Recent Trends in Combinatorics

We calculate 3-loop master integrals for heavy quark correlators and the 3-loop QCD corrections to the ρ-parameter. They obey non-factorizing differential equations of second order with more than three singularities, which cannot be factorized in Mellin-N space either. The solution of the homogeneous equations is possible in terms of convergent close integer power series as 2 F 1 Gauß hypergeometric functions at rational argument. In some cases, integrals of this type can be mapped to complete elliptic integrals at rational argument. This class of functions appears to be the next one arising in the calculation of more complicated Feynman integrals following the harmonic polylogarithms, generalized polylogarithms, cyclotomic harmonic polylogarithms, square-root valued iterated integrals, and combinations thereof, which appear in simpler cases. The inhomogeneous solution of the corresponding differential equations can be given in terms of iterative integrals, where the new innermost letter itself is not an iterative integral. A new class of iterative integrals is introduced containing letters in which (multiple) definite integrals appear as factors. For the elliptic case, we also derive the solution in terms of integrals over modular functions and also modular forms, using q-product and series representations implied by Jacobi's ϑ i functions and Dedekind's η-function. The corresponding representations can be traced back to polynomials out of Lambert-Eisenstein series, having representations also as elliptic polylogarithms, a q-factorial 1/η k (τ ), logarithms and polylogarithms of q and their q-integrals. Due to the specific form of the physical variable x(q) for different processes, different representations do usually appear. Numerical results are also presented.

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Iterated elliptic and hypergeometric integrals for Feynman diagrams

Ablinger

Blümlein

Freitas

et al. 2018

Journal of Mathematical Physics

118

117

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Families of Ramanujan-Type Congruences Modulo 4 for the Number of Divisors

Merca

2022

Axioms

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From Permutations to Horizontal Visibility Patterns of Periodic Series

Muñoz

Nuño

2021

The 7th International Conference on Time Series and Forecasting

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Periodic series of period T can be mapped into the set of permutations of [T−1]={1,2,3,…,T−1}. These permutations of period T can be classified according to the relative ordering of their elements by the horizontal visibility map. We prove that the number of horizontal visibility classes for each period T coincides with the number of triangulations of the polygon of T+1 vertices that, as is well known, is the Catalan number CT−1. We also study the robustness against Gaussian noise of the permutation patterns for each period and show that there are periodic permutations that better resist the increase of the variance of the noise.

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Recent Trends in Combinatorics

Cited by 3 publications

References 15 publications

Iterated elliptic and hypergeometric integrals for Feynman diagrams

Iterated elliptic and hypergeometric integrals for Feynman diagrams

Families of Ramanujan-Type Congruences Modulo 4 for the Number of Divisors

From Permutations to Horizontal Visibility Patterns of Periodic Series

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