Zhi-Wei Sun and Hao Pan (Nanjing)

Transcription

1 Aca Arih. 5(006, o., 39. IDENTITIES CONCERNING BERNOULLI AND EULER POLYNOMIALS Zhi-Wei Su ad Hao Pa (Najig Absrac. We esabish wo geera ideiies for Beroui ad Euer poyomias, which are of a ew ype ad have may cosequeces. The mos sriig resu i his paper is as foows: If is a posiive ieger, r + s + ad x + y + z, he we have where [ s r r + s x y y z s x y : ] + z x 0 ( ( s ( B (xb (y. I is ieresig o compare his wih he foowig propery of deermias: r s x y + s r y z + z x 0. Ouymmeric reaio impies he curious ideiies of Mii ad Maiyasevich as we as some ew oes for Beroui poyomias such as ( B (xb (x ( ( + B (xb.. Iroducio Le N {0,,,... } ad Z + {,, 3,... }. The we-ow Beroui umbers B ( N are raioa umbers defied by B 0 ad ( + B 0 ( Z Mahemaics Subjec Cassificaio. Primary B68; Secodary 05A9. The firs auhor is resposibe for commuicaios, ad pariay suppored by he Naioa Sciece Fud for Disiguished Youg Schoars (o ad a Key Program of NSF (o i Chia.

2 Z. W. SUN AND H. PAN Simiary, Euer umbers E ( N are iegers give by E 0 ad ( E 0 ( Z +. For N he Beroui poyomia B (x ad he Euer poyomia E (x are as foows: B (x ( B x ad E (x ( ( E x. Ceary B (0 B ad E (/ E /. Here are some basic properies of Beroui ad Euer poyomias we wi eed aer. B ( x ( B (x, (B (x x ; E ( x ( E (x, (E (x x. (The operaors ad are defied by (f(x f(x + f(x ad (f(x f(x + + f(x. Aso, B +(x ( + B (x ad E +(x ( + E (x. For a sequece {a } N of compex umbers, is dua sequece {a } N is give by a ( ( a ( N. I is we ow ha a a. I 003 Z. W. Su [S] deduced some combiaoria ideiies i dua sequeces. The sequeces {( B } N ad {( E (0} N are boh sef-dua sequeces (cf. [S], aer we wi mae use of his fac. I 978 H. Mii [Mi] discovered he foowig curious ideiy: B B ( ( B B ( H B for every 4, 5,..., where H + / + + /. I 997 Y. Maiyasevich [Ma] foud aoher ideiy of his ype: ( + ( + B B B B ( + B for ay 4, 5,.... These wo ideiies are of a deep aure. I fac, a ow proofs of hese ideiies by ohers are compicaed (cf. [Mi], [G] ad [DS]; for exampe, he approach of G. V. Due ad C. Schuber [DS] was eve moivaed by quaum fied heory ad srig heory.

3 IDENTITIES CONCERNING BERNOULLI AND EULER POLYNOMIALS 3 Recey he auhors [PS] preseed a ew mehod o hade such ideiies. Though heir approach oy ivoves differeces ad derivaives of poyomias, hey were abe o use he powerfu mehod o exed Mii s ad Maiyasevich s ideiies o ideiies cocerig he sums B (xb (y ad B (x B (y B (x B (y + B (y B (x (where is a posiive ieger. They aso haded simiaums reaed o Euer poyomias. Le be ay posiive ieger. As usua, ( z z(z (z + /! (ad ( z 0 eve if z N. Observe ha B (xb (y ( ( B (xb (y ad B (x B (y ( ( B (x B (y ( ( B (xb (y. im 0 Ispired by his observaio, here we ivesigae reaios amog he sums s ( P (xq (y wih P, Q {B, E}. Our cera resu is he foowig heorem. Theorem.. Le Z + ad x + y + z. (i If r + s +, he ( B (xe (z r ( ( B (ye (z r s ( E (ye (x. (.

4 4 Z. W. SUN AND H. PAN (ii If r + s +, he we have he symmeric reaio s r r + s + 0 (. x y y z z x where s : x y s ( B (xb (y. (.3 Remar.. I is ieresig o compare (. wih he foowig propery of deermias: 0 z x y r s x y + s r y z + z x. I view of K. Dicher s paper [D], he referee hough ha Theorem. migh have a geeraizaio ivovig sums of producs of m Beroui or Euer poyomias. Bu we are uabe o obai a compac exesio of Theorem. hough we have made a serious aemp. Coroary.. Le Z + ad e α, x, y be parameers. The α + + ( α + E (xe (y ( ( ( α + + α + ( B (x B (y E (x y ad ( α + (α + + B (xb (y ( α + + (α + ( B (xb (x y α + + α + + B (yb (x y. (.4 (.5 Proof. Le x x ad z x y. The x + y + z. Appyig Theorem.(i wih r α + +, s ad α we he ge (.4. (Noe ha ( ( ( z z+. By Theorem.(ii, α (α + + x y α α + + α (α +. y x y x y x

