A talk given at Institut Camille Jordan, Université Claude Bernard Lyon-I. (Jan. 13, 2005), and University of Wisconsin at Madison (April 4, 2006).

Transcription

1 A tal give at Istitut Caille Jorda, Uiversité Claude Berard Lyo-I (Ja. 13, 005, ad Uiversity of Wiscosi at Madiso (April 4, 006. SOME CURIOUS RESULTS ON BERNOULLI AND EULER POLYNOMIALS Zhi-Wei Su Departet of Matheatics Najig Uiversity Najig 10093, P. R. Chia E-ail: Hoepage: Abstract. I this tal we tell the story how the developets of soe curious idetities cocerig Beroulli (ad Euler polyoials fially led to the followig uified syetric relatio (of Z. W. Su ad H. Pa: If is a positive iteger, r + s + t ad x + y + z 1, the we have where [ ] [ s t t r r + s x y y z [ ] s t x y : ] [ ] r s + t z x 0 ( 1 ( s ( t B (xb (y. It is iterestig to copare this with the easy idetity r s t 0 r s t z x y r s x t y + s t y r z + t r z s x. We will also tal about soe cogrueces for Euler ubers ad q- Euler ubers. All papers of the speaer etioed i this survey are available fro his hoepage 1

2 ZHI-WEI SUN 1. Vo Ettigshause s idetity ad its geeralizatios Let N {0, 1,,... } ad Z + {1,, 3,... }. The well-ow Beroulli ubers B ( N are ratioal ubers defied by B 0 1 ad ( + 1 B 0 ( Z +. Siilarly, Euler ubers E ( N are itegers give by E 0 1 ad ( E 0 ( Z +. Beroulli ubers ad Euler ubers ca also be give by ad 0 0 B x! x e x 1 E x! ex e x + 1 ( e x 1 ( 1 x 0 ( e x + e x 1 ( x 1 ( x < π ( + 1! 0 x 1 ( x < π. (! It is well ow that B 3 B 5 0 ad E 1 E 3 E 5 0. For N the Beroulli polyoial B (x ad the Euler polyoial E (x are as follows: B (x ( B x ad E (x ( ( E x 1. Clearly B (0 B ad E (1/ E /. Here are soe well-ow properties of Beroulli ad Euler polyoials. B (1 x ( 1 B (x, B +1(x ( + 1B (x; E (1 x ( 1 E (x, E +1(x ( + 1E (x.

3 CURIOUS RESULTS ON BERNOULLI AND EULER POLYNOMIALS 3 I a boo of vo Ettigshause published i 187, the author obtaied that we ca copute B i ters of B, B +1,..., B 1 by the recursio: ( + 1 ( + + 1B + 0 ( 1,,.... (1.1 With the help of cotiued fractios, i 1995 M. Kaeo [Proc. Japa Acad. Ser. A. Math. Sci. 71(1995, ] rediscovered this. (The speaer thas Prof. T. Agoh for his iforig e that Kaeo repeated vo Ettigshause s discovery. I 001, by eployig certai itegrals over Z p, H. Moiyaa [Fiboacci Quart. 39(001, 85-88] exteded the vo Ettigshause idetity i the followig syetric for: ( + 1 ( 1 ( + + 1B + ( ( ( 1 ( + + 1B + providig that, N ad + > 0. I Noveber 001, the speaer foud Moiyaa s paper ad ased y studets to provide a iductio proof of (1. ad exted it to Beroulli polyoials. Soo, Hao Pa, oe of y studets, proved (1. by iductio The, o Dec. 1, 001, the speaer succeeded i givig the polyoial for of (1.: ( + 1 ( 1 ( + + 1B + (x ( ( 1 ( + + 1B + ( x ( 1 ( + + 1( + + x +. (1.3

