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Derivative polynomials and infinite series for squigonometric functions

Bart S. van Lith

TL;DR

The paper addresses computing derivatives and power-series coefficients for squigonometric functions, defined as generalizations of circles through $|x|^p+|y|^p=1$, via a recursive framework of derivative polynomials. It develops a systematic recursion for the derivative polynomials $Q_k(u)$, introduces coefficient triangles, and proves real-rootedness and interlacing of these polynomials, enabling algebraic evaluation of squigonometric functions at derivative-vanishing points. The authors then derive explicit MacLaurin and Taylor coefficients, present stable numerical strategies including continued fractions and in-place Horner-like evaluation, and provide practical algorithms for computing $\pi_p$ and Beta functions at rational arguments. These results yield an efficient, accurate approach to evaluating squigonometric functions on the real line and offer new connections to combinatorial coefficient structures and special constants.

Abstract

All squigonometric functions admit derivatives that can be expressed as polynomials of the squine and cosquine. We introduce a general framework that allows us to determine these polynomials recursively. We also provide an explicit formula for all coefficients of these polynomials. This also allows us to provide an explicit expression for the MacLaurin series coefficients of all squigonometric functions. We further discuss some methods that can compute the squigonometric functions up to any given tolerance over all of the real line.

Derivative polynomials and infinite series for squigonometric functions

TL;DR

The paper addresses computing derivatives and power-series coefficients for squigonometric functions, defined as generalizations of circles through , via a recursive framework of derivative polynomials. It develops a systematic recursion for the derivative polynomials , introduces coefficient triangles, and proves real-rootedness and interlacing of these polynomials, enabling algebraic evaluation of squigonometric functions at derivative-vanishing points. The authors then derive explicit MacLaurin and Taylor coefficients, present stable numerical strategies including continued fractions and in-place Horner-like evaluation, and provide practical algorithms for computing and Beta functions at rational arguments. These results yield an efficient, accurate approach to evaluating squigonometric functions on the real line and offer new connections to combinatorial coefficient structures and special constants.

Abstract

All squigonometric functions admit derivatives that can be expressed as polynomials of the squine and cosquine. We introduce a general framework that allows us to determine these polynomials recursively. We also provide an explicit formula for all coefficients of these polynomials. This also allows us to provide an explicit expression for the MacLaurin series coefficients of all squigonometric functions. We further discuss some methods that can compute the squigonometric functions up to any given tolerance over all of the real line.

Paper Structure

This paper contains 22 sections, 12 theorems, 83 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction: squigonometry
  2. Derivative polynomials
  3. A family of number triangles
  4. Algebraic values of squigonometric functions
  5. Power series and continued fractions
  6. MacLaurin series
  7. Taylor series around other points
  8. Continued fractions and numerical stability
  9. Explicit formulae for MacLaurin coefficients
  10. Factor sequences
  11. Rethinking the factorial
  12. Bounds on the number of nonzero factor sequences
  13. Computing squigonometric functions
  14. Choice of interval
  15. Single polynomial or piecewise?
  16. ...and 7 more sections

Key Result

Theorem 2.1

For any $n,m \in \mathbb{Z}$, $p \geq 2$ an integer and $k\geq 0$ an integer, we have where $Q_k(u)$ is a polynomial that satisfies the recursion together with the initial condition $Q_0(u) = 1$.

Figures (3)

  • Figure 1: Several squircles for $p=1$ (dotted), $p=2$ (solid), $p=4$ (dashed), and $p=20$ (dash-dot).
  • Figure 2: Squine (solid) and cosquine (dashed) for $p=4$.
  • Figure 3: Plot of $\ln|q^{(k)}_j|$ for $n=0$, $m=1$ and $p=4$. The red line indicates the location of the MacLaurin coefficients of the cosquine, and the yellow line indicates the location of the squine MacLaurin coefficients.

Theorems & Definitions (23)

  • Theorem 2.1
  • Remark 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • ...and 13 more