Generalized Log-sine integrals and Bell polynomials

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Abstract

In this paper, we investigate the integral of \(x^n\log^m(\sin(x))\) for natural numbers \(m\) and \(n\). In doing so, we recover some well-known results and remark on some relations to the log-sine integral \(\operatorname{Ls}_{n+m+1}^{(n)}(\theta)\). Later, we use properties of Bell polynomials to find a closed expression for the derivative of the central binomial and shifted central binomial coefficients in terms of polygamma functions and harmonic numbers.

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Massive Feynman diagrams and inverse binomial sums

, (2004)

When calculating higher terms of the epsilon-expansion of massive Feynman diagrams, one needs to evaluate particular cases of multiple inverse binomial sums. These sums are related to the derivatives of certain hypergeometric functions with respect to their parameters. Exploring this connection and using it together with an approach based on generating functions, we analytically calculate a number of such infinite sums, for an arbitrary value of the argument which corresponds to an arbitrary value of the off-shell external momentum. In such a way, we find a number of new results for physically important Feynman diagrams. Considered examples include two-loop two- and three-point diagrams, as well as three-loop vacuum diagrams with two different masses. The results are presented in terms of generalized polylogarithmic functions. As a physical example, higher-order terms of the epsilon-expansion of the polarization function of the neutral gauge bosons are constructed.