4 4 ZHI-WEI SUN This result appeared i the paper [K. J. Wu, Z. W. Su ad H. Pa, Fiboacci Quart. 4(004, 95-99]. O Dec. 7, 001 the speaer obtaied the followig result ore geeral tha (1.3: ( 1 providig x + y + z 1. ( x B + (y ( 1 ( x B + (z (1.4 Now let e explai how (1.4 was foud origially. Let, N. The, for ay h Z + we have As we have ( 1 ( 1 ( 1 ( h 1 (x + r + a r0 h 1 ( 1 (x + r (x + r + a r0 h 1 ( 1 (x + h 1 s (x + h 1 s + a ( 1 r0 s0 ( a h 1 ( x h + 1 a + s +. s0 h 1 (x + r + B ++1(x + h B ++1 (x, ( ( a B ++1(x + h B ++1 (x a B ++1( x a + 1 B ++1 ( x a + 1 h, + + 1

5 CURIOUS RESULTS ON BERNOULLI AND EULER POLYNOMIALS 5 i.e., f(a, x + h does ot deped o h Z + where ( f(a, x ( 1 a B ++1(x ( + ( 1 a B ++1(1 a x Therefore f(a, x f(a, 0 ad hece xf(a, x 0 which gives the idetity ( 1 ( a B + (x ( 1 O Dec. 10, 00, with help of the beta fuctio B(a, b 1 0 ( a B + (1 a x. x a 1 (1 x b 1 dx Γ(aΓ(b Γ(a + b, the speaer got the followig result: If, N ad x + y + z 1, the ( ( 1 x B ++1(y ( + ( 1 x B ++1(z ( ( x ++1 ( ( + we ca also replace Beroulli polyoials i (1.5 by correspodig Euler polyoials. If we tae partial derivative of (1.5 with respect to y ad view z 1 x y as a fuctio of y, we the obtai (1.4. For a sequece {a } N of coplex ubers, its dual sequece {a } N are give by a a, ( ( 1 a ( N. It is well ow that a. The sequeces {( 1 B } N ad {( 1 E (0} N are both self-dual sequeces. I Deceber 001, the speaer obtaied the followig geeral result.

6 6 ZHI-WEI SUN Theore 1.1 [Z. W. Su, Europea J. Cobi. 4(003, ]. Let {a } N be a sequece of coplex ubers. For N let ( A (x ( 1 a x ad A (x ( ( 1 a x. If, N ad x + y + z 1, the we have the idetity ( ( 1 x A ++1(y ( + ( 1 x A ++1 (z cosequetly ad ( 1 ( x ++1 a 0 ( ( +, ( x A + (y ( 1 ( + 1 ( 1 ( + + 1x +1 A + (y ( ( 1 x +1 ( + + 1A +(z (1.6 ( x A +(z (1.7 ( + + ( ( 1 +1 A ++1 (y + ( 1 +1 A ++1(z. (1.8 Quite recetly R. Chapa [Itegers 5(005] subsues these three idetities of Su ito a ifiite faily of idetities, ad J. X. Hou ad J. Zeg [Europea J. Cobi., i press, arxiv:ath.co/ ] got the q-aalogue of the above theore.

7 CURIOUS RESULTS ON BERNOULLI AND EULER POLYNOMIALS 7. Mii s ad Matiyasevich s idetities ad their polyoial fors I 1978 H. Mii [J. Nuber Theory 10(1978, 97-30] discovered the followig curious idetity which ivolves both a ordiary covolutio ad a bioial covolutio of Beroulli ubers: B B ( ( B B ( H B (.1 for every 4, 5,..., where H I the origial proof of this idetity, Mii showed that the two sides of (.1 are cogruet odulo all sufficietly large pries. I 198 Shiratai ad Yooyaa [Me. Fac. Sci. Kyushu Uiv. Ser. A 36(198, 73-83] gave aother proof of (.1 by p-adic aalysis. Ispired by Mii s wor, Matiyasevich foud the followig two idetities of the sae ature by the software Matheatica. ad B B l ( Bl l l B l H B ( + ( + B B B l B l ( + 1B (. l l for each 4, 5,.... Clearly the first oe is actually equivalet to Mii s

