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26
A decomposition of Riemann's zeta function
, 1997
"... proof, `Although this proof is not very long, it seems too complicated compared with the elegance of the statement. It would be nice to find a more natural proof': Unfortunately much the same can be said of the proof that I have presented here. Markett [7] and J. Borwein and Girgensohn [3] were ..."
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proof, `Although this proof is not very long, it seems too complicated compared with the elegance of the statement. It would be nice to find a more natural proof': Unfortunately much the same can be said of the proof that I have presented here. Markett [7] and J. Borwein and Girgensohn [3] were able to evaluate i(p 1 ; p 2 ; p 3 ) in terms of values of i(p) whenever p 1 + p 2 + p 3 6, and in terms of i(p) and i(a; b) whenever p 1 + p 2 + p 3 10  it would be interesting to know whether such `descents' are always possible or, as most researchers seem to believe, that there is only a small class of such sums that can be so evaluated. Proof of (1).
Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms
, 2013
"... In recent three–loop calculations of massive Feynman integrals within Quantum Chromodynamics (QCD) and, e.g., in recent combinatorial problems the socalled generalized harmonic sums (in short Ssums) arise. They are characterized by rational (or real) numerator weights also different from ±1. In th ..."
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In recent three–loop calculations of massive Feynman integrals within Quantum Chromodynamics (QCD) and, e.g., in recent combinatorial problems the socalled generalized harmonic sums (in short Ssums) arise. They are characterized by rational (or real) numerator weights also different from ±1. In this article we explore the algorithmic and analytic properties of these sums systematically. We work out the Mellin and inverse Mellin transform which connects the sums under consideration with the associated Poincaré iterated integrals, also called generalized harmonic polylogarithms. In this regard, we obtain explicit analytic continuations by means of asymptotic expansions of the Ssums which started to occur frequently in current QCD calculations. In addition, we derive algebraic and structural relations, like differentiation w.r.t. the external summation index and different multiargument relations, for the compactification of Ssum expressions. Finally, we calculate algebraic relations for infinite Ssums, or equivalently for generalized harmonic polylogarithms evaluated at special values. The corresponding algorithms and relations are encoded in the computer algebra package HarmonicSums.
2007a). Skewness for maximum likelihood estimators of the negative binomial distribution, Far East Journal of Theoretical Statistics
"... The probability generating function of one version of the negative binomial distribution being (p + 1 pt)k, we study elements of the Hessian and in particular Fisher's discovery of a series form for the variance of k̂, the maximum likelihood estimator, and also for the determinant of the Hessi ..."
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The probability generating function of one version of the negative binomial distribution being (p + 1 pt)k, we study elements of the Hessian and in particular Fisher's discovery of a series form for the variance of k̂, the maximum likelihood estimator, and also for the determinant of the Hessian. There is a link with the Psi function and its derivatives. Basic algebra is excessively complicated and a Maple code implementation is an important task in the solution process. Low order maximum likelihood moments are given and also Fisher's examples relating to data associated with ticks on sheep. Eciency of moment estimators is mentioned, including the concept of joint eciency. In an Addendum we give an interesting formula for the dierence of two Psi functions. 2000 Mathematics Subject Classication: 62E20. Key words and phrases: asymptotic skewness, asymptotic variance, Hessian, iterative schemes,
ON A RECIPROCITY LAW FOR FINITE MULTIPLE ZETA VALUES MARKUS KUBA AND HELMUT PRODINGER
, 905
"... Abstract. It was shown in [7, 9] that harmonic numbers satisfy certain reciprocity relations, which are in particular useful for the analysis of the quickselect algorithm. The aim of this work is to show that a reciprocity relation from [7, 9] can be generalized to finite variants of multiple zeta v ..."
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Abstract. It was shown in [7, 9] that harmonic numbers satisfy certain reciprocity relations, which are in particular useful for the analysis of the quickselect algorithm. The aim of this work is to show that a reciprocity relation from [7, 9] can be generalized to finite variants of multiple zeta values, involving a finite variant of the shuffle identity for multiple zeta values. We present the generalized reciprocity relation and furthermore a simple elementary proof of the shuffle identity using only partial fraction decomposition. We also present an extension of the reciprocity relation to weighted sums.
Quantization of the Optical Phase Space S 2 = {ϕ mod 2π, I> 0} in Terms of the Group SO ↑ (1, 2)
, 2003
"... The problem of quantizing properly the canonical pair “angle and action variables ”, ϕ and I, is almost as old as quantum mechanics itself and since decades an intensively debated but still unresolved issue in quantum optics. The present paper proposes a new approach to the problem, namely quantizat ..."
