1. Journal of … The multiple polylogarithm is the most generic member of a family of functions, to which others like the harmonic polylogarithm, Nielsen's generalized polylogarithm and the multiple zeta value belong. The coefficients of their expansions and their Mellin transforms are harmonic sums. Furthermore, we relate sums S to Nielsen’s polylogarithm. PolyLog [ n, z] gives the polylogarithm function. The polylogarithm of Negative Integer order arises in sums of the form. The polylogarithm … A r´esum´e of the earliest articles that consider the integral defining this function, from the late seventeenth century to the early nineteenth century, is presented. Harmonic sums of the variable are associated to Mellin transforms of weighted harmonic polylogarithms. The Nielsen-Ramanujan constanst are also beautiful: and . 3 and 11) showed and Ramanujan independently discovered (Berndt 1994) that (5) where is the Euler-Mascheroni Constant and is Soldner's Constant. and it is computed as Polylog[n,p,z] sometimes. e Nielsen generalized polylogarithm, introduced by the DanishmathematicianNielsNielsen,readsasfollows:, ( ) = ( 1) + 1 ( 1 )!! A brief summary of the definingequations and properties for the frequently utilized generalizations of the dilogarithm (polylogarithm, Nielsen's generalized polylogarithm, Lerch's transcendent) is also given. Nielsen Generalized Polylogarithm A generalization of the polylogarithm function defined by The function reduces to the usual polylogarithm for the case The Nielsen generalized polylogarithm is implemented as PolyLog [ n, p, z ]. $\begingroup$ "Could the Nielsen generalized polylogarithm function mentioned within the Mathematica help system be of any service?" eralizations of the dilogarithm (polylogarithm, Nielsen’s generalized polylogarithm, Jonqui`ere’s function, Lerch’s function) is also given. In the case of the harmonic polylogarithm m → may also contain negative integers. We also give some functional equations and evaluations for Nielsen polylogarithms in weights up to 8, and general families of identities in higher weight. As an application we revisit the limit law of the number of comparisons of the Quicksort algorithm: we reprove that the moments of the limit law are rational polynomials in the zeta values. A r¶esum¶e of the earliest articles that consider the … The Nielsen generalized polylogarithm is . Also, in the very interesting cases of arguments 0 or πone is looking at polylogarithms at 1 and −1, respectively. The polylogarithm function is defined by a power series in z, which is also a Dirichlet series in s: This definition is valid for arbitrary complex order s and for all complex arguments z with |z| < 1; it can be extended to |z| ≥ 1 by the process of analytic continuation. Primary definition (1 formula) © 1998–2021 Wolfram Research, Inc. 1 0 ln 1 ln (1 ) , inwhich isacomplexand and arepositiveintegers.Itis ageneralizationofthepolylogarithmfunction .Infact, +1 = ,1 .Inparticular,thevaluesof, ( ) for = In this paper, we present an implementation of the harmonic polylogarithm of Remiddi and Vermaseren [E. Remiddi, J.A.M. The Nielsen generalized polylogarithm, introduced by the Danish mathematician Niels Nielsen, reads as follows: (1)Sn,p(z)=(-1)n+p-1(n-1)!p!∫01lnn-1tlnp(1-zt)tdt, in which z is a The Wolfram Language supports zeta and polylogarithm functions of a complex variable in full generality, performing efficient arbitrary-precision evaluation and implementing extensive symbolic transformations. We present extensions to generalized Nielsen polylogarithms. Similar terminology is used for other polylogarithmic quantities such as Clausen and Glaisher functions of Nielsen type: Cl … 3. Polylogarithm. J. Request PDF | Relations for Nielsen polylogarithms | Polylogarithms appear in many diverse fields of mathematics. Nielsen (1965, pp. Recall that the classical Riemann zeta function and the prime zeta function have already being defined as - so far as I can tell, no; there isn't a straightforward relationship between the incomplete FD integral and Nielsen's polylogarithm. Another Formula due to Ramanujan which converges more rapidly is (6) α → is the expanded parameter string for the G-functions.The d i 's are positive or negative numbers indicating the signs of a small imaginary part of α i. gives the Nielsen generalized polylogarithm function . Mathematical function, suitable for both symbolic and numerical manipulation. . . . PolyLog [ n, z] has a branch cut discontinuity in the complex plane running from 1 to . For certain special arguments, PolyLog automatically evaluates to exact values. In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function Lis(z) of order s and argument z. Everyone of these functions can also be written as a multiple polylogarithm with specific parameters. It is a generalization of the polylogarithm … An integrand reconstruction method for three-loop amplitudes. ; it is evident that L k;d(z) = P j … The derivative is therefore given by. Furthermore, we relate sums S to Nielsen’s polylogarithm. They are a generalization of Nielsen's polylogarithms, satisfying a product algebra (the product of two hpl's is in turn a combination of hpl's) and forming a set closed under the transformation of the arguments x=1/z and x=(1-t)/(1+t). Zeta Functions and Polylogarithms: PolyLog[nu,p,z] (48 formulas)Primary definition (1 formula) Specific values (7 formulas) General characteristics (11 formulas) Only for special values of s does the polylogarithm reduce to an elementary function such as the natural logarithm or rational functions. The main result (Theorem 14) in On functional equations for Nielsen polylogarithms is to give explicit Li5 terms for this, from which one gets the above analytic identity, by following the same steps as to obtain the corresponding Li2 identity. Some New Transformation Properties of the Nielsen Generalized Polylogarithm Table 1 Time-consuming comparison of the algorithms ( 59 ), ( 4 ), ( 28 ), ( 30 ), ( 33 ), and ( 34 ) with PolyLog for different precistions ( , , and ). Zeta Functions & Polylogarithms. In quantum electrodynamics, polylogarithms of positive integer