The Miki-type identity for the Apostol-Bernoulli numbers

Transcription

1 Annales Mahemaicae e Informaicae 46 6 pp hp://ami.ef.hu The Mii-ype ideniy for he Aposol-Bernoulli numbers Orli Herscovici, Toufi Mansour Deparmen of Mahemaics, Universiy of Haifa, Haifa, Israel, orli.herscovici@gmail.com and mansour@univ.haifa.ac.il Submied Ocober 9, 5 Acceped Sepember 7, 6 Absrac We sudy analogues of he Mii, Maiyasevich, and Euler ideniies for he Aposol-Bernoulli numbers and obain he analogues of he Mii and Euler ideniies for he Aposol-Genocchi numbers. Keywords: Aposol-Bernoulli numbers; Aposol-Genocchi numbers; Mii ideniy; Maiyasevich ideniy; Euler ideniy MSC: 5A9; B68. Inroducion The Aposol-Bernoulli numbers are defined in [] as λe = B n λ n.. n= Noe ha a λ = his generaing funcion becomes e = n B n, n= where B n is he classical nh Bernoulli number. Moreover, B = B λ = while B = see [9]. The Genocchi numbers are defined by he generaing funcion e + = n G n, n= 97

2 98 O. Herscovici, T. Mansour which are closely relaed o he classical Bernoulli numbers and he special values of he Euler polynomials. I is nown ha G n = n B n and G n = ne, where E n is a value of he Euler polynomials evaluaed a someimes are called he Euler numbers [4,, ]. Liewise he Aposol-Bernoulli numbers, he Aposol-Genocchi numbers are defined by heir generaing funcion as λe + = G n λ n n=. wih G = G λ =. Over he years, differen ideniies were obained for he Bernoulli numbers for insance, see [3, 4, 6, 7,,, 6, 7]. The Euler ideniy for he Bernoulli numbers is given by see [6, 5] B B n = n + B n, n 4..3 = Is analogue for convoluion of Bernoulli and Euler numbers was obained in [] using he p-adic inegrals. The similar convoluion was obained for he generalized Aposol-Bernoulli polynomials in [3]. In 978, Mii [5] found a special ideniy involving wo differen ypes of convoluion beween Bernoulli numbers: = B B n n = B B n n = H n B n, n 4,.4 n where H n = n is he nh harmonic number. Differen inds of proofs of his ideniy were represened in [, 5, 8]. Gessel [8] generalized he Mii ideniy for he Bernoulli polynomials. Anoher generalizaion of he Mii ideniy for he Bernoulli and Euler polynomials was obained in [6]. In 997, Maiyasevich [, 4] found an ideniy involving wo ypes of convoluion beween Bernoulli numbers + n + B B n B B n = nn + B n..5 = = The analogues of he Euler, Mii and Maiyasevich ideniies for he Genocchi numbers were obained in []. In his paper, we represen he analogues of hese ideniies for he Aposol-Bernoulli and he Aposol-Genocchi numbers.. The analogues for he Aposol-Bernoulli numbers In our wor, we use he generaing funcions mehod o obain new analogues of he nown ideniies for he Aposol-Bernoulli numbers see [, 9]. I is easy o show ha λe a µe b = λµe a+b + λe a + µe b..

3 The Mii-ype ideniy for he Aposol-Bernoulli numbers 99 Le us ae a = x and b = x and muliply boh sides of he ideniy. by x. x x λe x µe x x = + λµe x+ x λe x + µe x x x = λµe x x + λe x + x.. µe x By using. and he Cauchy produc, we ge on he LH side of. x x λe x µe x = B n λ n x n B n µ n x n n= n= [ n ] n = B λ B n µ n x n,.3 n= = and on he RH side of. we obain x x λµe x x + λe x + x µe x = B n λµ xn n= x + B n λ n x n + B n µ n x n n= n= = B λµn xn n= [ n ] n + B λµb n λ n x n n= + = [ n n= = B λµb n µ n ] x n..4 By comparing he coefficiens of xn on lef.3 and righ.4 hand sides, we ge n = n B λb n µ = n B λµ + n = n B λµb n λ

