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22
Two notes on notation
 American Mathematical Monthly
, 1992
"... Mathematical notation evolves like all languages do. As new experiments are made, we sometimes witness the survival of the fittest, sometimes the survival of the most familiar. A healthy conservatism keeps things from changing too rapidly; a healthy radicalism keeps things in tune with new theoretic ..."
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Mathematical notation evolves like all languages do. As new experiments are made, we sometimes witness the survival of the fittest, sometimes the survival of the most familiar. A healthy conservatism keeps things from changing too rapidly; a healthy radicalism keeps things in tune with new theoretical emphases. Our mathematical language continues to improve, just as “the dism of Leibniz overtook the dotage of Newton ” in past centuries [4, Chapter 4]. In 1970 I began teaching a class at Stanford University entitled Concrete Mathematics. The students and I studied how to manipulate formulas in continuous and discrete mathematics, and the problems we investigated were often inspired by new developments in computer science. As the years went by we began to see that a few changes in notational traditions would greatly facilitate our work. The notes from that class have recently been published in a book [15], and as I wrote the final drafts of that book I learned to my surprise that two of the notations we had been using were considerably more useful than I had previously realized. The ideas “clicked ” so well, in fact, that I’ve decided to write this article, blatantly attempting to promote these notations among the mathematicians who have no use for [15]. I hope that within five years everybody will be able to use these notations in published papers without needing to explain what they mean.
Euler Sums and Contour Integral Representations
, 1998
"... This paper develops an approach to the evaluation of Euler sums that involve harmonic numbers, either linearly or nonlinearly. We give explicit formulæ for several classes of Euler sums in terms of Riemann zeta values. The approach is based on simple contour integral representations and residue comp ..."
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Cited by 28 (1 self)
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This paper develops an approach to the evaluation of Euler sums that involve harmonic numbers, either linearly or nonlinearly. We give explicit formulæ for several classes of Euler sums in terms of Riemann zeta values. The approach is based on simple contour integral representations and residue computations.
Harmonic Sums and Mellin Transforms up to twoloop Order, Phys. Rev. D60
, 1999
"... A systematic study is performed on the finite harmonic sums up to level four. These sums form the general basis for the Mellin transforms of all individual functions fi(x) of the momentum fraction x emerging in the quantities of massless QED and QCD up to two–loop order, as the unpolarized and polar ..."
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Cited by 23 (4 self)
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A systematic study is performed on the finite harmonic sums up to level four. These sums form the general basis for the Mellin transforms of all individual functions fi(x) of the momentum fraction x emerging in the quantities of massless QED and QCD up to two–loop order, as the unpolarized and polarized splitting functions, coefficient functions, and hard scattering cross sections for space and timelike momentum transfer. The finite harmonic sums are calculated explicitly in the linear representation. Algebraic relations connecting these sums are derived to obtain representations based on a reduced set of basic functions. The Mellin–transforms of all the corresponding Nielsen functions are calculated. For the study of the scaling violations of deepinelastic scattering structure functions and other hard scattering processes different techniques were developed [1, 2]. Due to mass factorization the processes can in general be described by a Mellinconvolution of the parton densities and coefficient functions or the hard scattering cross sections in the CM subsystem, respectively.
The Incomplete Gamma Functions Since Tricomi
 In Tricomi's Ideas and Contemporary Applied Mathematics, Atti dei Convegni Lincei, n. 147, Accademia Nazionale dei Lincei
, 1998
"... The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asy ..."
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Cited by 15 (1 self)
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The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asymptotic expansions, zeros, inequalities, computational methods, and applications.
A result in order statistics related to probabilistic counting
 Computing
, 1993
"... Abstract Zusammenfassung ..."
Algebraic aspects of multiple zeta values
 in ”Zeta Functions, Topology and Quantum Physics
, 2005
"... Multiple zeta values have been studied by a wide variety of methods. In this article we summarize some of the results about them that can be obtained by an algebraic approach. This involves “coding ” the multiple zeta values by monomials in two noncommuting variables x and y. Multiple zeta values ca ..."
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Multiple zeta values have been studied by a wide variety of methods. In this article we summarize some of the results about them that can be obtained by an algebraic approach. This involves “coding ” the multiple zeta values by monomials in two noncommuting variables x and y. Multiple zeta values can then be thought of as defining a map ζ: H 0 → R from a graded rational vector space H 0 generated by the “admissible words ” of the noncommutative polynomial algebra Q〈x,y〉. Now H 0 admits two (commutative) products making ζ a homomorphism–the shuffle product and the “harmonic ” product. The latter makes H 0 a subalgebra of the algebra QSym of quasisymmetric functions. We also discuss some results about multiple zeta values that can be stated in terms of derivations and cyclic derivations of Q〈x,y〉, and we define an action of QSym on Q〈x,y 〉 that appears useful. Finally, we apply the algebraic approach to relations of finite partial sums of multiple zeta value series. 1
A completely monotone function related to the Gamma function
 J. Comp. Appl. Math
, 1999
"... Abstract We show that the reciprocal of the function f (z) = log \Gamma (z + 1) z log z; z 2 C n] \Gamma 1; 0]; ..."
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Abstract We show that the reciprocal of the function f (z) = log \Gamma (z + 1) z log z; z 2 C n] \Gamma 1; 0];
Structural relations of harmonic sums and Mellin transforms at weight w=6, arXiv:0901.0837 [mathph
"... We derive the structural relations between the Mellin transforms of weighted Nielsen integrals emerging in the calculation of massless or massive single–scale quantities in QED and QCD, such as anomalous dimensions and Wilson coefficients, and other hard scattering cross sections depending on a sing ..."
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Cited by 8 (3 self)
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We derive the structural relations between the Mellin transforms of weighted Nielsen integrals emerging in the calculation of massless or massive single–scale quantities in QED and QCD, such as anomalous dimensions and Wilson coefficients, and other hard scattering cross sections depending on a single scale. The set of all multiple harmonic sums up to weight five cover the sums needed in the calculation of the 3–loop anomalous dimensions. The relations extend the set resulting from the quasishuffle product between harmonic sums studied earlier. Unlike the shuffle relations, they depend on the value of the quantities considered. Up to weight w = 5, 242 nested harmonic sums contribute. In the present physical applications it is sufficient to consider the subset of harmonic sums not containing an index i = −1, which consists out of 69 sums. The algebraic relations reduce this set to 30 sums. Due to the structural relations a final reduction of the number of harmonic sums to 15 basic functions is obtained. These functions can be represented in terms of factorial series, supplemented by harmonic sums which are algebraically reducible. Complete analytic representations are given for these 15 meromorphic functions in the complex plane
Generalized reciprocity laws for sums of harmonic numbers
, 2005
"... We present summation identities for generalized harmonic numbers, which generalize reciprocity laws discovered when studying the algorithm quickselect. Furthermore, we demonstrate how the computer algebra package Sigma can be used in order to find/prove such identities. We also discuss alternating h ..."
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Cited by 4 (2 self)
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We present summation identities for generalized harmonic numbers, which generalize reciprocity laws discovered when studying the algorithm quickselect. Furthermore, we demonstrate how the computer algebra package Sigma can be used in order to find/prove such identities. We also discuss alternating harmonic sums, as well as limiting relations. 1.
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 able ..."
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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).