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Families of Integrals of Polylogarithmic Functions † mathematics Article Families of Integrals of Polylogarithmic Functions † Anthony Sofo Victoria University, P. O. Box 14428, Melbourne City, Victoria 8001, Australia; [email protected] † Lovingly dedicated to Mia Eva Manca. Received: 30 December 2018; Accepted: 26 January 2019; Published: 3 February 2019 Abstract: We give an overview of the representation and many connections between integrals of products of polylogarithmic functions and Euler sums. We shall consider polylogarithmic functions with linear, quadratic, and trigonometric arguments, thereby producing new results and further reinforcing the well-known connection between Euler sums and polylogarithmic functions. Many examples of integrals of products of polylogarithmic functions in terms of Riemann zeta values and Dirichlet values will be given. Suggestions for further research are also suggested, including a study of polylogarithmic functions with inverse trigonometric functions. Keywords: Euler sums; zeta functions; Dirichlet functions; series representation; harmonic number MSC: 11M06; 11M32; 33B15 1. Introduction and Preliminaries It is well known that integrals of products of polylogarithmic functions can be associated with Euler sums, see Reference [1]. In this paper, we investigate the representations of integrals of the type 1 Z m x Liq (±x) Lit (±x) dx, 0 1 Z 1 Li (±x) Li x2 dx, x q t 0 for m ≥ −2, and for integers q and t. For m = −2, −1, 0, we give explicit representations of the integrals 1 R m in terms of Euler sums. For the case x Lit (−x) Liq (−x) dx, for m ≥ 0, we show that the integral 0 satisfies a certain recurrence relation. We also mention two specific integrals with a trigonometric argument in the polylogarithm of the form 1 Z sin px Lip (sin px) dx. 0 Some examples are highlighted, almost none of which are amenable to a computer mathematical package. This work extends the results given in Reference [1], where the author examined integrals with positive arguments of the polylogarithm. Devoto and Duke [2] also list many identities of lower order polylogarithmic integrals and their relations to Euler sums. Some other important sources of information on polylogarithm functions are the works of References [3] and [4]. In References [5] and [6], the authors explore the algorithmic and analytic properties of generalized harmonic Euler sums systematically, in order to compute the massive Feynman integrals which arise in quantum field Mathematics 2019, 7, 143; doi:10.3390/math7020143 www.mdpi.com/journal/mathematics Mathematics 2019, 7, 143 2 of 20 theories and in certain combinatorial problems. Identities involving harmonic sums can arise from their quasi-shuffle algebra or from other properties, such as relations to the Mellin transform 1 Z M[ f (z)](N) = dz zN f (z) , 0 where the basic functions f (z) typically involve polylogarithms and harmonic sums of lower weight. Applying the latter type of relations, the author in Reference [7] expresses all harmonic sums of the above type with weight w = 6, in terms of Mellin transforms and combinations of functions and constants of lower weight. In another interesting and related paper [8], the authors prove several identities containing infinite sums of values of the Roger’s dilogarithm function. defined on x 2 [0, 1], by 8 Li (x) + 1 ln x ln (1 − x) ; 0 < x < 1 <> 2 2 LR (x) = 0; x = 0 . :> z (2) ; x = 1 The Lerch transcendent, ¥ zm = F (z, t, a) ∑ t m =0 (m + a) is defined for jzj < 1 and < (a) > 0 and satisfies the recurrence F (z, t, a) = z F (z, t, a + 1) + a−t. The Lerch transcendent generalizes the Hurwitz zeta function at z = 1, ¥ 1 = F (1, t, a) ∑ t m =0 (m + a) and the polylogarithm, or de-Jonquière’s function, when a = 1, ¥ zm ( ) = 2 j j < < ( ) > j j = Lit z : ∑ t , t C when z 1; t 1 when z 1. m =1 m Let n 1 Z 1 1 − tn ¥ n H = = dt = g + y (n + 1) = , H := 0 n ∑ − ∑ ( + ) 0 r=1 r 0 1 t j=1 j j n be the nth harmonic number, where g denotes the Euler–Mascheroni constant, n m−1 1 n (m) 1 (−1) Z 1 − t H = = lnm−1 (t) dt (1) n ∑ m ( − ) − r=1 r m 1 ! 