5 IDENTITIES CONCERNING BERNOULLI AND EULER POLYNOMIALS 5 This is a equivae versio of (.5. Remar.. (.5 i he case α x y 0 yieds Maiyasevich s ideiy sice B + 0 for,, 3,.... Coroary.. Le > 0 be iegers. The + δ,0 E + (xe (y + + (( B +(x B +(y E (x y + (.6 (where δ,m aes or 0 accordig o wheher m or o, ad B + (xb (y + + (( B +(x + B +(y B (x y. + I paricuar, ( + ( + 8 ( + δ,0 ( + + (.7 E + (xe (x + B + (x ( + (.8 B + ad ( ( B + (xb (x + ( ( + + B + (xb. (.9 Proof. As ( + + ( + ad ( x + y + (x y,

6 6 Z. W. SUN AND H. PAN by Theorem.(i we have + ( B ( xe + (x y + + ( + ( B (ye + (x y + δ,0 ( ( ( E (ye + ( x + ( E (ye + ( x + δ,0 which ca be reduced o (.6. (.8 foows from (.6 i he case y x sice (( m E m (0 4( m+ B m+ /(m + for m,, 3,.... (I is ow ha (m+e m (x (B m+ (x m+ B m+ (x/ (cf. [AS] ad [S]. I igh of Theorem.(ii, ( [ x y ] + + y x y x y x + This is equivae o (.7. I he case y x, (.7 gives (.9 because (( m + B m B m for m 0,, 3,.... Remar.3. Puig 0 ad x / i (.8 ad oig ha B (/ ( B (see, e.g., [AS] ad [S], we he ge he foowig ideiy: ( + 8 ( ( + + E E + ( ( + B B + for ay Z +. Simiary, (.9 i he case x 0 yieds he foowig ew ideiy: ( B B for every 4, 5,.... ( ( + B B The foowig heorem ca be deduced from Theorem.. ( B

7 IDENTITIES CONCERNING BERNOULLI AND EULER POLYNOMIALS 7 Theorem.. Le, m, Z +, mi{m, } ad x + y + z. The ( m m ( m + ( ( m ( ( + ( ( m + B ++ (xe m (z B m ++ (ye (z ( E + (xe m (y (.0 ad m ( B m (xe + (z m m + ( m B m (ye + (z ( m m + E (ye m + (x. We aso have ( m m m δ,m ( m + ( ( m ( + ( ( m + ( m ( B + (xb m (z B m + (yb (z ( B + (xb m (y. (. (. Coroary.3 (Woodcoc [W]. Le m, Z +. The m m ( m ( B m B + ( ( B B m +. Proof. Simpy ae x y 0 ad z i (.. From Theorem. we ca aso deduce he foowig resu.

8 8 Z. W. SUN AND H. PAN Theorem.3. Le Z +, ad e, x, y, z be parameers wih x+y+z. The we have ( ( E (xe (y ( ( ( E (z B (xe (z + B (y ad ( ( (.3 ( ( E (x E (y ( E (z ( B (y ( B (x E (z E (z. (.4 Aso, ( ( ( B (x B (y B ( (z ( B (y ( B (z + ( B (x B (z. Coroary.4. Le Z + ad x + y + z. The ( + (( B (x B (y E (z + ( E (xe (y, ( B (x E (z ( ( B (y E (z ( E (ye (x H E (z ad ( + ( + ( B (x B (xb (y + B (z ( + ( B (y B (z + ( H ( + B (z. (.5 (.6 (.7 (.8

9 IDENTITIES CONCERNING BERNOULLI AND EULER POLYNOMIALS 9 Proof. Taig i Theorem.3 we immediaey ge (.6-(.8. Coroary.5. Le Z + ad x + y + z. The ( E (x E (y + H E (y ( E (z ( B (y + ( H E (zb (x ad ( ( E (x E (z + ( B (y E (y + H ( B (y E (z + We aso have ( B (x (B (y + ( B (z ( B (y B (z ( B (x H E (z. H B (y + ( B (z (.9 (.0 (. Remar.4. I he case x y 0 ad z, (. yieds Mii s ideiy. The ex secio is devoed o proofs of Theorems. ad.. Theorem.3 ad Coroary.5 wi be proved i Secio 3.. Proofs of Theorems.. Lemma.. Le P (x, Q(x C[x] where C is he fied of compex umbers. (i We have ad (P (xq(x P (x + (Q(x + (P (xq(x (. (P (xq(x P (x + (Q(x (P (xq(x. (. (ii If (P (x (Q(x, he P (x Q (x. If (P (x (Q(x, he P (x Q(x. Proof. The firs par ca be verified easiy. Par (ii is Lemma 3. of [PS]. The foowig emma has he same favor wih Theorem. of Su [S].

