ON THE MIKI AND MATIYASEVICH IDENTITIES FOR BERNOULLI NUMBERS

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1 #A7 INTEGERS 4 (4) ON THE MIKI AND MATIYASEVICH IDENTITIES FOR BERNOULLI NUMBERS Takash Agoh Departmet of Mathematcs, Tokyo Uversty of Scece, Noda, Chba, Japa agoh takash@ma.oda.tus.ac.jp Receved: 3/9/3, Revsed: /4/3, Accepted: /8/4, Publshed: 3/4/4 Abstract I ths paper, we study the well-kow Mk ad Matyasevch covoluto dettes for Beroull umbers ad deduce aalogues of ther dettes for umbers related to Beroull umbers.. Itroducto The Beroull umbers B appear may areas of mathematcs, most otably umber theory, the calculus of fte d ereces ad asymptotc aalyss wth mportat applcatos. They ca be defed by the geeratg fucto e X B! ( < ). (.) Oe ca easly see that B k+ ad ( ) k+ B k > for all k. Let Z p be the rg of p-adc tegers wth p a prme. The vo Staudt-Clause theorem asserts that B Z p f p -, ad pb Z p, more precsely pb (mod p) f p. Varous types of lear ad olear recurrece relatos for Beroull umbers have bee studed for a log tme. We ca fd a large umber of formulas the classcal books [7, 8] ad []. For specal type recurrece ad recprocty relatos for these umbers, see, e,g., [,, 3, 8, ]. As oe of may covoluto dettes for Beroull umbers, Mk [6] proved 978 the followg curous detty based o p-adc argumets. The author was partally supported by the Mstry of Educato, Scece, Sports ad Culture, Grat--Ad for Scetfc Research (C).

2 INTEGERS: 4 (4) Proposto. (Mk, 978). For 4, B B B B H B, (.) where H Sce ( ) + s the th harmoc umber., we may rewrte (.) as the form B B B B H B. (.3) I 5, Mk s detty was eteded by Gessel [] to the Beroull polyomals B ( ) ( ) defed by e e X B ( )! ( < ). Ideed, he proved the followg Proposto. (Gessel, 5). For, B ( )B ( ) ( ) B ( )B H B ( ) + B ( ). (.4) Sce B () B, detty (.) s gve as a specal case of (.4) where. Also the case / reduces to Faber ad Padharpade s detty show [7] (see also [6] observg t from a qute d eret vewpot) for the umbers defed by B (( )/ )B. Ideed, we have B B B B B O the other had, Matyasevch [5] dscovered the followg good compao detty for (.) wth the ad of computer software system Mathematca : Proposto.3 (Matyasevch, 997). For 4, + ( + ) B B B B ( + )B. (.5) There are several kds of proofs of (.), (.4) ad (.5) usg some tools from combatorcs, cotour tegrals, p-adc aalyss ad others, however we ow pay partcular atteto to Crabb s short ad tellgble proof of (.4) gve [5]. To.

3 INTEGERS: 4 (4) 3 prove (.4), he used a certa fuctoal equato related to the geeratg fucto of B ( ) stated above. Ths paper s cocered wth the Mk ad Matyasevch covoluto dettes. I Secto, as a prelmary we frst epla a umbral otato ad later troduce Euler-type dettes ad some requred fuctoal dettes. I Secto 3, we rederve (.) ad (.5) wth elemetary ad shorter proofs based o essetally the same dea as Crabb s. I Secto 4, we deal wth aalogues of ther covoluto dettes for the umbers B ( )B defed by the geeratg fucto e + X B! ( < ). (.6) We ote that these umbers B are closely related to the Geocch umbers defed by G ( )B B ( ). Therefore, all the results cocerg B gve below ca be epressed terms of Geocch umbers.. Prelmary I ths secto, we prepare some matters whch wll be eeded the later sectos. Gve two sequeces S {S } ad T {T } of umbers or fuctos, we use the followg umbral otato (for more detals o umbral calculus, see [9, 9,, ]). We ow defe for ay u, v R, (us + vt ) S T, (us + vt ) u v S T ( ). I other words, we epad (us +vt ) full by meas of the bomal theorem ad replace S ad T by S ad T (,,..., ), respectvely. For eample, we may wrte the most basc detty whch s usually attrbuted to Euler as, cosderg the sequece B {B }, (B + B) (B ), (B + B) ( )B B ( ). (.) If S() ad T () are the epoetal geeratg fuctos of the sequeces S ad T, respectvely, the we have S(u)T (v) X (us + vt )!, ad hece the followg dervatve epresso ca be gve: apple d (us + vt ) d S(u)T (v).