4 O. Herscovici, T. Mansour n + n B λµb n µ..5 = I follows from. ha B n = B n. I is well nown ha B =, bu from. we ge B =. Therefore, we concenrae he members, conaining he h index he cases = and = n, ou of he sums. The sum on he lef hand side of.5 can be rewrien as n = = n B λb n µ = n B λb n µ + n B λb n µ + n B n λb µ.6 = n B λb n µ + n δ,λ B n µ + n B n λδ,µ, = where δ p,q is he Kronecer symbol. On he righ hand side of.5 we have ha he firs sum can be rewrien as n n B λµb n λ = = n B λµb n λ = = = + n B λµb n λ + B n λµb λ.7 n B λµb n λ + n δ,λµ B n λ + B n λµδ,λ, and he second sum can be rewrien as n n B λµb n µ = = n B λµb n µ = + n B λµb n µ + B n λµb µ.8 = n B λµb n µ + n δ,λµ B n µ + B n λµδ,µ. = By subsiuing he deailed expressions.6.8 bac ino.5, we ge n B λb n µ + n δ,λ B n µ + n B n λδ,µ =

5 The Mii-ype ideniy for he Aposol-Bernoulli numbers = n B λµ + B n λµδ,λ + n B λµb n λ + n δ,λµ B n λ + = = + B n λµδ,µ. n B λµb n µ + n δ,λµ B n µ.9 By dividing boh sides of.9 by, we obain = n B λb n µ + δ,λ B n µ + B nλδ,µ = nb λµ + n B λµb n λ + δ,λµ B n λ. = + B nλµδ,λ + n B λµb n µ + δ,λµ B n µ = + B nλµδ,µ. We rewrie he. as where n B λb n µ = = nb λµ + n B λµb n λ + = = n B λµb n µ + A δ,. A δ = B nλµ B n µδ,λ + B n λ + B n µδ,λµ + B nλµ B n λδ,µ.. By inegraing. beween and wih respec o and using he formulae p q d = p!q!, p, q, p + q +!

6 O. Herscovici, T. Mansour we obain p+ p+ = = + d = n B λb n µd nb λµd + = which is equivalen o = p d = H p, p, n B λµb n λd n B λµb n µd +!n! B λb n µ n! = = nb λµ + B λµ B n λ n + = = B λµ B n µ n + A δ d, A δ d..3 By dividing boh sides of.3 by n and performing elemenary ransformaions of he binomial coefficiens of.3, we can sae he following resul. Theorem.. For all n, = B λ B n µ n = B λµ + = where A δ is given by.. B λµ B n λ + B n µ n + n We have o consider differen possible cases for λ and µ values. Example.. Le λ =, µ =. I follows from. ha A δ = A δ d,.4 B n + + B n + B n..5

7 The Mii-ype ideniy for he Aposol-Bernoulli numbers 3 Therefore, he inegraing of.5 beween and wih respec o gives A δ d = n n B n d + B n d + B n d = H n B n..6 By subsiuing.6 bac ino.4 and replacing all B by B consisenly wih he case condiion, we ge = B B n n = B B n n = B B n + H n n. Noe ha for even n 4, all summands, conaining odd-indexed Bernoulli numbers, equal zero. Thus, he sums mus be limied from = up o over even indexes only. Moreover, he erm B on he RH side disappears from he same reason. Now we have = B B n n = B B n n = H B n n n. In order o obain he Mii ideniy.4, le us consider he sum = Finally, using = B B n n = n = n B B n = n nown Mii ideniy.4 see [, 8, 5]. = n B B n n B B n. see [], we obain he Corollary.3. Le µ. For all n, he following ideniies are valid = = B B µ B n µ n B n µ n = Moreover, if λ, µ, λµ, hen = B λ B n µ n B µ B n + B n µ n B n µ = B µ + H n,.7 B n µ B + B n µ = B..8 n n = = B λµ B n λ + B n µ n = B λµ..9

8 4 O. Herscovici, T. Mansour Proof. In he case λ =, µ, we have from. ha A δ = B n µ. The inegraing beween and wih respec o gives A δ d = By subsiuing. ino.4, we obain = B λ B n µ n = = B λµ + B n µd = H B n µ.. B λµ B n λ + B n µ n + n H B n µ. By aing ino accoun ha λ = and B p = B p, we ge he ideniy.7. In order o prove.8, we suppose ha λ = µ. Then, from., we obain ha A δ = B n µ + B n µ. By inegraing of Aδ beween and wih respec o, we ge A δ d = B n µ d + By subsiuing. ino.4, we obain = B λ B n µ = B λµ + n = + B n µd = B n µ + B nµ.. n = B λµ B λµ B n λ n B n µ n + B nλ + B n λ n. By subsiuing λ = µ ino he las equaion and using he facs ha B p = B p and B =, we obain.8. Equaion.9 follows from he fac ha A δ = for λ, µ, λµ. By inegraing boh sides of.9 from o wih respec o and muliplying by n + n +, we obain he following resul, which is an analogue of he Maiyasevich ideniy.5. Theorem.4. For all n, n + B λb n µ = + = B λµ B n λ + B n µ nn + n + = B λµ. 6 n n + + B n µδ,λ + B n λδ,µ + B n λ + B n µδ,λµ.

