We consider finite iterated generalized harmonic sums weighted by the binomial 2k k in numerators and denominators. A large class of these functions emerges in the calculation of massive Feynman diagrams with local operator insertions starting at 3-loop order in the coupling constant and extends the classes of the nested harmonic, generalized harmonic, and cyclotomic sums. The binomially weighted sums are associated by the Mellin transform to iterated integrals over square-root valued alphabets. The values of the sums for N → ∞ and the iterated integrals at x = 1 lead to new constants, extending the set of special numbers given by the multiple zeta values, the cyclotomic zeta values and special constants which emerge in the limit N → ∞ of generalized harmonic sums. We develop algorithms to obtain the Mellin representations of these sums in a systematic way. They are of importance for the derivation of the asymptotic expansion of these sums and their analytic continuation to N ∈ C. The associated convolution relations are derived for real parameters and can therefore be used in a wider context, as, e.g., for multi-scale processes. We also derive algorithms to transform iterated integrals over root-valued alphabets into binomial sums. Using generating functions we study a few aspects of infinite (inverse) binomial sums. C 2014 AIP Publishing LLC. [http://dx

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.

We calculate convergent 3-loop Feynman diagrams containing a single massive loop equipped with twist τ = 2 local operator insertions corresponding to spin N . They contribute to the massive operator matrix elements in QCD describing the massive Wilson coefficients for deep-inelastic scattering at large virtualities. Diagrams of this kind can be computed using an extended version to the method of hyperlogarithms, originally being designed for massless Feynman diagrams without operators. The method is applied to Benz-and V -type graphs, belonging to the genuine 3-loop topologies. In case of the Vtype graphs with five massive propagators new types of nested sums and iterated integrals emerge. The sums are given in terms of finite binomially and inverse binomially weighted generalized cyclotomic sums, while the 1-dimensionally iterated integrals are based on a set of ∼ 30 square-root valued letters. We also derive the asymptotic representations of the nested sums and present the solution for N ∈ C. Integrals with a power-like divergence in N -space ∝ a N , a ∈ R, a > 1, for large values of N emerge. They still possess a representation in x-space, which is given in terms of root-valued iterated integrals in the present case. The method of hyperlogarithms is also used to calculate higher moments for crossed box graphs with different operator insertions.

We present our most recent results on the calculation of the heavy flavor contributions to deep-inelastic scattering at 3-loop order in the large Q 2 limit, where the heavy flavor Wilson coefficients are known to factorize into light flavor Wilson coefficients and massive operator matrix elements. We describe the different techniques employed for the calculation and show the results in the case of the heavy flavor non-singlet and pure singlet contributions to the structure function F 2 (x, Q 2 ).

We calculate the massive two-loop pure singlet Wilson coefficients for heavy quark production in the unpolarized case analytically in the whole kinematic region and derive the threshold and asymptotic expansions. We also recalculate the corresponding massless twoloop Wilson coefficients. The complete expressions contain iterated integrals with elliptic letters. The contributing alphabets enlarge the Kummer-Poincaré letters by a series of square-root valued letters. A new class of iterated integrals, the Kummer-elliptic integrals, are introduced. For the structure functions F 2 and F L we also derive improved asymptotic representations adding power corrections. Numerical results are presented.

The construction of Mellin-Barnes (MB) representations for non-planar Feynman diagrams and the summation of multiple series derived from general MB representations are discussed. A basic version of a new package AMBREv.3.0 is supplemented. The ultimate goal of this project is the automatic evaluation of MB representations for multiloop scalar and tensor Feynman integrals through infinite sums, preferably with analytic solutions. We shortly describe a strategy of further algebraic summation.

We calculate the polarized massive two-loop pure singlet Wilson coefficient contributing to the structure functions g 1 (x, Q 2 ) analytically in the whole kinematic region. The Wilson coefficient contains Kummer-elliptic integrals. We derive the representation in the asymptotic region Q 2 m 2 , retaining power corrections, and in the threshold region. The massless Wilson coefficient is recalculated. The corresponding twist-2 corrections to the structure function g 2 (x, Q 2 ) are obtained by the Wandzura-Wilczek relation. Numerical results are presented. arXiv:1904.08911v1 [hep-ph] 18 Apr 2019 is presented. The recalculation of the massless Wilson coefficient is necessary, since in Ref.[13] different schemes have been used in part. The corresponding massive Wilson coefficient is calculated in Section 4. The corresponding results for the twist-2 contributions to the structure function g 2 (x, Q 2 ) can be obtained by using the Wandzura-Wilczek relation [19], as has been shown for the massless quarkonic [20,21] and gluonic [22] cases, for diffractive scattering [23], non-forward scattering [24], and the target mass corrections [25,27]. Limiting cases are studied in Section 5 and numerical results are presented in Section 6. The conclusions are given in Section 7. Some Mellin convolutions appearing due to renormalization are listed in the Appendix. The Deep-inelastic Scattering Cross SectionThe scattering cross sections for deep-inelastic charged lepton scattering of polarized nucleons 1 For other γ 5 schemes see Refs. [16].

Extreme ocean waves are characterized by the energy concentration in a few chosen waves/modes. Frequency modulation due to the nonlinear resonances is one of the possible processes yielding the appearance of independent wave clusters which keep their energy. Energetic behavior of these clusters is defined by (1) integer solutions of the resonance conditions, and (2) coupling coefficients of the dynamical system on the wave amplitudes. General computation algorithms are presented which can be used for arbitrary 3-wave resonant system. Implementation in Mathematica is given for planetary ocean waves. Short discussion concludes the paper.