The Experts below are selected from a list of 189 Experts worldwide ranked by ideXlab platform
Oliver Schlotterer - One of the best experts on this subject based on the ideXlab platform.
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the structure of n point one loop open superstring amplitudes
Journal of High Energy Physics, 2014Co-Authors: Carlos R Mafra, Oliver SchlottererAbstract:In this article we investigate one-loop amplitudes in maximally supersymmetric superstring theory. The non-anomalous part of the worldsheet integrand is presented for any Number of massless open-string states. The polarization dependence is organized into the same BRST-invariant kinematic combinations which also govern the leading string correction to tree-level amplitudes. The dimensions of the bases for both the kinematics and the associated worldsheet integrals is found to be the unsigned Stirling Number $$ {S}_3^{n-1} $$ of first kind. We explain why the same combinatorial structures govern on the one hand finite one-loop amplitudes of equal helicity states in pure Yang-Mills theory and on the other hand the color tensors at order α′2 of the color-dressed tree amplitude.
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the structure of n point one loop open superstring amplitudes
arXiv: High Energy Physics - Theory, 2012Co-Authors: Carlos R Mafra, Oliver SchlottererAbstract:In this article we present the worldsheet integrand for one-loop amplitudes in maximally supersymmetric superstring theory involving any Number n of massless open string states. The polarization dependence is organized into the same BRST invariant kinematic combinations which also govern the leading string correction to tree level amplitudes. The dimensions of the bases for both the kinematics and the associated worldsheet integrals is found to be the unsigned Stirling Number S_3^{n-1} of first kind. We explain why the same combinatorial structures govern on the one hand finite one-loop amplitudes of equal helicity states in pure Yang Mills theory and on the other hand the color tensors at quadratic alpha prime order of the color dressed tree amplitude.
Carlos R Mafra - One of the best experts on this subject based on the ideXlab platform.
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the structure of n point one loop open superstring amplitudes
Journal of High Energy Physics, 2014Co-Authors: Carlos R Mafra, Oliver SchlottererAbstract:In this article we investigate one-loop amplitudes in maximally supersymmetric superstring theory. The non-anomalous part of the worldsheet integrand is presented for any Number of massless open-string states. The polarization dependence is organized into the same BRST-invariant kinematic combinations which also govern the leading string correction to tree-level amplitudes. The dimensions of the bases for both the kinematics and the associated worldsheet integrals is found to be the unsigned Stirling Number $$ {S}_3^{n-1} $$ of first kind. We explain why the same combinatorial structures govern on the one hand finite one-loop amplitudes of equal helicity states in pure Yang-Mills theory and on the other hand the color tensors at order α′2 of the color-dressed tree amplitude.
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the structure of n point one loop open superstring amplitudes
arXiv: High Energy Physics - Theory, 2012Co-Authors: Carlos R Mafra, Oliver SchlottererAbstract:In this article we present the worldsheet integrand for one-loop amplitudes in maximally supersymmetric superstring theory involving any Number n of massless open string states. The polarization dependence is organized into the same BRST invariant kinematic combinations which also govern the leading string correction to tree level amplitudes. The dimensions of the bases for both the kinematics and the associated worldsheet integrals is found to be the unsigned Stirling Number S_3^{n-1} of first kind. We explain why the same combinatorial structures govern on the one hand finite one-loop amplitudes of equal helicity states in pure Yang Mills theory and on the other hand the color tensors at quadratic alpha prime order of the color dressed tree amplitude.
Horst Wegner - One of the best experts on this subject based on the ideXlab platform.
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an almost accurate location of the maximum Stirling Number s of the second kind
Results in Mathematics, 2012Co-Authors: Horst WegnerAbstract:The Stirling Number of the second kind S(n, k) is the Number of ways of partitioning a set of n elements into k nonempty subsets. It is well known that the Numbers S(n, k) are unimodal in k, and there are at most two consecutive values K n such that (for fixed n) S(n, K n ) is maximal. We determine asymptotic bounds for K n , which are unexpectedly good and improve earlier results. The method used here shows a possible strategy for obtaining numerical bounds such that in almost all cases K n can be uniquely determined.
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on the location of the maximum Stirling Number s of the second kind
Results in Mathematics, 2009Co-Authors: Horst WegnerAbstract:The Stirling Number of the second kind S(n, k) is the Number of ways of partitioning a set of n elements into k nonempty subsets. It is well known that the Numbers S(n, k) are unimodal in k, and there are at most two consecutive values K n such that (for fixed n) S(n,K n ) is maximal. We determine numerical bounds for K n , and our result shows that in many cases K n can be uniquely determined.
Jacopo Daurizio - One of the best experts on this subject based on the ideXlab platform.