10 0 Z. W. SUN AND H. PAN Lemma.. Le {a } be a sequece of compex umbers, ad {a } be is dua sequece. Se A ( ( ( a ad A ( ( ( a (.3 for 0,,,.... Le Z +, r + s + ad x + y + z. The ( (( ( ( x A (y ( A (z 0. (.4 Proof. By Remar. of Su [S], ( A (z A (x + y ( x A (y. Therefore r ( x A (z r x x A (y ( r ( r x A (yc ( x A (y ( ( r where c r ( ( r + Thus (.4 foows. ( r + (by Vadermode s ideiy ( s (. Remar.. If we e a ( B for 0,,,..., he A ( A ( B (. Aso, A ( A ( E ( if a ( E (0 for 0,,,.... Proof of Theorem.. We fix y ad view z x y as a fucio i x.

11 IDENTITIES CONCERNING BERNOULLI AND EULER POLYNOMIALS (i Se P (x ( B (xe (z. The, by Lemma., (P (x coicides wih where ( (B (xe (z (B ( (x + (z x E (z ( (z B (x + + rσ Σ r s ( x E (z. r s ( x E (z. ( Appyig Lemma. ad Remar. we obai ha r (P (x ( ( (z B (y s + r ( x E (y. I foows ha (P (x (Q(x where r Q(x ( ( B (ye (z + r s ( E (ye (x. Thus P (x Q(x by Lemma.. (.. (ii Se P (x z x This is equivae o he desired ( B (xb (z.

12 Z. W. SUN AND H. PAN By Lemma., (B (xb (z (B (xb (z + B (x + (B (z for every 0,,...,. Thus x B (z ( B (x + (z (P (x rr(x s ( B (x + (z where r s R(x ( x B (z s r ( ( x B (z. Appyig Lemma. ad Remar. we obai ha (P (x r ( ( s s( ( x B (y r ( (z B (y I foows ha (P (x (Q (x where Q (x r s ( B (xb (y ( s r ( B (zb (y. r s ( B (xb (y s r ( B (zb (y. Thus P (x Q (x by Lemma..

13 IDENTITIES CONCERNING BERNOULLI AND EULER POLYNOMIALS 3 Observe ha P (x coicides wih ( B (xb (z ( ( B (xb (z r ( + s ( + B (xb (z + ( ( B (xb (z r ( s (r + (s + + B (xb (z ( z x ad s Q (x r ( B (xb (y r + s ( B (zb (y s r r s. x y y z Thus he equaiy P (x Q (x gives ha s r r + s + x y y z z x where (r + s. Repacig by we he obai he required ideiy (.. This cocudes he proof. Proof of Theorem.. Ceary m + Z +. By Theorem.(i, + ( m B (xe + (z + + ( + ( B (ye + (z + m ( E (ye (x. 0

14 4 Z. W. SUN AND H. PAN Tha is, m ( + m B + (xe (z + ( B + (ye (z + m m ( m E m (ye + (x + ( m ( m ( ( E m (ye + (x. Therefore (.0 foows. By Theorem.(i we aso have m ( B (xe (z m ( ( B (ye (z m ( E (ye (x m δ,m ( ( ( E (ye m+ (x m + m ( ( ( E (ye m + (x, m + δ,m which gives (. afer few rivia seps. I igh of Theorem.(ii, m m x y y z Tha is, m + z x m m ( m B + (xb m (y + m ( B m + (yb (z m + m m + ( B (zb (x. This is equivae o (.. We are doe..

15 IDENTITIES CONCERNING BERNOULLI AND EULER POLYNOMIALS 5 3. Proofs of Theorem.3 ad Coroary.5 Lemma 3.. Le be a oegaive ieger ad s be a parameer. The im 0 (( s + ( s ( s 0 < s. (3. I paricuar, im 0 (( ( ( H. (3. Proof. Observe ha ( s + ( s s + 0 < s ( s ( +. s 0 < So (3. foows. I he case s, (3. urs ou o be (3.. Proof of Theorem.3. (. i he case s yieds ha ( ( B (xe (z ( ( B (ye (z ( E (xe (y. For each 0,,..., we ceary have ( ( ( + ( + ( + ( +. Therefore ( ( E (xe (y ( ( ( ( B (y + E ( (z B (xe (z E (z B (y.

