Some Identities on Generalized Poly-Euler and Poly-Bernoulli Polynomials

Transcription

1 MATIMYÁS MATEMATIKA Joural of the Matheatcal Socety of the Phlppe ISSN Vol. 39 No pp Soe Idette o Geeralzed Poly-Euler ad Poly-Beroull Polyoal Roberto B. Corco Departet of Matheatc Cebu Noral Uverty Cebu Cty, Phlppe rcorco@yahoo.co Crta B. Corco Departet of Matheatc Cebu Noral Uverty Cebu Cty, Phlppe Jay M. Otola Departet of Matheatc Cebu Noral Uverty Cebu Cty, Phlppe Haa Jolay UFR de Mathéatque Laboratore Paul Palevé Uverté de Scece et Techologe de Llle Vlleeuve d Acq Cedex/Frace haa.olay@ath.uv-llle.fr Abtract I th paper, ore dette for geeralzed poly-euler ad poly-beroull polyoal wth three paraeter are obtaed coecto wth the Strlg uber of the ecod kd. Moreover, yetrzed geeralzato troduced to etablh certa dualty relato. Keyword: Poly-Euler polyoal, poly-beroull polyoal, geeratg fucto, Strlg uber Matheatc Subect Clafcato: B68, B73, 05A5 Itroducto Leoard Euler troduced Euler uber ad polyoal h dere to evaluate the alteratg u A

2 44 R. Corco, C. Corco, J. Otola, H. Jolay Euler uber are uually defed a coeffcet of the Taylor ere expao of the recprocal of hyperbolc coe fucto. More precely, coh t 2et e 2t + E t.. Oe ca ealy verfy that Euler uber wth odd dce are all zero. That, E 2 0,, 2,... O the other had, the eve-dexed oe are o-zero wth alteratg g uch a E 0, E 2, E 4 5, E 6 6, E 8 385, E Alot all geeralzato of Euler uber are baed o the above expao.. For tace, a ple geeralzato polyoal for gve by 2e xt e t + E x t where E x are the o-called Euler polyoal. Coequetly, th defto gve the followg detty E x + + E x 2x..2 A etoed above, the a obectve of troducg Euler polyoal to evaluate the alteratg u A. The, addg ad ubtractg detty.2 wth x, x 2,..., x yeld A 2 E + E for teger, 0. Aother geeralzato due to Oho ad Saak [7] whch defed by L k e 4t E k t 4t coh t where L k z z k,.3 are the o-called polylogarth ad the uber E k are called poly-euler uber. Recetly, H. Jolay et al. [6] have exteded thee uber polyoal for, deoted by E k x; a, b, c, whch are called geeralzed polyeuler polyoal. That, 2L k ab t a t + b t c xt E k x; a, b, c t. 0 Thee polyoal atfy the followg dette E k x; a, b, c E k x; a, b, c l a + l b E k l c E k a, bx x l c + l a l a + l b d dx Ek + x; a, b, c + l cek x; a, b, c

3 Soe Idette o Geeralzed Poly-Euler ad where E k ee [9, 6]. a, b E k 0; a, b, c ad E k 2L k ab t a t + b t 2L k e t + e t e xt x E k x;, e, e, whch ca be defed by E k a, b t E k x t O the other had, Beroull uber, deoted by B, frt appeared a faou treate publhed 73, eght year after the death of Jacob Beroull Thee uber appeared a part of the coeffcet whe Beroull expreed the u of power of coecutve teger p a polyoal, whch gve by p p + k0 k k p p p + B k p+ k k where p ad are potve teger ad B /2 ued. Beroull uber alo appeared the coputato of Rea zeta fucto ζ2p k k 2p 4p B 2p π 2p 22p! ad a coeffcet the Taylor ere expao of the hyperbolc taget fucto z e z z k B k, z < 2π. k! k0 Beroull uber have alo bee exteded polyoal for a coeffcet of the followg geeratg fucto ze xz e z B k x zk k! k0 where B k x are called the Beroull polyoal. Thee polyoal have bee exteded further by Kaeko [3] a coeffcet, deoted by B k, of the followg ere expao L k e x e x B k x, whch are called poly-beroull polyoal. Several properte of B k Kaeko [3] cludg oe explct ad recurve forula. are etablhed by I th paper, ore dette for geeralzed poly-euler polyoal are etablhed cotag Strlg uber of the ecod kd ad other kow cobatoral uber. Moreover, the geeralzed poly-beroull polyoal are defed ad oe eceary properte are obtaed parallel to thoe of geeralzed poly-euler polyoal.