8 8 ZHI-WEI SUN idetity (.1 sice 1 B B ( l ( B B ( Bl B l l l( l B B 1 l l ( Bl l l B l. ( ( 1 l l + 1 B l B l l I Jue 004, Due ad Schubert [arxiv:ath.nt/ ] preseted a ew approach to (.1 ad (. otivated by quatu field theory ad strig theory. Sice all previous proofs of Mii s idetity are o-atural ad coplicated, i May 004 H. Pa ad Z. W. Su developed a ew ethod which oly ivolves differeces ad derivatives of polyoials. Defie the operators ad by (f(x f(x + 1 f(x ad (f(x f(x f(x. It is well ow that (B (x x 1 ad (E (x x for 0, 1,,.... Lea.1 [H. Pa ad Z. W. Su, J. Cobi. Theory Ser A 113(006]. Let P (x, Q(x C[x] where C is the field of coplex ubers. (i If (P (x (Q(x the P (x Q (x. (ii If (P (x (Q(x the P (x Q(x. To illustrate the power of Lea.1, let us give a siple proof of Raabe s ultiplicatio forula: 1 r0 ( x + r B 1 B (x for Z + ad N.

9 Clearly CURIOUS RESULTS ON BERNOULLI AND EULER POLYNOMIALS 9 ad hece ( 1 ( x + r B r0 1 ( ( ( x + r + 1 x + r B B r0 ( x ( x ( x 1 B + 1 B ( 1 B (x 1 r0 B 1 ( x + r d 1 dx r0 ( x + r B for 1,, 3,..., this proves Raabe s forula. d dx (1 B (x 1 B 1 (x With help of Lea.1, Pa ad Su were able to exted Mii s idetity (.1 ad Matiyasevich s idetity (. to Beroulli polyoials. Theore.1 [H. Pa ad Z. W. Su, J. Cobi. Theory Ser. A 113(006]. Let > 1 be a iteger. The 1 1 B (xb (y ( l1 H 1 B (x + B (y ad B (xb (y l0 ( 1 Bl (x yb l (y + B l (y xb l (x l 1 l + B (x B (y (x y (.3 ( + 1 Bl (x yb l (y + B l (y xb l (x l + 1 l + B +1(x + B +1 (y (x y + B+(x B + (y (x y 3. (.4

10 10 ZHI-WEI SUN Lettig y ted to x, (.3 ad (.4 tur out to be 1 B (xb (x ( 1 Bl B l (x B (x ( l 1 l H 1 1 ad B (xb (x respectively. l l ( + 1 Bl B l (x l + 1 l + (.5 ( + 1B (x. (.6 Siilar to Theore.1 Pa ad Su proved the followig idetities ivolvig Euler polyoials. Theore. [H. Pa ad Z. W. Su, J. Cobi. Theory Ser. A 113(006]. Let be a positive iteger. The E (xe (y 4 + B+(x B + (y x y +1 ( + 1 El (x yb +1 l (y + E l (y xb +1 l (x. l l + 1 Also, ad l0 B (x ( ( Bl (x y l l 1 l1 B (xe (y E (y H E (y E (x E (y x y E l (y E l 1(y x E l (x, ( ( + 1 B l (x ye l (y E l 1(y x E l (x l + 1 l1 ( E (x + ( + 1 x y + E (y E +1(x E +1 (y (x y. (.7 (.8 (.9

