Centered Polygonal Lacunary Sequences

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.

Taming the Natural Boundary of Centered Polygonal Lacunary Functions—Restriction to the Symmetry Angle Space

“…Definitions, notation, and some theorems from References [14,15] are briefly collected here for the convenience of the reader.…”

“…(Note that the sum starts at n = 1 for convenience but not necessity.) Following References [14,15], we use the notation…”

“…It turns out that nearly all of the structural features of centered polygonal lacunary functions are independent of the choice of k (see References [14,15]). This is because the centered polygonal numbers are related to the triangular numbers (see Reference [24]) in a simple way.…”

“…Their special properties are mainly due to the unusual symmetry present in this family, compared to an arbitrary lacunary function (see References [14][15][16]). A class of infinite sequences associated with lacunary functions are called lacunary sequences and recent work has focused on exploring particular bounded sequences of numbers arising at the natural boundary of centered polygonal lacunary sequences (see References [14,15]). These p-sequences, as they are called, have been well characterized and this work has been significantly enhanced by the construction of graphs to represent the p-sequences (see Reference [14]).…”

“…A class of infinite sequences associated with lacunary functions are called lacunary sequences and recent work has focused on exploring particular bounded sequences of numbers arising at the natural boundary of centered polygonal lacunary sequences (see References [14,15]). These p-sequences, as they are called, have been well characterized and this work has been significantly enhanced by the construction of graphs to represent the p-sequences (see Reference [14]). The graphs that have arisen are interesting in and of themselves, especially in that they reveal self-similarity and scaling that allow for a renormalization approach (see Reference [15]).…”