A -ANALOG OF EULER S DECOMPOSITION FORMULA FOR THE DOUBLE ZETA FUNCTION DAVID M. BRADLEY Received 25 Feruary 2005 and in revised form 6 Septemer 2005 The doule zeta function was first studied y Euler in response to a letter from Goldach in 742. One of Euler s results for this function is a decomposition formula, which expresses the product of two values of the Riemann zeta function as a finite sum of doule zeta values involving inomial coefficients. Here, we estalish a -analog of Euler s decomposition formula. More specifically, we show that Euler s decomposition formula can eextendedtowhatmightereferredtoasa doule-zeta function in such a way that Euler s formula is recovered in the limit as tends to.. Introduction The Riemann zeta function is defined for Rs > y Accordingly, ζs,t:= n n n= s k= ζs:= n s.. n=, Rs >, Rs + t > 2,.2 kt is known as the doule zeta function. The sums.2, and more generally those of the form ζ s,s 2,...,s m := m k >k 2> >k m>0 j=, k sj j n R s j >n, n =,2,...,m,.3 j= have attracted increasing attention in recent years; see, for example, [3, 4, 5, 7, 8, 9, 0, 2, 4, 9]. The survey articles [6, 5, 22, 23, 25] provide an extensive list of references. In.3 the sum is over all positive integers k,...,k m satisfying the indicated ineualities. Copyright 2005 Hindawi Pulishing Corporation International Mathematics and Mathematical Sciences 2005:2 2005 3453 3458 DOI: 0.55/IJMMS.2005.3453
3454 A -analog of Euler s formula Note that with positive integer arguments, s > is necessary and sufficient for convergence. The prolem of evaluating sums of the form.2 for integers s>, t>0seemstohave een first proposed in a letter from Goldach to Euler [7] in 742. See also [6, 8]and [, page 253]. Among other results for.2, Euler proved that if s andt are positive integers, then the decomposition formula s a + t ζsζt = ζt + a,s a+ t t a + s ζs + a,t a.4 s holds. A cominatorial proof of Euler s decomposition formula.4 ased on the simplex integral representations [3, 4, 5, 6, 7] ζs = >x > >x s>0 ζs,t = >x > >x s+t>0 s i= s i= dx i dxs x i x s dx i dxs, x i x s s+t i=s+ dx i dxs+t, x i x s+t.5 and the shuffle multiplication rule satisfied y such integrals is given in [4, 0]. It is of course well known that.4 can also e proved algeraically y summing the partial fraction decomposition see [2, page 48] and [20, Lemma 3.] s x s c x t = a + t t t x s a c t+a + a + s s c s+a c x t a.6 over appropriately chosen integers x and c. See, e.g., [2]. With the general goal of gaining a more complete understanding of the myriad relations satisfied y the multiple zeta functions.3 inmind,a-analog of.3 was introduced in []as where Oserve that we now have ζ [ s,s 2,...,s m ] := m k >k 2> >k m>0 j= sj kj [ kj ] sj,.7 k [k] := j = k, 0<<..8 j=0 ζ s,...,s m = lim ζ [ s,...,s m ],.9 so that.7 represents a generalization of.3. The paper [] considers values of the multiple -zeta functions.7 and estalishes several infinite classes of relations satisfied y them. See also [3]. Here, we continue this general program of study y estalishing a -analog of Euler s decomposition formula.4.
2. Main result David M. Bradley 3455 Our -analog of Euler s decomposition formula naturally reuires only the m = and m = 2 cases of.7; specificallythe -analogs of.and.2given y ζ[s] = n>0 We also define, for convenience, the sum ϕ[s]:= s n [n] s, ζ[s,t] = n>k>0 n= n s n [n] s = n= s n t k [n] s [k] t. 2. n s n [n] s ζ[s]. 2.2 We can now state our main result. Theorem 2.. If s and t are positive integers, then ζ[s]ζ[t] = s t + s a t a a + t t a + s s t s ζ[t + a,s a ] ζ[s + a,t a ] 2.3 mins,t j= s + t j! s j!t j! j ϕ[s + t j]. j! Oserve that the limiting case = oftheorem2. reduces to Euler s decomposition formula.4. 3. A differential identity Our proof of Theorem 2. relies on the following identity. Lemma 3.. Let s and t e positive integers, and let x and y e nonzero real numers. Then, for all real such that x + y + xy 0, s x s y t = t + s a t a a + t t a + s s t s + y a + x t x s a x + y + xy t+a + x a + y s y t a x + y + xy s+a mins,t j= s + t j! s j!t j! j j! t j + ys j + x s+t j. x + y + xy 3.
3456 A -analog of Euler s formula Proof. Apply the partial differential operator s t 3.2 s! x t! y to oth sides of the identity xy = x + y + xy x + y +. 3.3 Oserve that in the limit as, Lemma 3. reduces to the identity s x s y t = a + t t t x s a x + y t+a + a + s s x + y s+a, 3.4 yt a from which the partial fraction identity.6 proved y induction in [20] trivially follows. 4. Proof of Theorem 2. First, oserve that if s>andt>, then from 2., ζ[s]ζ[t] = n= u+v=n s u [u] s t v [v] t, 4. where the inner sum is over all positive integers u and v such that u + v = n. Next,apply Lemma 3. with x = [u], y = [v], noting that then + x = u, + y = v, x + y + xy = [u + v]. 4.2 After interchanging the order of summation, there comes ζ[s]ζ[t] = s t + s a t a a + t t a + s s t s S[s,t,a,] S[t,s,a,] 4.3 mins,t j= s + t j! s j!t j! j T[s,t, j], j!