11 CURIOUS RESULTS ON BERNOULLI AND EULER POLYNOMIALS 11 Lettig y ted to x ad otig that E l (0 (1 l+1 B l+1 /(l + 1, we obtai fro (.7 (.9 the followig idetities. ( ( + E (xe (x 8 l B (xe (x B (x E (x l l ( + 1 l + 1 l ( l ( l 1 B l l B + l(x, (.10 l B l l E l(x H E (x, (.11 ( l + l 1 B l l E l(x ( + 1E (x. (.1 3. Woodcoc s idetity ad its geeralizatios I 1979 C. F. Woodcoc [J. Lodo Math. Soc. 0(1979, ] discovered that A 1, A 1, for, Z + (3.1 where A, 1 1 ( ( 1 B + B. (3. Thus 1 1 as oted by L. Euler. ( B B + B 1 A 1 1, A 1,1 B Usig Lea.1 H. Pa ad Z. W. Su proved i August 004 the followig theore which iplies the Woodcoc idetity.

12 1 ZHI-WEI SUN Theore 3.1 [H. Pa ad Z. W. Su, J. Cobi. Theory Ser. A 113(006]. Let, N ad x + y + z 1. The ( ( 1 B +1 (x + 1 B++1(y ( + ( 1 B +1 (x + 1 B++1(z Also, ad ( ( ( + + ( ( 1 ( 1 B++(x + + B +1(z + 1 B++(y ( ( 1 ( + ( 1 B+1(y + 1 B++(z + +. E (x B ++1(y ( E (x B ++1(z ( 1++1 E ++1 (x ( ( + E (ze (y ( E (x E ++1(y ( B +1 (x + 1 E++1(z ( 1 + ( ( + B++(x ( E++(z ( ( E (z B ++(y + +. (3.3 (3.4 (3.5 Fix y ad replace z i (1 by 1 x y. The, by taig differeces of both sides of (3.3 with respect to x, we ca get (1.5 agai. The siilar idetity for Euler polyoials is also iplied by Theore 3.1.

13 CURIOUS RESULTS ON BERNOULLI AND EULER POLYNOMIALS 13 If, Z + ad x + y + z 1, the we have the followig equivalet versio of (3.3: ( 1 ( B (x B +(y + + ( 1 ( B (x B +(z + ( 1+ ( 1!( 1! B + (x + B (z ( +! B(y. (3.3 Corollary 3.1 [H. Pa ad Z. W. Su, J. Cobi. Theory Ser. A 113(006]. Let x + y + z 1. Give, Z + we have the followig idetities: ad ( 1 ( 1 ( 1 ( 1 ( ( ( 1 ( 1 ( ( B (xb 1+ (y B (z B 1(y B (xb 1+ (z B (y B 1(z, E (xb + (y E 1(zE (y E (xb + (z E 1(yE (z ( ( E (xe 1+ (y B (xe + (z B (y E (z. (3.6 (3.7 (3.8 (3.6 ad (3.7 i the case x 1 t ad y z t yield the followig idetities siilar to the oe of Woodcoc. A 1, (t A 1, (t ad C, (t C, (t, (3.9

14 14 ZHI-WEI SUN where A, (t 1 ( ( 1 B + (tb (t B (t B (t (3.10 ad C, (t ( ( 1 B + (te (t E (te 1 (t. ( Uified idetities for Beroulli ad Euler polyoials Let be ay positive iteger. As usual, ( z z(z 1 (z + 1/! (ad ( z 0 1 eve if z N. Observe that ad B (xb (y 1 B (x B (y ( 1 ( 1 B (xb (y ( 1 ( 1 B (x B (y 1 ( t ( 1 B (xb (y. 1 1 li t 0 t Ispired by y above observatio, i Sept. 004 the speaer ad H. Pa ivestigated relatios aog the sus with P, Q {B, E}. 1 ( ( s t ( 1 P (xq (y Theore 4.1 [Z. W. Su ad H. Pa, arxiv:ath.nt/ ]. Let Z + ad x + y + z 1.