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The problem of quantizing properly the canonical pair “angle and action variables ”, ϕ and I, is almost as old as quantum mechanics itself and since decades an intensively debated but still unresolved issue in quantum optics. The present paper proposes a new approach to the problem, namely quantization in terms of the group SO(1, 2): The crucial point is that the phase space S 2 = {ϕ mod 2π, I> 0} has the global structure S 1 × R + (a simple cone) and cannot be quantized in the conventional manner. As the group SO(1, 2) acts transitively, effectively and Hamiltonlike on that space its irreducible unitary representations of the positive discrete series provide the appropriate quantum theoretical framework. The phase space S 2 has the conic structure of an orbifold R 2 /Z2. That structure is closely related to a Z2 gauge symmetry which corresponds to the center of a 2fold covering of SO(1, 2), the symplectic group Sp(2, R). The basic variables on the phase space are the 1
GENERATORS FOR VECTOR SPACES CONSISTING OF DOUBLE ZETA VALUES WITH EVEN WEIGHT
, 2008
"... Let DZk be the Qvector space consisting of double zeta values with weight k, and DMk be its quotient space divided by the space consisting of the zeta value ζ(k) and two products of zeta values with total weight k. When k is even, an upper bound for the dimension of DMk, which gives that of DZk, i ..."
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Let DZk be the Qvector space consisting of double zeta values with weight k, and DMk be its quotient space divided by the space consisting of the zeta value ζ(k) and two products of zeta values with total weight k. When k is even, an upper bound for the dimension of DMk, which gives that of DZk, is known. In this note, we obtain some sets of specific generators for DMk which represent the upper bound. These yield the corresponding sets and the upper bound for DZk.
unknown title
, 1995
"... The function y = Φα(x), the solution of y α e y = x for x and y large enough, has a series expansion in terms of lnx and lnlnx, with coefficients given in terms of Stirling cycle numbers. It is shown that this expansion converges for x> (αe) α for α ≥ 1. It is also shown that new expansions can b ..."
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The function y = Φα(x), the solution of y α e y = x for x and y large enough, has a series expansion in terms of lnx and lnlnx, with coefficients given in terms of Stirling cycle numbers. It is shown that this expansion converges for x> (αe) α for α ≥ 1. It is also shown that new expansions can be obtained for Φα in terms of associated Stirling numbers. The new expansions converge more rapidly and on a larger domain.
arXiv:1105.6063 [mathph] Harmonic Sums and Polylogarithms Generated by Cyclotomic Polynomials
, 2011
"... The computation of Feynman integrals in massive higher order perturbative calculations in renormalizable Quantum Field Theories requires extensions of multiply nested harmonic sums, which can be generated as real representations by Mellin transforms of Poincaré– iterated integrals including denomina ..."
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The computation of Feynman integrals in massive higher order perturbative calculations in renormalizable Quantum Field Theories requires extensions of multiply nested harmonic sums, which can be generated as real representations by Mellin transforms of Poincaré– iterated integrals including denominators of higher cyclotomic polynomials. We derive the cyclotomic harmonic polylogarithms and harmonic sums and study their algebraic and structural relations. The analytic continuation of cyclotomic harmonic sums to complex values of N is performed using analytic representations. We also consider special values of the cyclotomic harmonic polylogarithms at argument x = 1, resp., for the cyclotomic harmonic sums at N → ∞, which are related to colored multiple zeta values, deriving various of their relations, based on the stuffle and shuffle algebras and three multiple argument relations. We also consider infinite generalized nested harmonic sums at roots of unity which are related to the infinite cyclotomic harmonic sums. Basis representations are derived for weight w = 1,2 sums up to cyclotomy l = 20.
ON A RECIPROCITY LAW FOR FINITE MULTIPLE ZETA VALUES
"... ABSTRACT. It was shown in [11, 13] that harmonic numbers satisfy certain reciprocity relations, which are in particular useful for the analysis of the quickselect algorithm. The aim of this work is to show that a reciprocity relation from [11, 13] can be generalized to finite variants of multiple ze ..."
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ABSTRACT. It was shown in [11, 13] that harmonic numbers satisfy certain reciprocity relations, which are in particular useful for the analysis of the quickselect algorithm. The aim of this work is to show that a reciprocity relation from [11, 13] can be generalized to finite variants of multiple zeta values, involving a finite variant of the shuffle identity for multiple zeta values. We present the generalized reciprocity relation and furthermore a combinatorial proof of the shuffle identity based on partial fraction decomposition. We also present an extension of the reciprocity relation to weighted sums.
Mellin Moments of the Nexttonextto Leading Order Coefficient Functions for the DrellYan Process
, 2005
"... We calculate the Mellin moments of the nexttonextto leading order coefficient functions for the Drell–Yan and Higgs production cross sections. The results can be expressed in terms of multiple finite harmonic sums of maximal weight w = 4. Using algebraic and structural relations between harmonic ..."
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We calculate the Mellin moments of the nexttonextto leading order coefficient functions for the Drell–Yan and Higgs production cross sections. The results can be expressed in terms of multiple finite harmonic sums of maximal weight w = 4. Using algebraic and structural relations between harmonic sums one finds that besides the single harmonic sums only five basic sums and their derivatives w.r.t. the summation index contribute. This representation reduces the large complexity being present in x–space calculations and The Principle of Simplicity 1 is one of the guiding principles in physics [1]. Whenever possible one seeks for as simple as possible expressions, not only to obtain the result in a more compact form, but also to reveal the basic structures behind. This applies also to complex computations in particle physics. Without achieving suitable simplifications it is often impossible to undertake