4 INTEGERS: 4 (4) 4 As we metoed above, we use the otato B ( )B ( ). It s easy to costruct the followg fuctoal equatos from the geeratg fuctos (.) ad (.6) of B ad B, respectvely: e e + e + e e + ( ) e + + e + e, d d e. + Usg these, we ca mmedately produce aalogues of Euler s detty (.) for the sequeces B {B } ad B {B } by the dervatve method. Proposto.. For, we have (B + B ) B B B, (.) (B + B ) B + ( )B. (.3) We wll observe aga these dettes from a d eret drecto Secto 4. For, R, we preset the followg three types of ratoal fucto dettes: (a) X X X + + X + (,, + 6 ), X (b) X + X + X + X ( + 6 ), + X + (c) X X + X X ( 6 ). X + These dettes ca be easly cofrmed by drect calculatos. I the followg sectos, we utlze them to costruct some requred fuctoal equatos related to the geeratg fuctos (.) ad (.6) of B ad B, respectvely. 3. The Mk ad Matyasevch Idettes I ths secto, we frst preset a elemetary ad shorter proof of Proposto. by applyg a certa fuctoal equato costructed usg (a), whch s a slghtly modfed verso of Crabb s proof. Subsequetly, we gve a very short proof of Proposto.3 ad later we dscuss other types of covoluto dettes. I what follows, we assume that 4 ad s eve. Otherwse, both sdes of (.) vash because B k+ for all k ad so t s meagless. Proof of Proposto.. Put t, t ad X e (a), ad multply

5 INTEGERS: 4 (4) 5 both sdes by t( t). The we establsh the fuctoal equato t e t ( t) e ( t) t e t( t) + ( t) e t ( t) + t. e ( t) (3.) D eretate (3.) -tmes wth respect to by applyg Lebz s rule ad put. The we get (tb + ( t)b) t( t)b + ( t)(b + tb) + t(b + ( t)b), amely, t ( t) B B t( t)b ( t) t B B + t ( t) B B. (3.) If we gather the terms volvg B ( B B ), the, otg that B for a eve 4, above (3.) becomes t ( t) B B t + ( t) + B ( t)t + t( t) B B. (3.3) Dvde (3.3) by t( t) ad tegrate t betwee ad wth respect to t. The, makg use of the easly show formulas Z Z t m ( t) k m!k! dt (m + k + )! t m+ ( t) m+ dt t( t) Z (m, k ), t m t dt H m (m ), (3.4) we obta, sce B B ( ) ( )!( )! ( )! ( ) +, H B B B B B ( ). Ths s eactly the same as (.) f we dvde both sdes by.

6 INTEGERS: 4 (4) 6 It s also possble to deduce (.) by dvdg (3.3) by t ad tegratg betwee ad wth respect to t. Ideed, f we dvde (3.3) by t, the we have t t ( t) + B B ( t) B t t + t( t) B B. Itegratg ths betwee ad, we get, usg (3.4), B B + H + B + B B, whch s equvalet to (.3) (hece to (.)), sce H + H + /( + ). As a further applcato of (3.3), we ca easly reestablsh Matyasevch s detty (.5) wth a smpler ad shorter proof. Proof of Proposto.3. Itegratg drectly both sdes of (3.3) from to, we obta, usg (3.4), X B B + + B X B B ( + )( + ). Multply both sdes by ( + )( + ) to obta (.5). The followg dettes are also easy cosequeces from (3.3). Proposto 3.. For 4, we have B B ( + )B (Euler), (3.5) + B B B, (3.6) B B ( + )B, (3.7) 3 ( + )B B ( )B. (3.8) Proof. For Euler s detty (3.5), we have oly to dvde (3.3) by t ad put t. For (3.6), d eretate both sdes of (3.3) wth respect to t oly oce ad put t /. The we get ( )B B,