9 The Mii-ype ideniy for he Aposol-Bernoulli numbers 5 Example.5. Le λ =, µ =. Then, by using he fac ha B p = B p, we obain + n + B B n B B n = = = nn + B n + nn + n + B..3 6 Finally, by assuming ha n is even and n 4, we ge ha all erms, conaining odd indexed Bernoulli numbers, equal zero. Under his condiion he n s Bernoulli number on he RH side disappears, and he summaion limis are from ill n. Thus, we obain.5 see also []. Corollary.6. Le µ. Then, for all n, he following ideniies are valid: n + B B n µ = + = B µ B n + B n µ nn + n + n n + = B µ + B n µ,.4 6 n + B µ B n + n µ B B n µ + B n µ = = Moreover, if λ, µ, λµ, hen = nn + n + B + B n 6 µ + B nµ..5 n + B λb n µ = = + = B λµ B n λ + B n µ nn + n + B µ..6 6 Proof. By subsiuing λ = ino. and using he facs ha B p = B p and δ,µ = δ,λµ =, we obain.4. By subsiuing λ = µ ino. and using he fac ha δ,λ = δ,µ =, we obain.5. Equaion.6 follows from. by using he fac ha δ,λ = δ,µ = δ,λµ =. By dividing.9 by and subsiuing =, we obain he following analogue of he Euler ideniy.3. Theorem.7 The Euler ideniy analogue. For all n, B λµb n µ = nb λb µ nb λµ nb λµb λ = n B n µδ,λ B n λδ,µ B n µδ,λµ..7

10 6 O. Herscovici, T. Mansour Proof. By dividing.9 by, we obain = n B λb n µ + n δ,λ B n µ + B n λδ,µ = n B λµ + n B λµb n λ = + δ,λµ B n λ + B n λµδ,λ.8 + n B λµb n µ = Consider now he difference n + n δ,λµ B n µ + B n λµδ,µ. δ,λ B n µ B n λµδ,λ. I is obviously ha n δ,λ B n µ B n λµδ,λ n = δ,λ B n µ B n µδ,λ n j + j j= = δ,λ B n µ = δ,λ B n µ n j n. j j=.9 By subsiuing = ino.8 and using.9, we obain.7. Example.8. Le λ =, µ =. Then, by using he fac ha B =, we ge B B n = nb n nb. = Noe ha for n 4, he odd Bernoulli numbers equal o zero and, hus, only one of he members on he righ hand side will say. Therefore, by assuming ha n 4 and n is even, we obain he Euler ideniy.3 see also [, 6]. Corollary.9. For all n and µ, he following ideniies are valid: B µb n µ = n B n µ nb µ,.3 = = B B n µ = nb µb µ + nb B..3 µ

11 The Mii-ype ideniy for he Aposol-Bernoulli numbers 7 Moreover, if λ, µ, λµ, hen B λµb n µ = nb λb µ n + B λb λµ. = Ideniy.3 is obained by subsiuing λ = ino.7, and Ideniy.3 is obained in case λµ =. Noe ha here we use he fac ha B µ = µ and, herefore, B /µ = B µ Ideniies for he Aposol-Genocchi numbers Following he same echnique we used in he previous secion, we will obain he analogues of he Mii and Euler ideniies for he Aposol-Genocchi numbers. I is easy o show ha λe a + µe b + = λµe a+b λe a + µe b Le us ae a = x and b = x and muliply boh sides of he 3. by 4 x. We ge x λe x + x µe x + x x = λµe x x λe x + x, µe x + By using. and., we ge G n λ n x n G n µ n x n n= n= = B n λµ xn n= x G n λ n x n n= n= G n µ n x n. Therefore, by applying he Cauchy produc and exracing he coefficiens of xn, we obain n G λg n µ n = n = 4 nb λµ B λµg n λ n =

