Jounal of umbe Theoy 3 2 2259 227 Contents lists available at ScienceDiect Jounal of umbe Theoy www.elsevie.com/locate/jnt Sums of poducts of hypegeometic Benoulli numbes Ken Kamano Depatment of Geneal Education, Salesian Polytechnic, 4-6-8, Oyamagaoka, Machida-city, Tokyo, 94-25 Japan aticle info abstact Aticle histoy: Received 2 Januay 2 Revised 2 Apil 2 Communicated by Matthias Beck Keywods: Benoulli numbes Sums of poducts Confluent hypegeometic function We give a fomula fo sums of poducts of hypegeometic Benoulli numbes. This fomula is poved by using special values of multiple analogues of hypegeometic zeta functions. 2 Elsevie Inc. All ights eseved.. Intoduction and main esults The Benoulli numbes B n n =,, 2,... ae ational numbes defined by the geneating function x e x = B n n! xn x < 2π. n= The following well-known fomula fo sums of two poducts of Benoulli numbes is called Eule s fomula: n n B i B n i = nb n n B n n.. i i= Since B 2n+ = fon, Eq.. can be witten as n 2n B 2i B 2n 2i = 2n + B 2n n 2..2 2i i= E-mail addess: kamano@salesio-sp.ac.jp. 22-34X/$ see font matte 2 Elsevie Inc. All ights eseved. doi:.6/j.jnt.2.4.5
226 K. Kamano / Jounal of umbe Theoy 3 2 2259 227 These fomulas. and.2 have been genealized in many diections. Dilche [5] gave fomulas fo sums of poducts of Benoulli numbes =, 2, 3,..., which genealize.2. Agoh and Dilche [2,3] consideed the following types of sums of poducts of Benoulli numbes: i,...,i i + +i =n n! i! i! B m +i B m +i fo non-negative integes m i i and gave some fomulas fo them. Petojević [7] and Petojević andsivastava[8]studiedothetypesofsumsofpoductsofbenoullinumbes.manyesultson sums of poducts of analogues of Benoulli numbes ae also known. Fo example, sums of poducts of Calitz s q-benoulli numbes [4,9] and sums of poducts of Konecke s double seies [5] wee studied. In the aticle [5], Dilche also gave fomulas fo sums of poducts of Benoulli polynomials and Eule polynomials. Hee we state his fomula on Benoulli polynomials B n x: Theoem.. See [5, Theoem 3]. Let x,...,x R and y = x + +x. Then the following identity holds: i,...,i i + +i =n n = n! i! i! B i x B i x i= k= i [ ] i + k k i + k y k B n iy n i n,.3 whee [ k] ae unsigned Stiling numbes of the fist kind defined by xx + x + = [ ] k x k..4 k= Since B n = B n, we obtain the following fomula fo sums of poducts of the odinay Benoulli numbes by setting x = =x = in.3: i,...,i i + +i =n n! i! i! B i B i = n [ ] i n i B n i n..5 i= We note that this fomula.5 was poved by Vandive [2, Eq. 4] and it gives Eule s fomula. when = 2. It is known that fomulas like.,.2 and.3 can be poved by the method of multiple zeta functions e.g., [4,6,7]. Moe pecisely, these fomulas can be obtained by expessing a cetain multiple zeta function in two ways and compaing special values of them at non-negative integes. Chen [4, Theoems 3 5] gave some fomulas, which include.3, fo sums of poducts of genealized Benoulli polynomials and Eule polynomials by this method. The main esults of this pape, which is stated below, will be poved by essentially the same method.
