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Integrals of Polylogarithmic Functions with Negative Argument

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Integrals of Polylogarithmic Functions with Negative Argument Acta Univ. Sapientiae, Mathematica, 10, 2 (2018) 347{367 DOI: 10.2478/ausm-2018-0027 Integrals of polylogarithmic functions with negative argument Anthony Sofo Victoria University, Australia email: [email protected] Abstract. The connection between polylogarithmic functions and Eu- ler sums is well known. In this paper we explore the representation and many connections between integrals of products of polylogarithmic func- tions and Euler sums. We shall consider mainly, polylogarithmic functions with negative arguments, thereby producing new results and extending the work of Freitas. Many examples of integrals of products of polylog- arithmic functions in terms of Riemann zeta values and Dirichlet values will be given. 1 Introduction and preliminaries It is well known that integrals of products of polylogarithmic functions can be associated with Euler sums, see [16]. In this paper we investigate the repre- sentations of integrals of the type 1 m x Lit (-x) Liq (-x) dx; Z0 for m ≥ -2; and for integers q and t: For m = -2; -1; 0 we give explicit representations of the integral in terms of Euler sums and for m ≥ 0 we give a recurrence relation for the integral in question. We also mention two specific 2010 Mathematics Subject Classification: 11M06, 11M32, 33B15 Key words and phrases: polylogarithm function, Euler sums, zeta functions, Dirichlet functions 347 348 A. Sofo integrals with a different argument in the polylogarithm. Some examples are highlighted, almost none of which are amenable to a computer mathematical package. This work extends the results given by [16], who examined a simi- lar integral with positive arguments of the polylogarithm. Devoto and Duke [14] also list many identities of lower order polylogarithmic integrals and their relations to Euler sums. Some other important sources of information on poly- logarithm functions are the works of [19] and [20]. In [3] and [12] the authors explore the algorithmic and analytic properties of generalized harmonic Eu- ler sums systematically, in order to compute the massive Feynman integrals which arise in quantum field 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 M[f(z)](N) = dz zNf (z) ; Z0 where the basic functions f(z) typically involve polylogarithms and harmonic sums of lower weight. Applying the latter type of relations, the author in [6], 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 [17], the authors prove several identities containing infinite sums of values of the Roger's dilogarithm function. defined on x 2 [0:1], by 1 Li2 (x) + 2 ln x ln (1 - x) ; 0 < x < 1 L (x) = 0 ; x = 0 : R 8 < ζ (2) ; x = 1 The Lerch transcendent,: zm Φ (z; t; a) = t 1 m =0 (m + a) X is defined for jzj < 1 and < (a) > 0 and satisfies the recurrence Φ (z; t; a) = z Φ (z; t; a + 1) + a-t: The Lerch transcendent generalizes the Hurwitz zeta function at z = 1; 1 Φ (1; t; a) = t 1 m =0 (m + a) X Integrals of polylogarithmic 349 and the polylogarithm, or de-Jonqui`ere'sfunction, when a = 1; zm Lit (z) := t ; t 2 C when jzj < 1; < (t) > 1 when jzj = 1: 1 m m =1 X Let n 1 1 1 - tn n Hn = = dt = γ + (n + 1) = ;H0 := 0 r 1 - t 1 j (j + n) r=1 0 j=1 X Z X be the nth harmonic number, where γ denotes the Euler-Mascheroni constant, (m) n 1 th Hn = r=1 rm is the m order harmonic number and (z) is the digamma (or psi) function defined by P d Γ 0(z) 1 (z) := flog Γ(z)g = and (1 + z) = (z) + ; dz Γ(z) z moreover, 1 1 (z) = -γ + - : 1 n + 1 n + z n=0 X More generally a non-linear Euler sum may be expressed as, 0 t r 1 n qj m (±1) (αj) k H J(βk) np @ n n A n≥1 j=1 k=1 X Y Y where p ≥ 2; t; r; qj; αj; mk; βk are positive integers and