7 INTEGERS: 4 (4) 7 ad hece (3.6) s gve by usg (3.5). Net, puttg t / (3.3) ad multplyg t by, we have B B + ( )B B B, whch gves (3.7) by usg (3.6). Smlarly, f we put t /3 (3.3) ad multply t by 3 +, the 3 B B (3 + + )B whch leads to (3.8) usg (3.7). Ths completes the proof. 3 ( + )B B, The dettes (3.7) ad (3.8) ca also be obtaed by puttg t, 3 (3.3). 4. Aalogues of Propostos. ad.3 I ths secto, we study aalogues of the Mk ad Matyasevch covoluto dettes for the umbers B ( )B by the same argumets as that performed Secto 3. We frst prove the followg aalogues of Mk s detty (.): Proposto 4.. For 4, we have B B B B ( ) B + B B B ( ) B, (4.) H B. (4.) Proof. We frst prove (4.). Put t, t ad X e (b) ad multply t by t( t). The we get the fuctoal equato t e t + ( t) e ( t) + t e t( t) ( t) e t + t ( t). e ( t) + (4.3) If we d eretate (4.3) -tmes wth respect to ad put, the we obta for the sequeces B {B } ad B {B }, (tb + ( t)b ) t( t)b ( t)(b + tb ) t(b + ( t)b ),

8 INTEGERS: 4 (4) 8 amely, t ( t) BB t( t)b ( t)t + t( t) B B. Cosderg the obvous facts B, B ad B B for a eve 4, ths detty mples Dvdg (4.4) by t( t ( t) BB + (( t)t + t( t) ) B t), we have ( t)t + t( t) B B. t ( t) B B + t + ( t) B t + ( t) B B. (4.4) Itegratg ths betwee ad wth respect to t ad dvdg t by, we deduce, usg the frst formula (3.4), whch gves, sce ( B B ( ) + B ) B B +, + B B B, B B. Ths mples (4.) f we dvde ths by ad replace by. For the proof of (4.), we put t, t ad X e (c) ad multply t by t( t) to get the fuctoal equato t e t ( t) e ( t) + t e t( t) + ( t) + e t t ( t). e ( t) + (4.5)

9 INTEGERS: 4 (4) 9 D eretatg (4.5) tmes wth respect to ad puttg, we obta (tb + ( t)b ) t( t)b + ( t)(b + tb) t(b + ( t)b ), whch mples t ( t) B B t( t)b ( t) t B B t ( t) B B. Sce B, B ad B B, we ca wrte ths as t ( t) B B (( t) ( t) )B ( t) t B B t Dvdg (4.6) by t( t), we have t ( t) B B t B B ( t) B B. ( t) B t ( t) BB. (4.6) Smlarly to the above, tegratg betwee ad wth respect to t, we have, usg both formulas (3.4), B B B H B (B B ) ( ) B B, whch yelds (4.) dvdg by ad replacg by o the rght-had sde. Net, we deduce aalogues of Matyasevch s detty (.5) by makg aga use of (4.4) ad (4.6). Proposto 4.. For 4, we have + ( + ) BB + B B B, (4.7) + ( + ) B B B ( )( + ) B B. (4.8)