K. Kamano / Jounal of umbe Theoy 3 2 2259 227 226 Table Values of B,n. n 2 3 4 5 6 7 8 B,n 2 B 2,n 3 B 3,n 4 B 4,n 5 B 5,n 6 6 3 42 3 5 7 3 8 9 27 34 567 243 729 3 7 7453 4 6 56 896 92 768 4784 3 3 9 239 289 5689 75 875 2625 7875 9875 39375 82825 26 54 43 3 262 2646 3752 488988 598752 3575672 Fo a positive intege, Howad [2,3] defined genealized Benoulli numbes B,n n =,, 2,...as x /! e x T x = n= B,n x n,.6 n! whee T x is the Taylo polynomial of e x of degee, i.e. T x = m= xm /m!. When =, the numbes B,n ae nothing but the odinay Benoulli numbes B n. We list the numbes B,n fo 5 and n 8 in Table. Howad [3] himself efeed to B,n as A,n and gave many conguences about them. Fo example, he poved the conguence 2B 2,n mod 4 fo n > [2, Theoem 4.]. These numbes B,n wee evisited by Hassen and guyen [,] in the study of hypegeometic zeta functions, which ae defined in the next section, and they call B,n hypegeometic Benoulli numbes. Hee we explain why these numbes B,n ae called hypegeometic Benoulli numbes. Fo eal numbes a and b with b >, the confluent hypegeometic function F a, b; x is defined by the following infinite seies: F a, b; x = whee a n is the Pochhamme symbol defined by n= a n b n x n n!,.7 { aa + a + n n, a n = n =..8 The confluent hypegeometic function F a, b; x is a degeneate fom of Gauss s hypegeometic function 2 F a, b, c; x, and which aises as a solution of a cetain diffeential equation called Kumme s equation cf. [, Chapte 3]. By definition, we have e x T x = x F, + ; x/! fo any positive intege. Hence Eq..6 can be expessed as F, + ; x = n= B,n x n.9 n! and we may call B,n hypegeometic Benoulli numbes. As a genealization of the left-hand side of. and.5, we set the sums of poducts of hypegeometic Benoulli numbes as S, n := i,...,i i + +i =n n! i! i! B,i B,i.
2262 K. Kamano / Jounal of umbe Theoy 3 2 2259 227 fo positive integes and. Then Eule s fomula. can be witten as S,2 n = nb n n B n n.. The pupose of this pape is to give fomulas simila to. fo geneal S, n. The following is the main esult of this pape, and it will be poved by the method of multiple zeta functions. Main Theoem. Let and be positive integes. Fo any intege n, wehave S, n = i= A i; + n i n i!b,n i,.2 i whee A i; s Q[s] i ae polynomials defined by the following ecuence elation: A ; s =, A i; s = s A i; s + A i ; s + 2..3 Hee A i; s ae defined to be zeo fo i and i. It can be poved by induction on thatthedegeeofa i; + n n i is asa polynomial of n. Theefoe we obtain the following coollay. Coollay. Let and be positive integes. The numbe S, n has the following expession: S, n = F i nb,n i n,.4 i= whee each F i X Q[X] is a polynomial of degee, which depends only on and. This pape is oganized as follows. In Section 2 we eview hypegeometic zeta functions defined by Hassen and guyen, and define thei multiple analogues called multiple hypegeometic zeta functions. In Section 3 we pove the Main Theoem by the method of multiple hypegeometic zeta functions defined in Section 2. In the last Section 4 we give some examples. We give fomula.5 fom ou Main Theoem, and give explicit fomulas fo sums of poducts of hypegeometic Benoulli numbes fo = 2, 3 and 4. 2. Multiple hypegeometic zeta functions Let be a positive intege and s be a complex vaiable. We conside the following multiple zeta function: Z s = m,...,m Rs>. 2. m + +m s The ight-hand side of 2. is absolutely convegent fo Rs>. When =, the function Z s is nothing but the classical Riemann zeta function ζs. By the usual method, we can obtain the integal epesentation
K. Kamano / Jounal of umbe Theoy 3 2 2259 227 2263 Z s = x s Γs e x dx Rs>, 2.2 whee Γs is the gamma function. This is a simple genealization of the integal epesentation of the Riemann zeta function: ζs = x s Γs e x dx Rs>. 2.3 We note that the function Z s is a special case of the Banes multiple zeta function ζ s;α, ω defined by ζ s; α,ω = m,...,m Rs> ω m + +ω m + α s 2.4 fo ω = ω,...,ω C and α C with Rω i > i and Rα>. In fact, it is clea that Z s = ζ s;,,...,. The ight-hand side of 2.4 is also absolutely convegent fo Rs >. It is known that the Banes multiple zeta function ζ s;α, ω can be holomophically continued to the whole plane except fo simple poles at s =, 2,...,. Moeove, its values at negative integes can be expessed in tems of Benoulli numbes see, e.g., [6, Theoem ]. Fo a positive intege and a complex vaiable s C with Rs >, Hassen and guyen [] defined hypegeometic zeta functions ζ s as o equivalently, ζ s = Γ + x s+ 2 Γs + x F, + ; x ζ s = Γs + x s+ 2 dx, 2.5 dx. e x 2.6 T x The ight-hand side of 2.6 is convegent fo Rs >. When =, the ight-hand side of 2.6 coincides with the integal epesentation 2.3 of the Riemann zeta function, i.e. ζ s = ζs. Hassen and guyen [] poved that hypegeometic zeta functions can be meomophically continued to the whole plane and values of hypegeometic zeta functions at non-negative integes ae expessed by hypegeometic Benoulli numbes B,n. Theoem 2.. See [, Theoem 3.3]. The function ζ s is analytic on the whole plane except fo simple poles at {2, 3,...,} whose esidues ae Res ζ s = 2 n B, n 2 n. s=n 2 n Futhemoe, fo negative integes n less than 2, we have n ζ n = n + B, n.