0 1 t is the mth order harmonic number and y(z) is the digamma (or psi) function defined by d G0(z) 1 y(z) := flog G(z)g = and y(1 + z) = y(z) + , dz G(z) z moreover, ¥ 1 1 y(z) = −g + − . ∑ + + n=0 n 1 n z Mathematics 2019, 7, 143 3 of 20 More generally, a non-linear Euler sum may be expressed as n t qj r ! mk (±1) (aj) (bk) ∑ p ∏ Hn ∏ Jn n≥1 n j=1 k=1 where p ≥ 2, t, r, qj, aj, mk, bk are positive integers and !q + !m q n 1 m n (−1)j 1 (a) = (b) = Hn ∑ a , Jn ∑ b . j=1 j j=1 j If, for a positive integer, t r l = ∑ ajqj + ∑ bjmj + p, j=1 j=1 then we call it a l-order Euler sum. The polygamma function dk ¥ 1 y(k)(z) = fy(z)g = (− )k+1 k k 1 ! ∑ k+1 dz r=0 (r + z) and satisfies the recurrence relation (−1)k k! y(k)(z + 1) = y(k)(z) + . zk+1 The connection of the polygamma function with harmonic numbers is a (a+ ) (−1) H 1 = z (a + 1) + y(a) (z + 1) , z 6= f−1, −2, −3, ...g . (2) z a! and the multiplication formula is − 1 p 1 j y(k)(pz) = d p + y(k)(z + ) m,0 ln k+1 ∑ (3) p j=0 p for p is a positive integer and dp,k is the Kronecker delta. We define the alternating zeta function (or Dirichlet eta function) h (z) as + ¥ (−1)n 1 ( ) = = − 1−z ( ) h z : ∑ z 1 2 z z (4) n =1 n where h (1) = ln 2. If we put + (p) ¥ (−1)n 1 H = n S (p, q) : ∑ q , n =1 n in the case where p and q are both positive integers and p + q is an odd integer, Flajolet and Salvy [9] gave the identity ! p p p + i − 1 2S (p, q) = 1 − (−1) z (p) h (q) + 2 (−1) ∑ z (p + i) h (2k) i+2k=q p − 1 ! (5) q + j − 1 j +h (p + q) − 2 ∑ (−1) h (q + j) h (2k) , j+2k=p q − 1 Mathematics 2019, 7, 143 4 of 20 1 1 where h (0) = 2 , h (1) = ln 2, z (1) = 0, and z (0) = − 2 in accordance with the analytic continuation of the Riemann zeta function. We also know, from the work of Reference [10] that for odd weight w = (p + q) we have ! (p) p ¥ Hn p [ 2 ] p + q − 2j − 1 BW (p, q) = ∑ q = (−1) ∑ z (p + q − 2j) z (2j) n =1 n j =1 p − 1 p ! p+1 p [ ] p + q − 2j − 1 + 1 1 + (−1) z (p) z (q) + (−1) ∑ 2 z (p + q − 2j) z (2j) (6) 2 j =1 q − 1 ! !! ( + ) + p + q − 1 + p + q − 1 + z p q 1 + (−1)p 1 + (−1)p 1 , 2 p q where [z] is the integer part of z. It appears that some isolated cases of BW (p, q) , for even weight (p + q) , can be expressed in zeta terms, but in general, almost certainly, for even weight (p + q) , no general closed form expression exits for BW (p, q) . Two examples with even weight are ( ) ( ) ¥ H 2 1 ¥ H 4 37 n = 2 ( ) − ( ) n = ( ) − 2 ( ) ∑ 4 z 3 z 6 , ∑ 2 z 6 z 3 . n =1 n 3 n =1 n 12 There are also the two general forms of identities ( + ) ¥ H 2p 1 n = 2 ( + ) + ( + ) 2 ∑ 2p+1 z 2p 1 z 4p 2 , n =1 n and ( ) 2 ! ¥ H 2p (4p)!B n = − 2p ( ) 2 ∑ 2p 1 2 z 4p n =1 n 2 ((2p)!) B4p for p 2 N and where Bp are the signed Bernoulli numbers of the first kind. At even weight w = 8, neither BW (2, 6) nor BW (6, 2) currently have a reduced expression in terms of zeta values and their products. Using (1) we can express ( ) ¥ H 2 5 Z 1 ln x Li (x) BW (2, 6) = n = z (8) − 6 dx ∑ 6 − n =1 n 3 0 1 x and ( ) ¥ H 6 5 1 Z 1 ln5 x Li (x) BW (6, 2) = n = z (8) + 2 dx. ∑ 2 − n =1 n 3 5! 0 1 x The work in this paper extends the results of Reference [1] and later Reference [11], in which they gave identities of products of polylogarithmic functions with positive argument in terms of zeta functions. Other works, including References [12–23], cite many identities of polylogarithmic integrals and Euler sums. The following lemma was obtained by Freitas, [1]: Lemma 1. For q and t positive integers 1 (+,+) R Lit(x) Liq(x) I (q, t) = x dx 0 (7) q−1 j+1 q+1 = ∑j =1 (−1) z (t + j) z (q − j + 1) + (−1) EU (t + q) where EU (m) is Euler’s identity given in the next lemma. Mathematics 2019, 7, 143 5 of 20 Lemma 2. The following identities hold: for m 2 N. The classical Euler identity, as given by Euler, states ¥ Hn m EU (m) = ∑n =1 nm = ( 2 + 1)z (m + 1) (8) 1 m−2 − 2 ∑j=1 z (m − j) z (j + 1) .
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