10 INTEGERS: 4 (4) Proof. Drectly tegratg (4.4) betwee ad wth respect to t, we have X + B B + ( + )( + ) B B B ( + )( + ). By the same argumets as doe above, we obta from (4.6) X B B B + + B (B B ) ( + )( + ) X + B ( + )( + ) B. Multplyg (4.9) ad (4.) by ( + )( + ), we get the dettes dcated. (4.9) (4.) Euler-type dettes (.) ad (.3) Proposto. ca also be deduced from (4.4) ad (4.6), respectvely. Ideed, f we dvde (4.4) ad (4.6) by t (or t) ad put t (or t ), the we have for 4, B B B, (4.) BB ( )B, (4.) whch are equvalet to (.) ad (.3), respectvely. Sce B ( )B, subtractg (4.) from (4.), we get BB B. As easly see, t s also possble to derve ths detty multplyg (3.5) by ad subtractg t from (3.7). At the ed of ths paper, we would lke to meto that may terestg dettes related to Beroull, Euler ad other polyomals obtaed by usg umbral calculus ad p-adc tegral o Z p ca be foud [4, 3, 4]. Ackowledgmets. I thak K. Dlcher for hs careful readg of the frst verso of ths paper ad the aoymous referee for brgg some refereces o umbral calculus ad others to my atteto.

11 INTEGERS: 4 (4) Refereces [] T. Agoh, Recurreces for Beroull ad Euler umbers ad polyomals, Epo. Math. 8 (), [] T. Agoh ad K. Dlcher, Covoluto dettes ad lacuary recurreces for Beroull umbers, J. Number Theory, 4 (7), 5. [3] T. Agoh ad K. Dlcher, Recprocty relatos for Beroull umbers, Amer. Math. Mothly, 5 (8), [4] A. Bayad ad T. Km, Idettes volvg values of Berste, q-beroull, ad q-euler polyomals, Russ. J. Math. Phys. 8 (), [5] M. C. Crabb, The Mk-Gessel Beroull umber detty, Glasg. Math. J. 47 (5), [6] G. V. Due ad C. Schubert, Beroull umber dettes from quatum feld theory, IHES preprt P/4/3, 4 ( [7] C. Faber ad R. Padharpade, Hodge tegrals ad Gromov-Wtte theory, Ivet. Math. 39 (), [8] M.B. Gelfad, A ote o a certa relato amog Beroull umbers ( Russa), Bashkr. Gos. Uv. Uche. Zap. Ser. Mat. 3 (968), 5 6. [9] I. M. Gessel, Applcatos of the classcal umbral calculus, Algebra Uversals, 49 (3), [] I. M. Gessel, O Mk s detty for Beroull umbers, J. Number Theory, (5), [] H. W. Gould ad J. Quatace, Beroull umbers ad a ew bomal trasform detty, J. Iteger Seq. 7 (4), Art [] E. R. Hase, A Table of Seres ad Products, Pretce-Hall, Eglewood Cl s, NJ, 975. [3] T. Km, D. S. Km, T. Masour, S. -H. Rm ad M. Schork, Umbral calculus ad She er sequeces of polyomals, J. Math. Phys. 54 (3), [4] D. S. Km ad T. Km, Some dettes of Beroull ad Euler polyomals arsg from umbral calculus, Adv. Stud. Cotemp. Math. (Kyugshag), 3 (3), [5] Y. Matyasevch, Idettes wth Beroull umbers, Joural/Beroull/berull.htm, 997. [6] H. Mk, A relato betwee Beroull umbers, J. Number Theory, (978), [7] N. Nelse, Traté élémetare des ombres de Beroull, Gauther-Vllars, Pars, 93. [8] L. Saalschütz, Vorlesuge über de Beroullsche Zahle, hre Zusammehag mt de Secate-Coe cete ud hre wchtgere Aweduge, Verlag vo Julus Sprger, Berl, 893. [9] T. I. Robso, Formal calculus ad umbral calculus, Electro. J. Comb. 7 (), #R95, 3. [] S. Roma, More o the umbral calculus, wth emphass o the q-umbral calculus, J. Math. Aal. Appl. 7 (985), 54. [] S. Roma, The Umbral Calculus, Dover Publ. Ic. New York, 5.