2264 K. Kamano / Jounal of umbe Theoy 3 2 2259 227 Remak. The ight-hand side of 2.5 can be defined fo any eal numbe >, hence ζ s can be defined fo any eal numbe >. Hassen and guyen [9] focused paticulaly on the function ζ s 2 and gave some analytic popeties of it. In this pape we only conside the case whee is a positive intege. ow we intoduce multiple hypegeometic zeta functions. Let and be positive integes. Fo s C with Rs> +, multiple hypegeometic zeta functions ae defined as Γ + ζ, s = Γs + x s+ 2 dx. x 2.7 F, + ; x The ight-hand side of 2.7 is absolutely convegent if Rs> +. When =, the function ζ, s is the odinay hypegeometic zeta function ζ s. When =, the function ζ, s is equal to Z s because x F, 2; x = e x. Theefoe we can say that ou functions ζ, s ae multiple analogues of hypegeometic zeta functions. To investigate the sums of poducts of hypegeometic Benoulli numbes, it is convenient to teat the following modified zeta function: ζ, s := Γ2 s Γ + ζ,s. 2.8 Since Γ sγ s = π/ sinπ s, the function ζ, s has an expession ζ, s = sinπs + x s+ 2 π x dx 2.9 F, + ; x fo Rs> +. Let us show that the function ζ, s can be continued to an entie function by the contou integal method. Fo a complex vaiable s, we put I, s = w F, + ; w w s+ dw w. 2. γ Hee the contou γ is taken to be along the eal axis fom to δ>, counteclockwise aound the cicle of adius δ with cente at the oigin, and then along the eal axis fom δ to. Welet w have agument π when we ae going towads the oigin and agument π when we ae going towads. Moeove we suppose that δ is sufficiently small such that thee ae no oots of w F, + ; w inside the cicle of adius δ with cente at the oigin except fo the tivial zeo w =. Then the integal 2. conveges fo all s and I, s defines an entie function. We emak that the integal in 2. is independent of the choice of δ by Cauchy s theoem. Poposition 2.2. i We have I, s = ζ, s fo Rs> +. Theefoe, the function ζ, s can be continued to an entie function. ii Fo an intege n, we have I, + n = { +n S, n/n! n, n <. 2.