12 8 O. Herscovici, T. Mansour n B λµg n µ n. 3. = Now we divide 3. by and hen inegrae wih respec o from o. By using he facs ha B =, B =, and G = G =, we obain he following saemen, ha is an analogue of he Mii ideniy.4 for he Aposol-Genocchi numbers. Theorem 3.. For all n, = G λ G n µ n + = Example 3.. Le λ = µ =. Then = G G n n + 4 B λµ G n λ + G n µ n = 4B λµ n G nλ + G n µ δ,λµ. = B G n n = 4B 4G n n. Le us suppose now ha n 4 and n is even. Then, he facs ha boh odd indexed Bernoulli and Genocchi numbers equal zero imply = G G n n + 4 = B G n n = 4G n n. Muliplying boh sides of his equaion by n and using n = n yield = G G n n + 4 = B G n n n n = + n = 4G n n. By dividing boh sides by and replacing he indexes by n and vice versa, we obain he following analogue of he Mii ideniy.4 for he Genocchi numbers and = G G n + = G B n = G n n. Noe ha his coincides wih [, Proposiion 4.] for he numbers B n, which are defined as G n = B n. Corollary 3.3. Le µ. For n, = G G n µ n + = B µ G n + G n µ n = 4B µ,

13 The Mii-ype ideniy for he Aposol-Bernoulli numbers 9 G µ G n µ n + = = = 4B G n n µ + G nµ. B G n µ + G n µ n Moreover, if λ, µ, λµ, hen = G λ G n µ n + = B λµ G n λ + G n µ n = 4B λµ. In order o obain he analogues of he Euler ideniy, we divide 3. by and subsiue =. Theorem 3.4. For all n, B λµg n µ = nb λµ G λ ng λg µ G n µδ,λµ. = Example 3.5. Le λ = µ =. Then, since G =, we obain B G n = nb n G G n. = By using he fac ha all odd indexed Bernoulli and Genocchi numbers saring from n = 3 disappear, we obain for all even n 4, = B G n = G n, where he summaion is over even indexed numbers see also []. Here are some ideniies of he Euler ype for he Aposol-Genocchi numbers following from Theorem 3.4. Corollary 3.6. Le λ. For n, = = = Moreover, if λ, µ, λµ, hen = B λg n λ = nb λ ng λ, B λg n = nb λ G λ ng λg, B G n λ = ng λg λ. B λµg n µ = nb λµ G λ ng λg µ.

14 O. Herscovici, T. Mansour Here we used he facs ha B = and G λ = G λ. Anoher series of he ideniies of he Mii and he Euler ypes for he Aposol-Genocchi numbers can be obained in he same manner, when he following, easily proved, equaion λe a µe b + = λµe a+b + + λe a µe b + is aen as a basis for he generaing funcion approach. The following resul may be proved in he same way as Theorem 3.. Le us ae a = x and b = x and muliply boh sides of he wo las ideniies by 4 x. We ge x λe x = x µe x + x x + λe x x µe x + x λµe x Again, we use. and. and apply he Cauchy produc in order o exrac he coefficiens of xn on boh sides of 3.3. Thus, we obain n B λg n µ n = n = ng λµ + G λµb n λ n = n G λµg n µ n. 3.4 = Now we divide boh equaions by and hen inegrae wih respec o from o. By using he facs ha B =, B =, and G = G =, we obain he following saemen, ha is anoher analogue of he Mii ideniy for he Aposol-Genocchi numbers. Theorem 3.7. For all n, = B λ G n µ n = G λµ Example 3.8. Le λ = µ =. Then, for all n, = B G n n = G B n G n n B n λ G n µ n = G λµ + G nµ n H δ,λ. 3.5 = G + G n n H. I is nown ha he Genocchi and Bernoulli numbers are relaed as G n = n B n

15 The Mii-ype ideniy for he Aposol-Bernoulli numbers see []. By subsiuing his ideniy ino he difference B n G n under he second summaion, we obain = B G n n = G B n n B n n = G + G n n H. Noe ha for n 3, he odd-indexed Bernoulli and Genocchi numbers disappear, herefore, le us assume now ha n is even and n 4. Thus, we have = B G n n = G n B n n Using he binomial ideniy = n leads o = B G n n = G n B n n n = G n n H. = G n n H. We replace by n under he second summaion. Finally, using he noaion G n = B n, proposed in [], and dividing boh sides by lead o he saemen 4. of [, Proposiion 4.] = B B n n = Corollary 3.9. Le µ. For all n, = B G n µ n = B G µ B n n = B n n H. B n G n µ n = G µ + G nµ n H. Due o he asymmery of λ and µ in he 3.5, we ge he following corollary of he Theorem 3.7. Corollary 3.. Le λ. For all n, = = B λ G n n B λ G n λ n = = Moreover, if λ, µ, λµ, hen = B λ G n µ n = G λ B n λ G n n = G λ, G B n λ G n λ = G. 3.6 n G λµ B n λ G n µ n = Gλµ.