q m 0 n 1 0 n 1 q 1 m (-1)j+1 H(α) = ; J(β) = : n @ jα A n @ jβ A j=1 j=1 X X If, for a positive integer t r λ = αjqj + βjmj + p; j=1 j=1 X X then we call it a λ-order Euler sum. The polygamma function k (k) d k+1 1 (z) = k f (z)g = - (-1) k! k+1 dz 1 r=0 (r + z) X 350 A. Sofo and has the recurrence (-1)k k! (k)(z + 1) = (k)(z) + : zk+1 The connection of the polygamma function with harmonic numbers is, (-1)α H(α+1) = ζ (α + 1) + (α) (z + 1) ; z 6= f-1; -2; -3; :::g : (1) z α! and the multiplication formula is p-1 1 j (k)(pz) = δ ln p + (k)(z + ) (2) m;0 pk+1 p j=0 X for p a positive integer and δp;k is the Kronecker delta. We define the alter- nating zeta function (or Dirichlet eta function) η (z) as n+1 (-1) 1-z η (z) := z = 1 - 2 ζ (z) (3) 1 n n =1 X where η (1) = ln 2: If we put n+1 (p) (-1) Hn S (p; q) := q ; 1 n n =1 X in the case where p and q are both positive integers and p + q is an odd integer, Flajolet and Salvy [15] gave the identity: p + i - 1 2S (p; q) = (1 - (-1)p) ζ (p) η (q) + 2 (-1)p ζ (p + i) η (2k) p - 1 i+2k=q X q + j - 1 + η (p + q) - 2 (-1)j η (q + j) η (2k) ; (4) q - 1 j+2k=p X 1 1 where η (0) = 2 ; η (1) = ln 2; ζ (1) = 0; and ζ (0) = - 2 in accordance with the analytic continuation of the Riemann zeta function. We also know, from the work of [11] that for odd weight (p + q) we have p (p) [ 2 ] Hn p p + q - 2j - 1 BW (p:q) = q = (-1) ζ (p + q - 2j) ζ (2j) 1 n p - 1 n =1 j =1 X X (5) Integrals of polylogarithmic 351 [ p ] 1 2 p + q - 2j - 1 + 1 + (-1)p+1 ζ (p) ζ (q) + (-1)p ζ (p + q - 2j) ζ (2j) 2 q - 1 j =1 X ζ (p + q) p + q - 1 p + q - 1 + 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 ex- pression exits for BW (p:q) : (at least at the time of writing this paper). Two examples with even weight are (2) (4) Hn 2 1 Hn 13 4 = ζ (3) - ζ (6) ; 4 = ζ (8) : 1 n 3 1 n 12 n =1 n =1 X X The work in this paper extends the results of [16] and later [25], in which they gave identities of products of polylogarithmic functions with positive argument in terms of zeta functions. Other works including, [1], [4], [8], [10], [13], [18], [22], [23], [24], cite many identities of polylogarithmic integrals and Euler sums, but none of these examine the negative argument case. The following result was obtained by Freitas, [16]. Lemma 1 For q and t positive integers 1 q-1 Li (x) Li (x) t q dx = (-1)j+1 ζ (t + j) ζ (q - j + 1) + (-1)q+1 EU (t + q) x j =1 Z0 X where EU (m) is Euler's identity given in the next lemma. The following lemma will be useful in the development of the main theorem. Lemma 2 The following identities hold: for m 2 N: Euler's identity states m-2 Hn EU (m) = m = (m + 2)ζ (m + 1) - ζ (m - j) ζ (j + 1) : (6) 1 n n =1 j=1 X X 352 A. Sofo For p a positive even integer, Hn p HE (p) = p = (ζ (p + 1) + η (p + 1)) - (ζ (p) + η (p)) ln 2 1 (2n + 1) 2 n =1 X p -1 (7) 1 2 - (ζ (p + 1 - 2j) + η (p + 1 - 2j)) (ζ (2j) + η (2j)) : 2 j=1 X For p a positive odd integer, Hn p HO (p) = p = (ζ (p + 1) + η (p + 1)) - (ζ (p) + η (p)) ln 2 1 (2n + 1) 4 n =1 X p-1 ! 1 1 + (-1) 2 p + 1 p + 1 - ζ + η (8) 4 2 2 2 1 b - (ζ (p - 2j) + η (p - 2j)) (ζ (2j + 1) + η (2j + 1)) 2 j=1 X p-1 h p-1 i 1+(-1) 2 where η (z) is the Dirichlet eta function, b = 4 - 2 and [z] is the greatest integer less than z: For p and t positive integers we have (-1)n+1 F (p; t) = t 1 p n =1 n (n + 1) Xp p + t - r - 1 = (-1)p-r η (r) p - r (9) r =1 X t p + t - s - 1 + (-1)p+1 (1 - η (s)) ; t - s s =1 X 1 p+1 p + t - 1 G (p; t) = t = (-1) 1 p p n =1 n (n + 1) Xp p + t - r - 1 + (-1)p-r ζ (r) p - r (10) r =2 X t p + t - s - 1 + (-1)p ζ (s) ; t - s s =1 X Integrals of polylogarithmic 353 and Hn p+1 p + t - 2 HG (p; t) = t = (-1) ζ (2) 1 p p - 1 n =1 n (n + 1) Xp p + t - r - 1 + (-1)p-r EU (r) p - r (11) r =2 X t p + t - s - 1 + (-1)p (EU (s) - ζ (s + 1)) : t - s s =2 X Proof.
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