4 46 R. Corco, C. Corco, J. Otola, H. Jolay 2 Geeralzed Poly-Euler Polyoal We tart our dcuo th ecto by troducg frt the volved cobatoral uber. {[ ] { }} Strlg uber are defed par, deoted by,, by ea of the followg relato [ ] x x 2. { } x x 2.2 [ ] { } where the uber ad are called Strlg uber of the frt ad ecod kd, repectvely, ad x the well-kow Pochaer ybol for the fallg factoral defed by x xx x 2... x +. Oe ca ealy wrte 2. a [ ] x x [ ] [ ] where the uber are called the gle Strlg uber of the frt kd ad x the well-kow Pochaer ybol for the rg factoral defed by x xx + x x +. [ ] The gle Strlg uber of the frt kd ca be terpreted a the uber of poble perutato { } of a -et wth oepty cycle ad the Strlg uber of the ecod kd ca be terpreted a the uber of way to partto a -et to oepty ubet. Thee uber atfy the followg expoetal geeratg fucto log + t [ ] t 2.3! e t { } t!. 2.4 For a detaled dcuo of thee uber, oe ay ee [5]. Before we troduce reult th ecto, let u defe frt forally the geeralzed poly-euler polyoal. Defto 2.. [6, H. Jolay et al.] For ay potve uber a, b, c ad ay real uber x wth k Z ad 0, the geeralzed poly-euler polyoal are defed by 2L k ab t a t + b t c xt E k x; a, b, c t. 2.5

5 Soe Idette o Geeralzed Poly-Euler ad Soe dette o geeralzed poly-euler polyoal are expreed ter of Strlg uber of the ecod kd. Such dette have appeared Theore 2.6 of [8] but wth c e. That, E k x; a, b E k x; a, b, e. More precely, 2L k ab t a t + b t e xt The ad dette are gve the followg theore. E k x; a, b t. 2.6 Theore 2.2. [8] For ay potve uber a, b ad ay real uber x wth k Z ad 0, the polyoal E k x; a, b atfy the followg explct forula { } l E k x; a, b E k l l ; a, bx 2.7 E k x; a, b E k x; a, b E k x; a, b l { } l l l l l+ l0 l λ 0 E k l 0; a, bx 2.8 { l + } E k l 0; a, bb x 2.9 λ E ; k a, bh x; λ, 2.0 where t e t e xt B x t ad λ e t e xt λ H x; λ t. Here, we derve oe dette for E k x; a, b, c whch are parallel to thoe Theore 2.2. The dette are gve the followg theore. Theore 2.3. For ay potve uber a, b, c ad ay real uber x wth k Z ad 0, the geeralzed poly-euler polyoal atfy the followg relato E k x; a, b, c E k x; a, b, c E k x; a, b, c E k x; a, b, c { } l log c l l { } l log c l l l { l l + l+ l0 l λ l 0 E k l log c; a, bx 2. E k l 0; a, bx 2.2 } E k l 0; a, bb x log c 2.3 λ E k ; a, bh x log c; λ. 2.4 Proof: For relato 2., ote that 2.5 ca be wrtte a E k x; a, b, c t 2L k ab t a t + b t e t log c x

6 48 R. Corco, C. Corco, J. Otola, H. Jolay Ug Newto Boal Theore ad the expoetal geeratg fucto for 2.4, we have { } E k x; a, b, c t 2L k ab t a t + b t x + e t log c x et log c 2L k ab t! a t + b t { } t log c x { { } l log c l l l Coparg coeffcet coplete the proof of 2.. For relato 2.2, we ca wrte 2.5 a t log c e E k log c; a, b t E k l log c; a, bx } t E k x; a, b, c t 2L k ab t a t + b t e t log c + x Ug Newto Boal Theore ad the expoetal geeratg fucto for 2.4, we have { } E k x; a, b, c t 2L k ab t a t + b t x e t log c e t log c 2L k ab t x! a t + b t { } t log c x { { } l log c l l l Coparg coeffcet coplete the proof of 2.2. For relato 2.3, equato 2.5 ca be wrtte a E k x; a, b, c t E k 0; a, b t E k l 0; a, bx } t e t t xt log c e 2Lk ab t! e t a t + b t! t