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on the interplay between hypergeometric series fourier legendre expansions and euler sums
Bollettino Della Unione Matematica Italiana, 2019Co-Authors: Marco Cantarini, Jacopo DaurizioAbstract:In this work we continue the investigation, started in Campbell et al. (On the interplay between hypergeometric functions, complete elliptic integrals and Fourier–Legendre series expansions, arXiv:1710.03221 , 2017), about the interplay between hypergeometric functions and Fourier–Legendre ( $$\text {FL}$$ ) series expansions. In the section “Hypergeometric series related to $$\pi ,\pi ^2$$ and the lemniscate constant”, through the FL-expansion of $$[x(1-x)]^\mu $$ (with $$\mu +1\in \frac{1}{4}{\mathbb {N}}$$ ) we prove that all the hypergeometric series $$\begin{aligned}&\sum _{n\ge 0}\frac{(-1)^n(4n+1)}{p(n)}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \right] ^3,\quad \sum _{n\ge 0}\frac{(4n+1)}{p(n)}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \right] ^4,\\&\quad \sum _{n\ge 0}\frac{(4n+1)}{p(n)^2}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \right] ^4,\; \sum _{n\ge 0}\frac{1}{p(n)}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \right] ^3,\; \sum _{n\ge 0}\frac{1}{p(n)}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \right] ^2 \end{aligned}$$ return rational multiples of $$\frac{1}{\pi },\frac{1}{\pi ^2}$$ or the lemniscate constant, as soon as p(x) is a polynomial fulfilling suitable symmetry constraints. Additionally, by computing the FL-expansions of $$\frac{\log x}{\sqrt{x}}$$ and related functions, we show that in many cases the hypergeometric $$\phantom {}_{p+1} F_{p}(\ldots , z)$$ function evaluated at $$z=\pm 1$$ can be converted into a combination of Euler sums. In particular we perform an explicit evaluation of $$\begin{aligned} \sum _{n\ge 0}\frac{1}{(2n+1)^2}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \right] ^2,\quad \sum _{n\ge 0}\frac{1}{(2n+1)^3}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \right] ^2. \end{aligned}$$ In the section “Twisted hypergeometric series” we show that the conversion of some $$\phantom {}_{p+1} F_{p}(\ldots ,\pm 1)$$ values into combinations of Euler sums, driven by FL-expansions, applies equally well to some twisted hypergeometric series, i.e. series of the form $$\sum _{n\ge 0} a_n b_n$$ where $$a_n$$ is a Stirling Number of the first kind and $$\sum _{n\ge 0}b_n z^n = \phantom {}_{p+1} F_{p}(\ldots ;z)$$ .
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on the interplay between hypergeometric series fourier legendre expansions and euler sums
arXiv: Number Theory, 2018Co-Authors: Marco Cantarini, Jacopo DaurizioAbstract:In this work we continue the investigation about the interplay between hypergeometric functions and Fourier-Legendre ($\textrm{FL}$) series expansions. In the section "Hypergeometric series related to $\pi,\pi^2$ and the lemniscate constant", through the FL-expansion of $\left[x(1-x)\right]^\mu$ (with $\mu+1\in\frac{1}{4}\mathbb{N}$) we prove that all the hypergeometric series $$ \sum_{n\geq 0}\frac{(-1)^n(4n+1)}{p(n)}\left[\frac{1}{4^n}\binom{2n}{n}\right]^3,\quad \sum_{n\geq 0}\frac{(4n+1)}{p(n)}\left[\frac{1}{4^n}\binom{2n}{n}\right]^4,$$ $$\quad \sum_{n\geq 0}\frac{(4n+1)}{p(n)^2}\left[\frac{1}{4^n}\binom{2n}{n}\right]^4,\; \sum_{n\geq 0}\frac{1}{p(n)}\left[\frac{1}{4^n}\binom{2n}{n}\right]^3,\; \sum_{n\geq 0}\frac{1}{p(n)}\left[\frac{1}{4^n}\binom{2n}{n}\right]^2 $$ return rational multiples of $\frac{1}{\pi},\frac{1}{\pi^2}$ or the lemniscate constant, as soon as $p(x)$ is a polynomial fulfilling suitable symmetry constraints. Additionally, by computing the FL-expansions of $\frac{\log x}{\sqrt{x}}$ and related functions, we show that in many cases the hypergeometric $\phantom{}_{p+1} F_{p}(\ldots , z)$ function evaluated at $z=\pm 1$ can be converted into a combination of Euler sums. In particular we perform an explicit evaluation of $$ \sum_{n\geq 0}\frac{1}{(2n+1)^2}\left[\frac{1}{4^n}\binom{2n}{n}\right]^2,\quad \sum_{n\geq 0}\frac{1}{(2n+1)^3}\left[\frac{1}{4^n}\binom{2n}{n}\right]^2. $$ In the section "Twisted hypergeometric series" we show that the conversion of some $\phantom{}_{p+1} F_{p}(\ldots,\pm 1)$ values into combinations of Euler sums, driven by FL-expansions, applies equally well to some twisted hypergeometric series, i.e. series of the form $\sum_{n\geq 0} a_n b_n$ where $a_n$ is a Stirling Number of the first kind and $\sum_{n\geq 0}b_n z^n = \phantom{}_{p+1} F_{p}(\ldots;z)$.
Marco Cantarini - One of the best experts on this subject based on the ideXlab platform.
