Details

In general, a trigonometric polynomial is an element of a bivariate polynomial ring, , subject to the following constraints:

,

,

,

which identify , . For any trigonometric polynomial of degree , we can obtain a normal form by reducing powers of to write

,

or .

Integrals of the form

arise in many contexts, including classical physics. Avoiding binomial coefficients and equations of motion, here we focus on cases in which

,

,

.

These conditions allow us to introduce integers , , such that

,

,

with rational identity

.

If in the limit when , then clearly goes to simultaneously. These few easy examples are chosen to obviously satisfy the convergence criterion, as while diverges to .

This recipe leads to an infinite set of integral approximations to , which we would like to start ranking by quality. As in [1], the quality of rational approximant (with ) is determined by the equation

, ( if ).

The function behaves stochastically on the scale of small variations, but overall tends to a stable limit with some variance. We would like to find the value

,

but direct integration becomes prohibitively difficult when the degree of is large. With , integration time already reaches the 1(s) scale. Fortunately, we have a general solution to the problem of quickly computing high-order terms, which reduces iteration to a linear calculation.

Every trigonometric polynomial contributes to an integral generating function

,

with formal parameter . Period is a rational integral, so satisfies a linear differential equation [3],

,

with a matrix of integers.

The source code contains a degree-bounded algorithm TrigPicardFuchs that must return a positive result with . We omit details of the proof here, but emphasize that the top-left block of the check matrix is always nondegenerate, which guarantees solvability and halting on success for any trigonometric polynomial . The outputs can be checked against HolonomicFunctions'Annihilate [4]. In timing tests, TrigPicardFuchs competes quite well against Annihilate.

Once the differential equation is known, we can rigorously derive nonlinear recurrences,

,

usually determining the integers by validating guessed recursions against the Frobenius method (see Frobenius Method (Wolfram MathWorld)).

Finally, if we redefine as an integral over the interval , the differential equation must be modified to nonhomogeneous form,

,

where the coefficients account for the rational part of , only up to a sign and scale given by the free parameter . We call the a checksum of the integral approximation.

These calculations take us through some fun areas of mathematics, but we should not delay the data analysis. Iterating to , it seems that our approximation reaches a higher quality than , which appears in Beukers's article and in the On-Line Encyclopedia of Integer Sequences (cf. A006139 [5]).

To certainly reach , we must work with a more complicated integral,

,

,

.

This integral is Beukers's final example, and also appears in the On-Line Encyclopedia of Integer Sequences alongside the appropriate differential equations and linear recurrences (cf. A123178, A305997 and A305998 [6–8]). More work remains to be done. In particular, will a trigonometric polynomial ever lead to an approximation of ?

References

[1] F. Beukers, "A Rational Approach to Pi," Nieuw archief voor wiskunde, Serie 5, 1(4), 2000 pp. 372–379. (Jul 24, 2018) dspace.library.uu.nl/handle/1874/26398.

[2] S. B. Ekhad and D. Zeilberger, "Searching for Apery-Style Miracles [Using, Inter-Alia, the Amazing Almkvist-Zeilberger Algorithm]." arxiv.org/abs/1405.4445.

[3] P. Lairez, "Computing Periods of Rational Integrals," Mathematics of Computation, 85, 2016 pp. 1719–1752. doi:10.1090/mcom/3054.

[4] C. Koutschan. "HolonomicFunctions: A Mathematica Package for Dealing with Multivariate Holonomic Functions, Including Closure Properties, Summation, and Integration." (Jul 24, 2018) www.risc.jku.at/research/combinat/software/ergosum/RISC/HolonomicFunctions.html.

[5] N. J. A. Sloane. The On-Line Encyclopedia of Integer Sequences. oeis.org/A006139.

[6] N. J. A. Sloane. The On-Line Encyclopedia of Integer Sequences. oeis.org/A123178.

[7] N. J. A. Sloane. The On-Line Encyclopedia of Integer Sequences. oeis.org/A305997.

[8] N. J. A. Sloane. The On-Line Encyclopedia of Integer Sequences. oeis.org/A305998.