7 Soe Idette o Geeralzed Poly-Euler ad { } + t + +! { } + t + +! { { } l + t l+ l0 { { l l + l0 l+! B x log c t! B x log c t! Ek l +! B x log ce k l Coparg coeffcet coplete the proof of 2.3. For relato 2.4, ote that 2.5 ca be wrtte a } E k l 0; a, bb x log c E k 0; a, b t! t t! 0; a, b! t } t l t! 0; a, b l!! t } t E k x; a, b, c t λ log c e t λ 2Lk ab t e t ext λ λ a t + b t H x log c; λ t 0 λ λ 0 λ λ 0 λ 0 λ 2L k ab t a t + b t H x log c; λ t e t E k ; a, b t H x log c; λe k ; a, b t λ E k ; a, bh x log c; λ t Coparg coeffcet coplete the proof of 2.4. I partcular, whe c e 2., 2.2, 2.3 ad 2.4 yeld 2.7, 2.8, 2.9 ad Geeralzed Poly-Beroull Polyoal Parallel to the defto of geeralzed poly-euler polyoal 2.5, we have the followg geeralzato of poly-beroull uber. Defto 3.. For ay potve uber a, b, c ad ay real uber x wth k Z ad 0, the geeralzed poly-beroull polyoal are defed by L k ab t b t a t c xt. B k x; a, b, c t. 3. Oe ca ealy prove the followg theore ug the ae arguet dervg the dette Theore 2.3.

8 50 R. Corco, C. Corco, J. Otola, H. Jolay Theore 3.2. For ay potve uber a, b, c ad ay real uber x wth k Z ad 0, the geeralzed poly-beroull polyoal atfy the followg dette. { } l B k x; a, b, c log c l B k l log c; a, bx l l { } l B k x; a, b, c log c l B k l l 0; a, bx l { } B k l l + x; a, b, c l+ B k l 0; a, bb x log c l0 l B k x; a, b, c λ λ B k ; a, bh x log c; λ The ext theore cota a explct forula for B k x; a, b, c. 0 Theore 3.3. Explct Forula For ay potve uber a, b, c ad ay real uber x wth k Z ad 0,, we have B k x; a, b, c k x l c l a + l b. 3.2 >0 0 Proof: Frt, we eed to expad the followg expreo ug the defto of polylogarth.3 a follow L k ab t ab t b t a t b t ab t k >0 b t ab t k >0 Ug the Boal Theore, we have L k ab t b t a t b t >0 >0 By akg ue of equato 3., we get B k x; a, b, c t L k ab t b t a t k >0 0 >0 k 0 k 0 xt l c e k 0 e e t lab e t l a++ l b. tx l c l a + l b x l c l a + l b t

9 Soe Idette o Geeralzed Poly-Euler ad... 5 By coparg the coeffcet of t o both de, the proof copleted. The ext theore cota a expreo of B k x; a, b, c a polyoal x. Before we eto the theore, let u troduce the otato B k a, b whch equvalet to B k 0; a, b, c. More precely, the uber B k a, b are defed a coeffcet of the followg geeratg fucto L k ab t b t a t B k a, b t. Theore 3.4. The geeralzed poly-beroull polyoal atfy the followg relato B k x; a, b, c l c B k a, bx 3.3 Proof: Ug 3., we have B k x; a, b, c t 0 L k ab t b t a t c xt xt l c e xt l c B k a, b t!! 0 l c B k a, bx 0 B k a, b t Coparg the coeffcet of t, we obta the dered reult. Note that, whe a c e ad b, Defto 3. reduce to L k e t e t e xt B k x; e,, e t. 3.4 where B k x; e,, e equvalet to B k theore gve a relato betwee B k t. x, the poly-beroull polyoal. The followg x; a, b, c ad B k x. Theore 3.5. The geeralzed poly-beroull polyoal atfy the followg relato x l c l b B k x; a, b, c l a + l b B k 3.5 l a + l b Proof: Ug 3., we have B k x; a, b, c t L k ab t l c b t ab t ext x l c l b e l ab t l ab L k e t l ab + e l a + l b B k Coparg the coeffcet of t, we obta the dered reult. t l ab x l c l b l a + l b t.

