[Paper Review] Special functions, transcendentals and their numerics

Jakob Ablinger, J. Blümlein+2 more|arXiv (Cornell University)|Jan 1, 2017

Advanced Mathematical Identities24 references5 citations

TL;DR

This paper investigates cyclotomic polylogarithm constants evaluated at unity up to weight 2 for cyclotomies k ≤ 12, demonstrating that all PSLQ-identified relations for k ≤ 6 are analytically derivable from known special function identities, including dilogarithm functional equations and Clausen function identities. For cyclotomy 12, new relations beyond prior literature are found and analytically proven using Ramanujan's identities and Kummer's formula for complex dilogarithms.

ABSTRACT

Cyclotomic polylogarithms are reviewed and new results concerning the special constants that occur are presented. This also allows some comments on previous literature results using PSLQ.

Motivation & Objective

• To analyze special constants arising from numerical implementations of cyclotomic polylogarithms in particle physics.
• To determine whether PSLQ searches yield new relations among cyclotomic constants beyond known mathematical identities.
• To extend the understanding of special values of generalized polylogarithms, particularly at x=1, for higher cyclotomies.
• To establish a complete basis of constants for efficient numerical evaluation in high-precision amplitude calculations.
• To investigate whether cyclotomic polylogarithms of order 12 introduce new algebraic or transcendental relations not present in lower cyclotomies.

Proposed method

• Utilized iterated integral definitions of cyclotomic polylogarithms with indices (k,l) derived from cyclotomic polynomials Φk(x).
• Applied variable transformations such as x = (1−t)/(1+t) to improve series convergence for x near 1, enabling stable numerical evaluation.
• Employed the PSLQ algorithm with high-precision arithmetic (thousands of digits) to detect potential linear relations among constants.
• Used known functional equations for the dilogarithm, including Kummer’s formula for complex arguments and Ramanujan’s identities.
• Leveraged symbolic computation tools (HarmonicSums and Sigma) to eliminate known shuffle, stuffle, and integration-by-parts identities.
• Mapped cyclotomic polylogarithms to generalized polylogarithms via factorization, enabling use of established special function identities.

Experimental results

Research questions

• RQ1 Are all PSLQ-identified linear relations among cyclotomic polylogarithm constants at x=1 for k ≤ 6 already derivable from known mathematical identities?
• RQ2 Do cyclotomic polylogarithms of order 12 introduce new algebraic or transcendental relations not present in lower cyclotomies?
• RQ3 Can the special values of dilogarithms with complex arguments be fully reduced using known functional equations?
• RQ4 What is the minimal set of basis constants required to express all weight-2 cyclotomic polylogarithm constants at x=1?
• RQ5 To what extent do PSLQ searches provide new information beyond what is already known from analytic number theory?

Key findings

• All PSLQ-identified relations among cyclotomic constants at x=1 for cyclotomies k ≤ 6 are analytically provable using known identities, including Ramanujan's dilogarithm identities and Kummer's formula.
• For cyclotomy 12, new relations were discovered that are not derivable from previous literature, indicating the presence of previously unexplored algebraic structures.
• The constants π, log 2, log 3, log(√3−1), log(2−√3), Cl₂(π/3), Cl₂(π/6), and various real parts of Li₂ with complex arguments form a complete basis for weight-2 constants at x=1.
• The use of the transformation x = (1−t)/(1+t) enables fast-converging series expansions for x near 1, crucial for numerical stability in high-precision amplitude calculations.
• Ramanujan's identities were essential in proving the irreducibility of certain combinations of dilogarithm values, such as Li₂(−1/3) and Li₂(1/9).
• The study confirms that the symbol map captures all known relations among generalized polylogarithms, and no new relations were found beyond those already known for k ≤ 6.