David M. Bradley 3457 where S[s,t,a,] = = T[s,t, j] = n= u+v=n n= 5. Final remarks t+a n [n] t+a n= u+v=n s u t v t u av [u] s a [u + v] t+a = n u= s a u [u] s a n= u+v=n = ζ[t + a,s a ], s u t v t ju s jv [u + v] s+t j = n= u+v=n t+a u+v s a u [u + v] t+a [u] s a s+t j u+v [u + v] s+t j = ϕ[s + t j]. 4.4 In [24], Zhao gives a much more complicated formula for the product ζ[s]ζ[t]. Zhao s formula is derived using the -shuffle rule[6, ] satisfied y the Jackson -integral analogs of the representations.5. Of course from [], we also have the very simple -stuffle formula ζ[s]ζ[t] = ζ[s,t]+ζ[t,s]+ζ[s + t]+ ζ[s + t ] in which s> and t> need not e integers. Acknowledgment It is a pleasure to thank the four anonymous referees for a thorough reading of the manuscript and for their helpful comments and suggestions. References [] B. C. Berndt,Ramanujan s Noteooks. Part I, Springer, New York, 985. [2] D. Borwein, J. M. Borwein, and R. Girgensohn, Explicit evaluation of Euler sums, Proc. Edinurgh Math. Soc. 2 38 995, no. 2, 277 294. [3] J.M.Borwein,D.M.Bradley,andD.J.Broadhurst,Evaluations of k-fold Euler/Zagier sums: a compendium of results for aritrary k, Electron. J. Comin. 4 997, no. 2, Research Paper 5, approx. 2, The Wilf Festschrift Philadelphia, Pa, 996. [4] J. M. Borwein, D. M. Bradley, D. J. Broadhurst, and P. Lisoněk, Cominatorial aspects of multiple zeta values, Electron. J.Comin.5998, no., Research Paper 38, 2. [5], Special values of multiple polylogarithms, Trans. Amer. Math. Soc. 353 200, no. 3, 907 94. [6] D. Bowman and D. M. Bradley, Multiple polylogarithms: a rief survey, -Series with Applications to Cominatorics, Numer Theory, and Physics Urana, Ill, 2000 B. C. Berndt and K. Ono, eds., Contemp. Math., vol. 29, American Mathematical Society, Rhode Island, 200, pp. 7 92. [7], The algera and cominatorics of shuffles and multiple zeta values, J. Comin. Theory Ser. A 97 2002, no., 43 6. [8], Resolution of some open prolems concerning multiple zeta evaluations of aritrary depth, CompositioMath.39 2003, no., 85 00. [9] D.Bowman,D.M.Bradley,andJ.H.Ryoo,Some multi-set inclusions associated with shuffle convolutions and multiple zeta values, European J. Comin. 24 2003, no., 2 27. [0] D. M. Bradley, Duality for finite multiple harmonic -series,discrete Math. 300 2005, no. 3, 44 56.
3458 A -analog of Euler s formula [], Multiple -zeta values, J. Algera283 2005, no. 2, 752 798. [2], Partition identities for the multiple zeta function, Zeta Functions, Topology, and Quantum Physics T. Aoki, S. Kanemitsu, M. Nakahara, and Y. Ohno, eds., Springer Series: Developments in Mathematics, vol. 4, Springer, New York, 2005, pp. 9 29. [3], On the sum formula for multiple -zeta values, to appear in Rocky Mountain J. Math., http://arxiv.org/as/math.qa/04274. [4] D. J. Broadhurst and D. Kreimer, Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops, Phys. Lett. B 393 997, no. 3-4, 403 42. [5] P. Cartier, Fonctions polylogarithmes, nomres polyzêtas et groupes pro-unipotents [Polylogarithm functions, polyzeta numers and pro-unipotent groups], Astérisue 282 2002, viii, 37 73, Séminaire Bouraki, 53 eme année, 2000/200, Exp. No. 885. [6] L. Euler, Meditationes circa singulare serierum genus, NoviComm. Acad. Sci. Petropol. 20 775, 40 86, reprinted in Opera Omnia, ser. I, 5, B. G. Teuner, Berlin 927, 27 267. [7], Briefwechsel, vol., Birkhäuser, Basel, 975. [8] L. Euler and C. Goldach, Briefwechsel 729 764, Akademie, Berlin, 965. [9] T. Q. T. Le and J. Murakami, Kontsevich s integral for the Homfly polynomial and relations etween values of multiple zeta functions, Topology Appl. 62 995, no. 2, 93 206. [20] C. Markett, Triple sums and the Riemann zeta function, J. Numer Theory 48 994, no. 2, 3 32. [2] N. Nielsen, Die Gammafunktion. Band I. Handuch der Theorie der Gammafunktion. Band II. Theorie des Integrallogarithmus und verwandter Transzendenten, Chelsea, New York, 965. [22] M. Waldschmidt, Valeurs zêta multiples. Une introduction [Multiple zeta values: an introduction], J. Théor. Nomres Bordeaux 2 2000, no. 2, 58 595. [23], Multiple polylogarithms: an introduction, Numer Theory and Discrete Mathematics Chandigarh, 2000, Trends Math., Birkhäuser, Basel, 2002, pp. 2. [24] J. Zhao, -multiple zeta functions and -multiple polylogarithms, http://arxiv.org/as/math.qa/ 0304448, v2, May 2003. [25] V. V. Zudilin, Algeraic relations for multiple zeta values, Uspekhi Mat. Nauk 58 2003, no. 349, 3 32 Russian, translation in Russian Math. Surveys 58 2003, no., 29. David M. Bradley: Department of Mathematics & Statistics, University of Maine, 5752 Neville Hall, Orono, ME 04469-5752, USA E-mail addresses: radley@math.umaine.edu; dradley@memer.ams.org
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