16 6 Z. W. SUN AND H. PAN This proves (.3. Now we come o prove (.4 ad view s r as a fucio i r. I igh of (., s ( E (ye (x ( (( ( ( E (z B (x ( B (y r ( ( (( ( r s E (z B (x ( B (y + ( E (z ( ( ( s. r By Lemma 3., im r 0 ( ( s ( r ( im r 0 r (( r + ( ( As i he proof of (.3, we aso have ( ( ( + + (. for every 0,,...,. Thus, by eig r 0 we ge from he above ha ( ( ( E (ye (x ( E (z + + (( ( B (x E (z ( B (y, which is equivae o (.4. Now we ur o prove (.5. Le us view s r as a fucio i r. The ] ] im r 0 [ s x y [ x y ( ( ( ( ( B (x B (y. + B (xb (y

17 IDENTITIES CONCERNING BERNOULLI AND EULER POLYNOMIALS 7 O he oher had, im r 0 ( s r r y z ( ( + z x ( B (y B (z ( B (x B (z + ( B (zr where ( ( ( r R im ( r + ( r 0 r ( ( ( ( r + im ( r 0 r (( ( ( r + im r 0 r ( ( (. Appyig (. we he ge (.5 from he above. The proof of Theorem.3 is ow compee. Proof of Coroary.5. We ca easiy ge (. by cacuaig he imiaio of he ef had side of (.5 mius he righ had side of (.5 as eds o 0. Thus i remais o show (.9 ad (.0. (.3 ca be rewrie i he form ( ( E (x E (y + ( E (y (( ( B (xe (z + ( B (xe (z + ( ( B (y ( E (z + B (y. Leig 0 we ge ha ( B (xe (z + ( B (y ( E (y. (3.3

18 8 Z. W. SUN AND H. PAN Thus ( ( E (x E (y (( ( + ( ( E (z B (y. Leig 0 we he have ( im 0 im 0 B (xe (z + B (y ( E (xe (y + ( ( ( ( ( Observe ha ( ( ( ( Therefore ( ( ( ( H B (y ( B (y H + (( ( ( E (z B (y B (xe (z + B (y ( H. ( ( im 0 ( 0<< ( E (xe (y + ( + H ( B (xe (z ( ( ( (H H. ( E (z B (y ( (H H B (xe (z ( H E (zb (x ( H E (y + This proves (.9. ( H E (zb (x.

19 IDENTITIES CONCERNING BERNOULLI AND EULER POLYNOMIALS 9 We ca reformuae (.4 as foows: ( ( ( E (z E (x E (y + ( E (z ( ( E (y ( ( ( B (y E (z + B (y ( (( ( B (x E (z ( B (x E (z. I view of (3.3, ( B (x E (z ( B (x E (z ( ( B (y + E (y E (z. Therefore ( ( E (x E (y ( E (z ( ( ( B (y E (z ( ( + B(x E (z ( E (y + ( E ( (z (.

20 0 Z. W. SUN AND H. PAN Leig 0 we obai ha ( ( E (x E (y E (z H ( B (y I foows ha + E (z ( ( + H ( ( B (x E (y + E (z. E (z ( ( E (x E (y + ( B (y ( (H H B (x E (z + H ( ( ( E (y + E (z B (x H E (z + H R E (z where R ( B (x E (z + ( E (y + E (z (E (z + ( B (y (by (3.3. This proves (.0. We are doe. Acowedgme. commes. The auhors ha he referee for his/her hepfu Refereces [AS] M. Abramowiz ad I. A. Segu (eds., Hadboo of Mahemaica Fucios, Dover Pubicaios, New Yor, 97. [D] K. Dicher, Sums of producs of Beroui umbers, J. Number Theory 60 (996, 3 4.

21 IDENTITIES CONCERNING BERNOULLI AND EULER POLYNOMIALS [DS] G. V. Due ad C. Schuber, Beroui umber ideiies from quaum fied heory, prepri, 004, arxiv:mah.nt/ [G] I. M. Gesse, O Mii s ideiy for Beroui umbers, J. Number Theory 0 (005, [Ma] Y. Maiyasevich, Ideiies wih Beroui umbers, 997, hp://ogic.pdmi. ras.ru/ yuma/joura/beroui/berui.hm. [Mi] H. Mii, A reaio bewee Beroui umbers, J. Number Theory 0 (978, [PS] H. Pa ad Z. W. Su, New ideiies ivovig Beroui ad Euer poyomias, J. Combi. Theory Ser. A 3 (006, [S] Z. W. Su, Iroducio o Beroui ad Euer poyomias, a a give a Taiwa, 00, hp://pweb.ju.edu.c/zwsu/bere.pdf. [S] Z. W. Su, Combiaoria ideiies i dua sequeces, Europea J. Combi. 4 (003, [W] C. F. Woodcoc, Covouios o he rig of p-adic iegers, J. Lodo Mah. Soc. 0 (979, Deparme of Mahemaics Najig Uiversiy Najig 0093 Peope s Repubic of Chia E-mai: (Zhi-Wei Su zwsu@ju.edu.c (Hao Pa haopa79@yahoo.com.c