15 CURIOUS RESULTS ON BERNOULLI AND EULER POLYNOMIALS 15 (i If r + s + t, the we have the syetric relatio [ ] [ s t t r r + s x y y z ] [ ] r s + t 0 (4.1 z x where [ ] s t x y : ( ( s t ( 1 B (xb (y. (4. (ii If r + s + t 1, the ( ( r s ( 1 B (xe (z ( ( r t ( 1 ( 1 B (ye (z r 1 ( ( s ( 1 l t E l (ye 1 l (x. l 1 l l0 (4.3 I the case s t 1, Theore 4.1 yields that ( + B (xb (y ( + (( 1 B (x + B (yb (x y (4.4 ad E (xe 1 (y ( + 1 (( 1 B (x B (y E (x y. (4.5 Note that (4.4 i the case x y 0 yields Matiyasevich s idetity sice B l+1 0 for l 1,, 3,....

16 16 ZHI-WEI SUN We ca also deduce fro Theore 4.1 the followig result: If Z + ad x + y + z 1, the ( 1 B (x 1 (B (y + ( 1 B (z 1 ( 1 B (y 1 1 B (z H 1 B (y + ( 1 B (z. I the case x y 0 ad z 1, this yields Mii s idetity. (4.6 Let l,, Z +, l i{, } ad x + y + z 1. By Theore 4.1(i, [ ] [ ] [ ] l l l x y y z z x where + l Z +. It follows that ( 1 ( ( + 1 l 1 ( ( + ( 1 l ( l 1 l l ( ( 1 B l+ (xb (z B l+ (yb (z ( B l+ (xb (y. l I the case x y 0 ad l z 1, this yields Woodcoc s idetity 1 ( ( 1 B B 1+ 1 ( ( 1 B B (4.7 Oe ca also deduce the vo Ettigshause idetity fro Theore 4.1(i. As + + ad (1 x + y + (x y 1, Theore 4.1(i iplies the followig ew idetity: ( B (xb (y ( + 1 (( 1 B (x + B (y B (x y. ( (4.8

17 CURIOUS RESULTS ON BERNOULLI AND EULER POLYNOMIALS 17 I particular, ( B (xb (x ( ( B (xb. ( Soe cogrueces for Euler ubers ad q-euler ubers Euler ubers odulo a odd iteger are trivial. I fact, for ay N ad q Z + we have ( E q + 1 l0 ( El l l q l E E ( 1 (od q ad ( ( 1 E ( 1 q E q + 1 q 1 ( (( 1 j E j + 1 ( ( 1 j+1 E j j0 ( ( 1 j j + 1, q 1 j0 therefore q 1 E ( 1 j (j + 1 (od q providig q. (5.1 j0 It is atural to deterie Euler ubers odulo powers of two. However, this is a difficult tas sice 1/ is ot a -adic iteger. I a recet paper I deteried Euler ubers odulo powers of two i the followig explicit way.

18 18 ZHI-WEI SUN Theore 5.1 [Z. W. Su, J. Nuber Theory 115(005, ]. Let Z +. If N is eve, the E 3 1 3j + 1 ( 1 j 1 (j (od (5. j0 where α deotes the greatest iteger ot exceedig a real uber α, oreover for ay positive odd iteger we have the cogruece +1 ( 1 ( 1/ 1 j0 4 E ( 1 j 1 (j + 1 j + ( 1/ (od. (5.3 Note that ( /4 is a odd iteger if N is eve. Let, l N be eve. If ( l (i.e., ( l but +1 ( l where Z +, the (E E l by Theore 5.1. I other words, for ay Z + we have E E l (od l (od. (5.4 Ufortuately this discovery of the speaer repeated earlier wor. I 1875 M. A. Ster [J. Reie Agew. Math. 79(1875, 67 98] stated that E + s E + s (od s+1 for ay, s N ad gave a brief setch of a proof, the Frobeius aplified Ster s setch i I 1979 R. Ervall said that he could ot uderstad Frobeius proof ad provided his ow proof ivolvig ubral calculus. I 000 a iductio proof of the result was give by S. Wagstaff. The proof of Theore 5.1 depeds heavily o the followig explicit cogrueces for Beroulli ad Euler polyoials.