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on the interplay between hypergeometric series fourier legendre expansions and euler sums
Bollettino Della Unione Matematica Italiana, 2019Co-Authors: Marco Cantarini, Jacopo DaurizioAbstract:In this work we continue the investigation, started in Campbell et al. (On the interplay between hypergeometric functions, complete elliptic integrals and Fourier–Legendre series expansions, arXiv:1710.03221 , 2017), about the interplay between hypergeometric functions and Fourier–Legendre ( $$\text {FL}$$ ) series expansions. In the section “Hypergeometric series related to $$\pi ,\pi ^2$$ and the lemniscate constant”, through the FL-expansion of $$[x(1-x)]^\mu $$ (with $$\mu +1\in \frac{1}{4}{\mathbb {N}}$$ ) we prove that all the hypergeometric series $$\begin{aligned}&\sum _{n\ge 0}\frac{(-1)^n(4n+1)}{p(n)}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \right] ^3,\quad \sum _{n\ge 0}\frac{(4n+1)}{p(n)}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \right] ^4,\\&\quad \sum _{n\ge 0}\frac{(4n+1)}{p(n)^2}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \right] ^4,\; \sum _{n\ge 0}\frac{1}{p(n)}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \right] ^3,\; \sum _{n\ge 0}\frac{1}{p(n)}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \right] ^2 \end{aligned}$$ return rational multiples of $$\frac{1}{\pi },\frac{1}{\pi ^2}$$ or the lemniscate constant, as soon as p(x) is a polynomial fulfilling suitable symmetry constraints. Additionally, by computing the FL-expansions of $$\frac{\log x}{\sqrt{x}}$$ and related functions, we show that in many cases the hypergeometric $$\phantom {}_{p+1} F_{p}(\ldots , z)$$ function evaluated at $$z=\pm 1$$ can be converted into a combination of Euler sums. In particular we perform an explicit evaluation of $$\begin{aligned} \sum _{n\ge 0}\frac{1}{(2n+1)^2}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \right] ^2,\quad \sum _{n\ge 0}\frac{1}{(2n+1)^3}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \right] ^2. \end{aligned}$$ In the section “Twisted hypergeometric series” we show that the conversion of some $$\phantom {}_{p+1} F_{p}(\ldots ,\pm 1)$$ values into combinations of Euler sums, driven by FL-expansions, applies equally well to some twisted hypergeometric series, i.e. series of the form $$\sum _{n\ge 0} a_n b_n$$ where $$a_n$$ is a Stirling Number of the first kind and $$\sum _{n\ge 0}b_n z^n = \phantom {}_{p+1} F_{p}(\ldots ;z)$$ .
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on the interplay between hypergeometric series fourier legendre expansions and euler sums
arXiv: Number Theory, 2018Co-Authors: Marco Cantarini, Jacopo DaurizioAbstract:In this work we continue the investigation about the interplay between hypergeometric functions and Fourier-Legendre ($\textrm{FL}$) series expansions. In the section "Hypergeometric series related to $\pi,\pi^2$ and the lemniscate constant", through the FL-expansion of $\left[x(1-x)\right]^\mu$ (with $\mu+1\in\frac{1}{4}\mathbb{N}$) we prove that all the hypergeometric series $$ \sum_{n\geq 0}\frac{(-1)^n(4n+1)}{p(n)}\left[\frac{1}{4^n}\binom{2n}{n}\right]^3,\quad \sum_{n\geq 0}\frac{(4n+1)}{p(n)}\left[\frac{1}{4^n}\binom{2n}{n}\right]^4,$$ $$\quad \sum_{n\geq 0}\frac{(4n+1)}{p(n)^2}\left[\frac{1}{4^n}\binom{2n}{n}\right]^4,\; \sum_{n\geq 0}\frac{1}{p(n)}\left[\frac{1}{4^n}\binom{2n}{n}\right]^3,\; \sum_{n\geq 0}\frac{1}{p(n)}\left[\frac{1}{4^n}\binom{2n}{n}\right]^2 $$ return rational multiples of $\frac{1}{\pi},\frac{1}{\pi^2}$ or the lemniscate constant, as soon as $p(x)$ is a polynomial fulfilling suitable symmetry constraints. Additionally, by computing the FL-expansions of $\frac{\log x}{\sqrt{x}}$ and related functions, we show that in many cases the hypergeometric $\phantom{}_{p+1} F_{p}(\ldots , z)$ function evaluated at $z=\pm 1$ can be converted into a combination of Euler sums. In particular we perform an explicit evaluation of $$ \sum_{n\geq 0}\frac{1}{(2n+1)^2}\left[\frac{1}{4^n}\binom{2n}{n}\right]^2,\quad \sum_{n\geq 0}\frac{1}{(2n+1)^3}\left[\frac{1}{4^n}\binom{2n}{n}\right]^2. $$ In the section "Twisted hypergeometric series" we show that the conversion of some $\phantom{}_{p+1} F_{p}(\ldots,\pm 1)$ values into combinations of Euler sums, driven by FL-expansions, applies equally well to some twisted hypergeometric series, i.e. series of the form $\sum_{n\geq 0} a_n b_n$ where $a_n$ is a Stirling Number of the first kind and $\sum_{n\geq 0}b_n z^n = \phantom{}_{p+1} F_{p}(\ldots;z)$.