K. Kamano / Jounal of umbe Theoy 3 2 2259 227 2265 Poof. i We decompose the integal path as follows: I, s = δ x F, + ; x e s+ log x π i dx + + w =δ w F, + ; w w s+ dw w x F, + ; x e s+ log x+π i dx δ x x. 2.2 Unde the condition Rs> +, the second tem of 2.2 vanishes when δ tends to zeo. Theefoe we have I, s = eπ is+ e π is+ x s+ 2 x F, + ; x dx = sinπs + x s+ 2 π x F, + ; x dx = ζ, s and the assetion holds. ii When s is an intege, the fist and thid tems of 2.2 cancel each othe. Theefoe I, + n = w =δ = n F, + ; w n w n dw w w =δ m= S, m w m n dw m! w. 2.3 By the esidue theoem, this value is equal to +n S, n/n! fo n and equal to zeo fo n <. Since ζ, s = Γ2 s Γ + ζ, s, we can obtain the following theoem on the function ζ, s. This theoem is a genealization of Theoem 2.. Theoem 2.3. i The function ζ, s defined by 2.7 can be holomophically continued to the whole plane except fo possible simple poles at s = +, + 2,..., +. The esidue of ζ, s at s = + n n is! n! n! S, n. 2.4
2266 K. Kamano / Jounal of umbe Theoy 3 2 2259 227 ii Let u be an intege with u.thenwehave ζ, u = u! u! S, + u. + u! Poof. i By definition, we have ζ, s = Γ2 s Γ + ζ, s. The gamma facto Γ2 s has simple poles at s = + i fo i and ζ, s vanishes when s = + i fo i + by Poposition 2.2ii. Theefoe ζ, s has possible simple poles at s = +, + 2,..., +. The esidue of ζ, s at s = + n n isequalto lim s + n ζ, s s +n = lim s + n Γ2 s! ζ, s s +n Γn + 2 s = lim n + s s +n 2 s n + s! ζ, s = n n!! ζ, + n. By Poposition 2.2ii again, this value is equal to! n! n! S, n. ii When s = u u, we obtain fom Poposition 2.2ii that ζ, u = Γ 2 u Γ + ζ, u = u! u! S, + u + u! and this completes the poof. Remak 2. Since B, =, we have S, = foany and. Thus the function ζ, s has a pope simple pole at s = + with esidue! /!. The autho does not know whethe o not the esidue 2.4 of ζ, s at s = + n vanishes fo some, and n. 3. Poof of the Main Theoem In this section we pove ou Main Theoem. Fist we give the following ecuence elation of ζ, s: Lemma 3.. Fo 2,wehave ζ, s = s ζ, s + ζ, s +. 3.
K. Kamano / Jounal of umbe Theoy 3 2 2259 227 2267 Poof. It suffices to show 3. fo Rs> + because both sides of 3. ae holomophically continued to the whole plane. We ecall the following popeties of the confluent hypegeometic function: d x F, + ; x = x F, ; x, dx 3.2 F, ; x = + x F, + ; x 3.3 fo any >. Then, fo any s C with Rs> +, wehave x s+ 2 x F, + ; x dx = = x F, ; x x F, + ; x xs dx + x F, + ; x + x s dx x s x F, + ; x dx. x F, + ; x xs dx Calculating the fist tem by integation by pats, we obtain that x s+ 2 x F, + ; x dx = s x s 2 x F, + ; x dx x s x F, + ; x dx. Theefoe ζ, s = sinπs + x s+ 2 π x F, + ; x dx = s sinπs + x s + 2 π x F, + ; x sinπs + + x s ++ 2 π x F, + ; x dx dx
2268 K. Kamano / Jounal of umbe Theoy 3 2 2259 227 = s ζ, s + ζ, s + and this completes the poof. Poposition 3.2. Fo,wehave ζ, s = i= A i; s ζ, s + i. 3.4 Poof. We pove 3.4 by induction on. Thecase = is clea. We assume that the case holds. Then, by Lemma 3., we have ζ, s = = s s + 2 2 ζ, s + ζ, s + 2 2 2 i= 2 i= A i; s ζ, s 2 + i A i; s + ζ, s + 2 + i = s 2 A i; s ζ, s + i + i= = i= A i ; s + ζ, s + i s A i; s + A i ; s + i= ζ, s + i = i= A i; s ζ, s + i. Hence the case also holds and this completes the poof. We ae now in a position to pove the Main Theoem. Poof of the Main Theoem. By Poposition 3.2, we have ζ, + n = = i= i= A i; + n ζ, n i A i; + n +n i B,n i n i! 3.5
K. Kamano / Jounal of umbe Theoy 3 2 2259 227 2269 fo n. On the othe hand, by Poposition 2.2ii, we have ζ, + n = I, + n = n+ S, n/n!. 3.6 By compaing 3.5 and 3.6, we obtain that S, n = i= A i; + n i n i!b,n i 3.7 i fo n and this poves.2. 4. Examples In this section we give some examples. Let us fist conside the case = inthemaintheoem. In this case we can give the known fomula.5. We ecall some popeties of Stiling numbes, which ae easily poved by.4: [ ] =, 4. [ ] [ ] [ ] + + =, k 4.2 k k + k + cf. [8]. These fomulas ae used to pove the following lemma. Lemma 4.. Fo and i,wehave [ ] A Γs i; s = i!γs + + i. 4.3 Poof. We pove the lemma by induction on. ByA ; s = and 4., Eq. 4.3 holds fo = and i =. We assume that 4.3 holds fo some and all i =,,...,. Then, by the ecuence elation.3 and the inductive assumption, we have A + s i; s = A i; s + A i ; s = s [ ] Γs i!γs + + i [ ] Γs + i!γs + + i [ ] [ ] Γs = + i i +!Γs + + + i.