19 CURIOUS RESULTS ON BERNOULLI AND EULER POLYNOMIALS 19 Theore 5.. Let a Z ad, Z +. Let q > 1 be a iteger relatively prie to. (i [Z. W. Su, Discrete Math. 6(003, 53-76] We have ( ( 1 x + a B q 1 ( a + j j0 q B (x + 1 (x + a + j 1 (od q. (5.5 (ii [Z. W. Su, J. Nuber Theory 115(005, ] If q, the +1 q 1 ( x + a E ( 1a ( a + j ( 1 j 1 q j0 E (x + 1 (x + a + j (od q. (5.6 By the way we etio the followig observatio of the speaer [Nuber Theory: Traditio ad Moderizatio, Spriger, 006]: If N, a, Z ad the +1 ( x + a E ( 1a E (x Z[x]. (5.7 As usual we let (a; q 0 < (1 aq for every N, where a epty product is regarded to have value 1 ad hece (a; q 0 1. For N we set [] q 1 q 1 q 0 < this is the usual q-aalogue of. For ay, N, if the we call [ ] q 0<r [r] q 0<s [s] q 0<t [t] q q, (q; q (q; q (q; q

20 0 ZHI-WEI SUN a q-bioial coefficiet, if > the we let [ ] [ li q 1 ]q (. It is easy to see that [ ] [ ] [ ] 1 1 q + q q 1 q q 0. Obviously we have for all, 1,, 3,.... By this recursio, each q-bioial coefficiet is a polyoial i q with iteger coefficiets. H. Pa ad Z. W. Su defied q-euler ubers E (q ( N by x ( E (q (q; q 0 0 q ( x 1. (q; q Multiplyig both sides by 0 q( x /(q; q, we obtai the recursio [ ] q ( E (q δ,0 ( N q which iplies that E (q Z[q]. Note that li q 1 E (q E. The usual way to defie a q-aalogue of Euler ubers is as follows: x ( x 1 Ẽ (q. (q; q (q; q 0 0 It is easy to see that Ẽ(q q ( E (1/q. Recetly, with the help of cyclotoic polyoials, V.J.W. Guo ad J. Zeg [Europea J. Cobi., i press] proved that if,, s, t N, s t ad t the Ẽ (q q Ẽ (q ( od This is a partial q-aalogue of Ster s result. s r0 (1 + q rt. Here is a coplete q-aalogue of the classical result of Ster.

21 CURIOUS RESULTS ON BERNOULLI AND EULER POLYNOMIALS 1 Theore 5.3 (H. Pa ad Z. W. Su, Acta Arith., to appear. Let, s, t N ad t. The E (q E +s t(q [ s ] q t (od (1 + q[ s ] q t. (5.8 A ey tool i the proof of Theore 5.3 is the followig lea. Lea 5.1. For ay N we have E (q 1 [ ] ( q; q 1 0 0< The Salié ubers S ( N are give by x S! cosh x cos x ex + e x ( e ix + e ix x / (! 0 I 1965 Carlitz proved that S for ay N. q E ( (q. (5.9 0 ( 1 x H. Pa ad Z. W. Su defied q-salié ubers by x q ( 1 x / ( 1 q ( x S (q. (q; q (q; q (q; q (!. Here is a q-aalogue of Carlitz s result equivalet to a cojecture of Guo ad Zeg ad proved by Pa ad Su [Acta Arith., to appear]: If N the ( q; q 0< (1 + q divides S (q i the rig Z[q]. A ey tool i the proof is the followig recursio siilar to that i Lea 5.1. S (q 0< [ ] ( 1 ( q; q 1 q S ( (q (od ( q; q. It follows fro the followig deep cogruece due to Pa ad Su: [ ] ( 1 q ( 1 0 (od ( q; q ( l Z +l 0 provided that l Z,, N ad. q