vertical line, may also be partially understood in the framework of our forthcoming work on a class of differential q-difference equations, namely, on pantograph type equations (meanwhile see Kato and McLeod (1971) [6]).