227 K. Kamano / Jounal of umbe Theoy 3 2 2259 227 Using 4.2, we obtain [ ] + i; s = + Γs + i!γs + + + i, A and this means 4.3 holds fo the case +. ow we can deduce fomula.5. In fact, we obtain fom Lemma 4. that [ ] A Γ n i; n = i!γ n + + i [ ] = n + i + n + i! [ ] i+ = n i n + i! [ ] i+ n i! = i! n! fo n. By this equation and ou Main Theoem, we get S, n = A n i; n i i!b n i i i= [ ] n = + i n i B n i i= and this poves fomula.5. We end this pape with examples of fomulas fo sums of poducts of hypegeometic Benoulli numbes: S,2 n = n B,n + nb,n n. S,3 n = 2 2 n n 2B,n n4 3n + 2B,n + 2nn B,n 2 S,4 n = 6 3 n n 2n 3B,n n 2. + 6n 3 n 2 22 + 8 + 3n2 + 3 + B,n nn n + 8 + 5B,n 2 + 6nn n 2B,n 3 n 3. Refeences [] M. Abamowitz, I.A. Stegun Eds., Handbook of Mathematical Functions with Fomulas, Gaphs, and Mathematical Tables, Dove, 965. [2] T. Agoh, K. Dilche, Convolution identities and lacunay ecuences fo Benoulli numbes, J. umbe Theoy 24 27 5 22. [3] T. Agoh, K. Dilche, Highe-ode ecuences fo Benoulli numbes, J. umbe Theoy 29 29 837 847.
K. Kamano / Jounal of umbe Theoy 3 2 2259 227 227 [4] K.-W. Chen, Sums of poducts of genealized Benoulli polynomials, Pacific J. Math. 28 23 39 52. [5] K. Dilche, Sums of poducts of Benoulli numbes, J. umbe Theoy 6 996 23 4. [6] M. Eie, A note on Benoulli numbes and Shintani genealized Benoulli polynomials, Tans. Ame. Math. Soc. 348 996 7 36. [7] M. Eie, K.F. Lai, On Benoulli identities and applications, Rev. Mat. Ibeoameicana 4 998 67 23. [8] R.L. Gaham, D.E. Knuth, O. Patashnik, Concete Mathematics: A Foundation fo Compute Science, 2nd edition, Addison Wesley, 994. [9] A. Hassen, H.D. guyen, The eo zeta function, Int. J. umbe Theoy 3 27 439 453. [] A. Hassen, H.D. guyen, Hypegeometic Benoulli polynomials and Appell sequences, Int. J. umbe Theoy 4 28 767 774. [] A. Hassen, H.D. guyen, Hypegeometic zeta functions, Int. J. umbe Theoy 6 2 99 26. [2] F.T. Howad, A sequence of numbes elated to the exponential function, Duke Math. J. 34 967 599 65. [3] F.T. Howad, Some sequences of ational numbes elated to the exponential function, Duke Math. J. 34 967 7 76. [4] T. Kim, Sums of poducts of q-benoulli numbes, Ach. Math. 76 2 9 95. [5] T. Machide, Sums of poducts of Konecke s double seies, J. umbe Theoy 28 28 82 834. [6] K. Ota, Deivatives of Dedekind sums and thei ecipocity law, J. umbe Theoy 98 23 28 39. [7] A. Petojević, ew sums of poducts of Benoulli numbes, Integal Tansfoms Spec. Funct. 9 28 5 4. [8] A. Petojević, H.M. Sivastava, Computation of Eule s type sums of the poducts of Benoulli numbes, Appl. Math. Lett. 22 29 796 8. [9] J. Sato, Sums of poducts of two q-benoulli numbes, J. umbe Theoy 74 999 73 8. [2] H.S. Vandive, An aithmetical theoy of the Benoulli numbes, Tans. Ame. Math. Soc. 5 942 52 53.