PolyLog [ n, p, z] gives the Nielsen generalized polylogarithm function. The coefficients of their expansions and their Mellin transforms are harmonic sums. An integrand reconstruction method for three-loop amplitudes. Only for special values of s does the polylogarithm reduce to an elementary function such as the natural logarithm or rational functions. The Nielsen generalized polylogarithm, introduced by the Danish mathematician Niels Nielsen, reads as follows: in which is a complex and and are positive integers. = ( 1)k logk(1 z) k! 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 ]. The harmonic polylogarithms (hpl's) are introduced. The Nielsen-Ramanujan constanst are also beautiful: and . 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! Also, in the very interesting cases of arguments 0 or πone is looking at polylogarithms at 1 and −1, respectively. Accommodative amplitude using the minus lens at different near distances. Nielsen (1965, pp. 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. 2012-08-01. 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. PubMed Central. Request PDF | Relations for Nielsen polylogarithms | Polylogarithms appear in many diverse fields of mathematics. - so far as I can tell, no; there isn't a straightforward relationship between the incomplete FD integral and Nielsen's polylogarithm. Critical references to details concerning these functions and their applications are listed. 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. Similar terminology is used for other polylogarithmic quantities such as Clausen and Glaisher functions of Nielsen type: Cl … We consider the maxim In this paper, we present an implementation of the harmonic polylogarithm of Remiddi and Vermaseren [E. Remiddi, J.A.M. where is the Riemann Zeta Function. Polylogarithm. Harmonic sums can be simplied through algebraic and structural relations. J. The derivative is therefore given by. Furthermore, we relate sums S to Nielsen’s polylogarithm. NASA Astrophysics Data System (ADS) Badger, Simon; Frellesvig, Hjalte; Zhang, Yang. Nielsen Description Nielsen[i,j, x] denotes Nielsen's polylogarithm. (2014) Some New Transformation Properties of the Nielsen Generalized Polylogarithm. The polylogarithm … We also found similar identities for − ϕ ± 1, ϕ − 1. eralizations of the dilogarithm (polylogarithm, Nielsen’s generalized polylogarithm, Jonqui`ere’s function, Lerch’s function) is also given. 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. and it is computed as Polylog[n,p,z] sometimes. Furthermore, we relate sums S to Nielsen’s polylogarithm. A 15 (2000) 725, hep-ph/9905237] for Mathematica. 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 ). 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. Zeta — Riemann and generalized Riemann zeta function. Properties of the cumulants of the Quicksort limit law are also discussed. 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 Wolfram Language supports zeta and polylogarithm functions of a complex variable in full generality, performing efficient arbitrary-precision evaluation and implementing extensive symbolic transformations. 1 0 ln 1 ln (1 ) , inwhich isacomplexand and arepositiveintegers.Itis ageneralizationofthepolylogarithmfunction .Infact, +1 = ,1 .Inparticular,thevaluesof, ( ) for = 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 This latter fact provides a remarkable proof of the Wallis Formula . 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 § [ … where is an Eulerian Number . We present extensions to generalized Nielsen polylogarithms. It contains an implementation of the product algebra, the derivative properties, series expansion and numerical evaluation. 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). (2013) Analytic and algorithmic aspects of generalized harmonic sums and polylogarithms. 3. It is a generalization of the polylogarithm … 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 A r¶esum¶e of the earliest articles that consider the … We show that, when viewed modulo and products of lower weight functions, the weight Nielsen polylogarithm satisfies the dilogarithm five-term relation. Journal of … 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. 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. 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! Harmonic sums of the variable are associated to Mellin transforms of weighted harmonic polylogarithms. 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. See also: SimplifyPolyLog. 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. Modern Phys. 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. 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. A comment on the restriction on the indices of the MPL and the MZV as defined in eqs. such as . 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). 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. 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. with, ,… Finally, we are going to define higher order prime zeta functions. The Nielsen generalized polylogarithm is . 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 Vermaseren, Int. Recall that the classical Riemann zeta function and the prime zeta function have already being defined as Zeta Functions and Polylogarithms: PolyLog[nu,p,z] (48 formulas)Primary definition (1 formula) Specific values (7 formulas) General characteristics (11 formulas) In quantum electrodynamics, polylogarithms of positive integer order arise in the calculation of processes represented by higher-order Feynman diagrams. Zeta Functions & Polylogarithms. Primary definition (1 formula) © 1998–2021 Wolfram Research, Inc. PolyLog [ n, z] gives the polylogarithm function. e Nielsen generalized polylogarithm, introduced by the DanishmathematicianNielsNielsen,readsasfollows:, ( ) = ( 1) + 1 ( 1 )!! 1. Another Formula due to Ramanujan which converges more rapidly is (6) 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. 3 and 11) showed and Ramanujan independently discovered (Berndt 1994) that (5) where is the Euler-Mascheroni Constant and is Soldner's Constant. $\endgroup$ – J. M.'s ennui ♦ Oct 29 '12 at 1:32 Only for special values of s does the polylogarithm reduce to an elementary function such as the natural logarithm or rational functions. While x and the entries of x → can be arbitrary complex numbers, n, p and the entries of m → must be positive integers. Keyphrases note stirling series nielsen polylogarithm multiple zeta function asymptotic expansion Everyone of these functions can also be written as a multiple polylogarithm with specific parameters. We also give some functional equations and evaluations for Nielsen polylogarithms in weights up to 8, and general families of identities in higher weight. In the case of the harmonic polylogarithm m → may also contain negative integers. 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. The polylogarithm of Negative Integer order arises in sums of the form. International Journal of Mathematics and Mathematical Sciences 2014, 1-10. 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. 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. ; it is evident that L k;d(z) = P j … $\begingroup$ "Could the Nielsen generalized polylogarithm function mentioned within the Mathematica help system be of any service?" 1. 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
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