order arise in the calculation of processes represented by higher-order Feynman diagrams . The polylogarithm function is equivalent to the Hurwitz zeta function — either function can be expressed in terms of the other — and both functions are special cases of the Lerch transcendent. International Journal of Mathematics and Mathematical Sciences 2014, 1-10. A brief summary of the defining equations and properties for the frequently used generalizations of the dilogarithm (polylogarithm, Nielsen's generalized polylogarithm, Jonquière's function, Lerch's function) is also given. The harmonic polylogarithms (hpl's) are introduced. Z 1 0 logk 1(t)logd(1 zt) t dt By de nition of the generating function of the Stirling cycle numbers X n k n k zn n! Nielsen Description Nielsen[i,j, x] denotes Nielsen's polylogarithm. The Nielsen generalized polylogarithm, introduced by the Danish mathematician Niels Nielsen, reads as follows: in which is a complex and and are positive integers. The name of the function comes from the fact that it may also be defined as the repeated integral of itself: thus the dilogarithm is an integral of a function involving the logarithm, and so on. For nonpositive integer orders s, the polylogarithm is a rational function . when negative). It is often convenient to define such as . Properties of the cumulants of the Quicksort limit law are also discussed. We also found similar identities for − ϕ ± 1, ϕ − 1. This latter fact provides a remarkable proof of the Wallis Formula . Critical references to details concerning these functions and their applications are listed. We show that, when viewed modulo and products of lower weight functions, the weight Nielsen polylogarithm satisfies the dilogarithm five-term relation. Special cases are usual polylogarithms and usual Nielsen integrals A high degree of simplication can be achieved expressing the results in terms of harmonic sums. A brief summary of the de¯ning equations and properties for the frequently used gen-eralizations of the dilogarithm (polylogarithm, Nielsen’s generalized polylogarithm, Jonquiµere’s function, Lerch’s function) is also given. We consider the maxim Relation to Nielsen’s polylogarithm Nielsen’s polylogarithm L k;d(z) is de ned by L k;d(z) = ( k1) 1+d (k 1)!d! A brief summary of the defining equations and properties for the frequently used generalizations of the dilogarithm (polylogarithm, Nielsen's generalized polylogarithm, Jonquière's function, Lerch's function) is also given. Preprint typeset in JHEP style - PAPER VERSION IPPP/11/56 DCPT/11/112 From polygons and symbols to polylogarithmic functions Claude Duhr Institute for Particle Physics Phenomenology, University of Durham arXiv:1110.0458v1 [math-ph] 3 Oct 2011 Durham, DH1 3LE, U.K. and Institut f¨ ur theoretische Physik, ETH Z¨ urich, Wolfgang-Paulistr. Accommodative amplitude using the minus lens at different near distances. See also: SimplifyPolyLog. Vermaseren, Int. PubMed Central. = ( 1)k logk(1 z) k! A 15 (2000) 725, hep-ph/9905237] for Mathematica. PolyLog [ n, p, z] gives the Nielsen generalized polylogarithm function. with, ,… Finally, we are going to define higher order prime zeta functions. Further reductions in weight 4 and 5, focusing on the so-called Grassmannian polylogarithm, are investigated in [ ] , whereas identities and reductions involving the so-called Nielsen polylogarithms in weights 5 through 8 are investigated in [ ] (also using the clean single-valued version established in § [ … Harmonic sums can be simplied through algebraic and structural relations. be of Nielsen type (these are polylogarithms of height one; with the height of a polylogarithm indexed by a 1;:::;a k de ned as the number of indices j= 1;:::;k such that a j >1). A comment on the restriction on the indices of the MPL and the MZV as defined in eqs. In quantum statistics, the polylogarithm function appears as the closed form of integrals of the Fermi–Dirac distribution and the Bose–Einstein distribution, and is also known as the Fermi–Dirac integral or the Bose–Einstein integral. It contains an implementation of the product algebra, the derivative properties, series expansion and numerical evaluation. In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function Li s ( z) of order s and argument z. While x and the entries of x → can be arbitrary complex numbers, n, p and the entries of m → must be positive integers. (2013) Analytic and algorithmic aspects of generalized harmonic sums and polylogarithms. where is the Riemann Zeta Function. 1. Keyphrases note stirling series nielsen polylogarithm multiple zeta function asymptotic expansion A brief summary of the defining equations and properties for the frequently used generalizations of the dilogarithm (polylogarithm, Nielsen's generalized polylogarithm, Jonquière's function, Lerch's function) is also given. $\endgroup$ – J. M.'s ennui ♦ Oct 29 '12 at 1:32 (2014) Some New Transformation Properties of the Nielsen Generalized Polylogarithm. NASA Astrophysics Data System (ADS) Badger, Simon; Frellesvig, Hjalte; Zhang, Yang. Stack Exchange network consists of 177 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.. Visit Stack Exchange In quantum electrodynamics, polylogarithms of positive integer order arise in the calculation of processes represented by higher-order Feynman diagrams. Nielsen polylogarithms at special values Of particular interest are the values of Clausen and Glaisher functions at the special arguments π/3,π/2and 2π/3. (Note that the Notation is also used for the Polylogarithm.) Zeta — Riemann and generalized Riemann zeta function. where is an Eulerian Number . 2012-08-01. Γ ⁡ (z): gamma function, ζ ⁡ (s, a): Hurwitz zeta function, π: the ratio of the circumference of a circle to its diameter, e: base of natural logarithm, i: imaginary unit, (a, b): open interval, Li s ⁡ (z): polylogarithm, a: real or complex parameter, s: complex variable and z: complex variable Only for special values of s does the polylogarithm reduce to an elementary function such as the natural logarithm or rational functions. Modern Phys.

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