16 O. Herscovici, T. Mansour By dividing 3. and 3.4 by and hen subsiuing =, we obain he following analogue of he Euler ideniy. Theorem 3.. For all n, G λµg n µ = ng λµ + n G n λµδ,λ 3.7 = = + nb λ G λµ G µ. Example 3.. Le λ = µ =. Then G G n = ng + n G n. = By using he fac ha all odd indexed Bernoulli and Genocchi numbers saring from n = 3 disappear, we obain a more familiar form for all even n 4, G G n = n G n, where he summaion is over even indexed numbers see also []. Corollary 3.3. Le λ and n. Then he following ideniies are valid G λg n λ = ng λ + n G n λ, 3.8 = G λg n = ng λ + nb λg λ G, 3.9 = G G n λ = ng + nb λ G λ G. 3. = = Moreover, if λ, µ, λµ, hen G λµg n µ = ng λµ + nb λg λµ G µ. Proof. Replacing λ and µ in 3.7, and subsiuing µ = lead o = G λg n λ = ng λ + n G n λ + n G λ G λ. 3. The las summand equals zero, and we obain he ideniy 3.8. By subsiuing µ = ino 3.7 we obain 3.9. Subsiuing µ = λ ino.4 and using he fac ha + B λ = B λ lead o 3.. The second summand on he RH of he 3.7 disappears since λ, and we obain 3..

17 The Mii-ype ideniy for he Aposol-Bernoulli numbers 3 Remar 3.4. As i was menioned above, he classical Bernoulli and Genocchi numbers are conneced via he following relaionship G n = n B n. I is easy o see ha also he Aposol-Bernoulli and Aposol-Genocchi numbers saisfy G n λ = B n λ. Moreover, he Aposol-Bernoulli numbers saisfy B n λ = B n λ and B n+λ = B n+ λ for λ. In he same manner, he Aposol- Genocchi numbers saisfy G n λ = G n λ and G n+λ = G n+ λ for n >. These relaionships allow o obain new ideniies from hose considered in he curren paper. Acnowledgemen. The research of he firs auhor was suppored by he Minisry of Science and Technology, Israel. References [] Agoh, T., On he Mii and Maiyasevich ideniies for Bernoulli numbers, Inegers 4 4 A7. [] Aposol, T.M., On he Lerch zea funcion, Pacific J. Mah [3] Bayad, A., Kim, T., Ideniies for he Bernoulli, he Euler and he Genocchi numbers and polynomials, Adv. Sud. Con. Mah [4] Chang, C.-H., Ha, C.-W., On recurrence relaions for Bernoulli and Euler numbers, Bull. Ausral. Mah. Soc [5] Crabb, M.C., The Mii-Gessel Bernoulli number ideniy, Glasgow Mah. J [6] Dilcher, K., Sums of producs of Bernoulli numbers, J. Number Theory [7] Dunne, G.V., Schuber, C., Bernoulli number ideniies from quanum field heory and opological sring heory, arxiv:mah/466v, 4. [8] Gessel, I., On Mii s ideniy for Bernoulli numbers, J. Number Theory [9] He, Y., Wang, C., Some formulae of producs of he Aposol-Bernoulli and Aposol- Euler polynomials, Discr. Dyn. Naure Soc.,,. [] Hu, S., Kim, D., Kim, M.-S., New ideniies involving Bernoulli, Euler and Genocchi numbers, Adv. Diff. Eq [] Jolany, H., Sharifi, H., Alielaye, E., Some resuls for he Aposol-Genocchi polynomials of higher-order, Bull. Malays. Mah. Sci. Soc [] Kim, T., Rim, S.H., Simse, Y., Kim, D., On he analogs of Bernoulli and Euler numbers, relaed ideniies and zea and L-funcions, J. Korean Mah. Soc [3] Luo, Q.-M., Srivasava, H.M., Some generalizaions of he Aposol-Bernoulli and Aposol-Euler polynomials, J. Mah. Anal. Appl [4] Maiyasevich, Y., Ideniies wih Bernoulli numbers, hp://logic.pdmi.ras.ru/ yuma/personaljournal/ ideniybernoulli/bernulli.hm, 997.

18 4 O. Herscovici, T. Mansour [5] Mii, H., A relaion beween Bernoulli numbers, J. Number Theory [6] Pan, H., Sun, Z.-W., New ideniies involving Bernoulli and Euler polynomials, arxiv:mah/47363v, 4. [7] Simse, Y., Kim, T., Kim, D., A new Kim s ype Bernoulli and Euler numbers and relaed ideniies and zea and L-funcions, arxiv:mah/67653v, 6.