User:Dom walden/Meromorphic Functions with Multiple Dominant Singularities

Examples of periodic fluctuations

From Flajolet and Sedgewick.

The generating function ${\frac {1}{1-z^{3}}}=1+z^{3}+z^{6}+z^{9}+\cdots$ has radius of convergence $1$, but the coefficients cancel out completely for $n\equiv \pm 1{\pmod {3}}$. It has three singularities of equal modulus: $1,e^{i{\frac {2}{3}}\pi },e^{i{\frac {4}{3}}\pi }$.

The generating function ${\frac {1}{1+z^{2}}}=1-z^{2}+z^{4}-z^{6}+z^{8}-\cdots$ also has radius of convergence $1$, but the coefficients completely cancel out for odd $n$ and oscillate between $1$ and $-1$ for even $n$. It has two singularities of equal modulus: $\pm i$.

The periodic behaviour is even more interesting when the two above generating functions are added together:

${\frac {1}{1+z^{2}}}+{\frac {1}{1-z^{3}}}=2-z^{2}+z^{3}+z^{4}+z^{8}+z^{9}-z^{10}+2\,z^{12}-z^{14}+z^{15}+z^{16}+z^{20}+z^{21}-z^{22}+2z^{24}-z^{26}+z^{27}+z^{28}+\cdots$

The coefficients are:

${\begin{cases}0&n\equiv 1,5,6,7,11,{\pmod {12}}\\-1&n\equiv \pm 2{\pmod {12}}\\2&n\equiv 0{\pmod {12}}\\1&{\text{otherwise}}\end{cases}}$

Span

For a generating function $F(z)=\sum f_{n}z^{n}$ the support of $F$ is defined

$Supp(F)=\{n:f_{n}\neq 0\}$

$F$ is said to admit a span of $d$ if for some $r$

$Supp(F)\subseteq \{r,r+d,r+2d,\cdots \}$

The largest possible span is called the period.

Daffodil lemma

Applications

• regular languages (AC sec V.3)

References

• Flajolet, Philippe; Sedgewick, Robert (2009). Analytic Combinatorics (PDF). Cambridge University Press.
• Mishna, Marni (2020). Analytic Combinatorics: A Multidimensional Approach. Taylor & Francis Group, LLC.