- XSummer - Transcendental Functions and Symbolic Summation in Form

Harmonic sums and their generalizations are extremely useful in the evaluation of higher-order perturbative corrections in quantum field theory. Of particular interest have been the so-called nested sums,where the harmonic sums and their generalizations appear as building blocks, originating for example from the expansion of generalized hypergeometric functions around integer values of the parameters. In this Letter we discuss the implementation of several algorithms to solve these sums by algebraic means, using the computer algebra system Form.Comment: 21 pages, 1 figure, Late

Formulas for Generalized Two-Qubit Separability Probabilities

To begin, we find certain formulas $Q(k,\alpha)= G_1^k(\alpha) G_2^k(\alpha)$, for $k = -1, 0, 1,...,9$. These yield that part of the total separability probability, $P(k,\alpha)$, for generalized (real, complex, quaternionic,\ldots) two-qubit states endowed with random induced measure, for which the determinantal inequality $|\rho^{PT}| >|\rho|$ holds. Here $\rho$ denotes a $4 \times 4$ density matrix, obtained by tracing over the pure states in $4 \times (4 +k)$-dimensions, and $\rho^{PT}$, its partial transpose. Further, $\alpha$ is a Dyson-index-like parameter with $\alpha = 1$ for the standard (15-dimensional) convex set of (complex) two-qubit states. For $k=0$, we obtain the previously reported Hilbert-Schmidt formulas, with (the real case) $Q(0,\frac{1}{2}) = \frac{29}{128}$, (the standard complex case) $Q(0,1)=\frac{4}{33}$, and (the quaternionic case) $Q(0,2)= \frac{13}{323}$---the three simply equalling $P(0,\alpha)/2$. The factors $G_2^k(\alpha)$ are sums of polynomial-weighted generalized hypergeometric functions $_{p}F_{p-1}$, $p \geq 7$, all with argument $z=\frac{27}{64} =(\frac{3}{4})^3$. We find number-theoretic-based formulas for the upper ($u_{ik}$) and lower ($b_{ik}$) parameter sets of these functions and, then, equivalently express $G_2^k(\alpha)$ in terms of first-order difference equations. Applications of Zeilberger's algorithm yield "concise" forms, parallel to the one obtained previously for $P(0,\alpha) =2 Q(0,\alpha)$. For nonnegative half-integer and integer values of $\alpha$, $Q(k,\alpha)$ has descending roots starting at $k=-\alpha-1$. Then, we (C. Dunkl and I) construct a remarkably compact (hypergeometric) form for $Q(k,\alpha)$ itself. The possibility of an analogous "master" formula for $P(k,\alpha)$ is, then, investigated, and a number of interesting results found.Comment: 78 pages, 5 figures, 15 appendices, to appear in Adv. Math. Phys--verification in arXiv:1701.01973 of 8/33-two-qubit Hilbert-Schmidt separability probability conjecture note

Calculation of Massless Feynman Integrals using Harmonic Sums

A method for the evaluation of the epsilon expansion of multi-loop massless Feynman integrals is introduced. This method is based on the Gegenbauer polynomial technique and the expansion of the Gamma function in terms of harmonic sums. Algorithms for the evaluation of nested and harmonic sums are used to reduce the expressions to get analytical or numerical results for the expansion coefficients. Methods to increase the precision of numerical results are discussed.Comment: 30 pages, 6 figures; Minor typos corrected, references added. Published in Computer Physics Communication