Abstract
Lacunary functions based on centered polygonal numbers have interesting features which are distinct from general lacunary functions. These features include rotational symmetry of the modulus of the functions and a notion of polished level sets. The behavior and characteristics of the natural boundary for centered polygonal lacunary sequences are discussed. These systems are complicated but, nonetheless, well organized because of their inherent rotational symmetry. This is particularly apparent at the so-called symmetry angles at which the values of the sequence at the natural boundary follow a relatively simple 4 p -cycle. This work examines special limit sequences at the natural boundary of centered polygonal lacunary sequences. These sequences arise by considering the sequence of values along integer fractions of the symmetry angle for centered polygonal lacunary functions. These sequences are referred to here as p-sequences. Several properties of the p-sequences are explored to give insight in the centered polygonal lacunary functions. Fibered spaces can organize these cycles into equivalence classes. This then provides a natural way to approach the infinite sum of the actual lacunary function. It is also seen that the inherent organization of the centered polygonal lacunary sequences gives rise to fractal-like self-similarity scaling features. These features scale in simple ways.
Highlights
Analytic functions play a predominant role in physics
The zeros of the cpn-based lacunary functions | f N (z)| are the zeros encountered along the radial segment from the origin to the natural boundary for segments at integer multiples of the pth symmetry angle
The special characteristics of the cpns lead to true rotational symmetry of | f N (z)| and to the polished nature of its main unity contour. This was compared with other lacunary sequences which exhibit only quasi-rotational symmetry and are not polished
Summary
Analytic functions play a predominant role in physics. The isolated singularities of what are otherwise analytic functions often carry much information about the function itself and thereby provide physical insight. The goal of this current work is to provide general insight into lacunary functions, but, more to discuss features of what are referred to in this work as centered polygonal lacunary functions This is the family of lacunary functions where the active terms in the Taylor series are those in which the powers are centered polygonal numbers (cpns) [25]. This work proceeds in investigating the structure of the limiting values at the natural boundary of centered polygonal lacunary sequences along symmetry angles. These special infinite sequences will be referred to here as p-sequences. Yamada and Ikeda have recently investigated the use of Padé approximate methods for speeding up the convergence of summations for lacunary functions [10,28]
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