10 52 R. Corco, C. Corco, J. Otola, H. Jolay The followg theore cota a dervatve forula for geeralzed poly-beroull polyoal. Theore 3.6. The geeralzed poly-beroull polyoal atfy the followg relato Proof: Hece, d dx Bk + x; a, b, c + l cbk x; a, b, c 3.6 Ug 3., we have d dx Bk x; a, b, c t d dx Bk x; a, b, c t d + dx Bk + xt l c e tl cl k ab t b t a t l cb k x; a, b, c t. x; a, b, ct l cb k x; a, b, c t. Coparg the coeffcet of t, we obta the dered reult. The followg corollary edately follow fro Theore 3.6 by takg c e. brevty, let u deote B k x; a, b, e by B k x; a, b. Corollary 3.. The geeralzed poly-beroull polyoal are Appell polyoal the ee that d dx Bk + x; a, b + Bk x; a, b 3.7 Coequetly, ug the characterzato of Appell polyoal [8, 9, 20], the followg addto forula ca ealy be obtaed. Corollary 3.2. The geeralzed poly-beroull polyoal atfy the followg addto forula B k x + y; a, b B k x; a, by 3.8 However, we ca derve the addto forula for B k x; a, b, c a follow B k x + y; a, b, c t 0 L k ab t b t a t c x+yt L k ab t b t a t c xt c yt B k x; a, b, c t y l c t y l c B k t x; a, b, c. Coparg the coeffcet of t yeld the followg reult. 0 For

11 Soe Idette o Geeralzed Poly-Euler ad Theore 3.7. The geeralzed poly-beroull polyoal atfy the followg addto forula B k x + y; a, b, c l c B k x; a, b, cy. 0 Note that whe x 0, the above addto forula yeld B k y; a, b, c 0 l c B k 0; a, b, cy whch exactly the forula Theore 3.4 that expree B k y; a, b, c a polyoal y wth the uber l c B k 0; a, b, c a coeffcet. 4 Syetrzed Geeralzato The yetrzed geeralzato of poly-euler polyoal wth paraeter a, b ad c, deoted by D x, y; a, b, c, ha bee defed [8] a follow D x, y; a, b, c l a + l b k0 k y l c + l a E k x; a, b, c 4. k l a + l b where, 0. Thee polyoal atfy the followg double geeratg fucto ad explct forula D x, y; a, b, c t u D x, y; a, b, c 2 0! 2e y l c+l a l a+l b u x l c+l a e l a+l b t e t+u e t e t + e t + e u e t+u, 4.2! 2 { l l l0 0 l c x a +2 b + l l c x a + b l l a + l b l } y l c + 2 l a + l b r0 l a + l b r r { r The a purpoe of troducg yetrzed geeralzato to etablh certa dualty relato. However, for the yetrzed geeralzato of poly-euler polyoal, dualty relato ot poble. I th ecto, we defe the yetrzed geeralzato of poly- Beroull polyoal wth three paraeter, deoted by C oe properte parallel to that of D }. x, y; a, b, c, ad etablh x, y; a, b, c cludg the dualty relato. Now, let u troduce the followg defto of C x, y; a, b, c. Defto 4.. For, 0, we defe C x, y; a, b, c l a + l b k0 k B k l b x; a, b, c y l c. k l a + l b 4.3

12 54 R. Corco, C. Corco, J. Otola, H. Jolay The followg theore preet the double geeratg fucto for C x, y; a, b, c whch parallel to the followg double geeratg fucto obtaed by Kaeko [3] k0 Theore 4.2. For, 0, we have Proof: LHS C x, y; a, b, c t u By ug the defto of C l a + l b By puttg l k, we get LHS k0 l0 l b y l c e l a+l bu l b y l c e l a+l bu l b y l c e l a+l bu l b y l c e l a+l bu e k B k t u k k! e t+u e t + e u. 4.4 et+u! ex l c+ l a B k x; a, b, c l a + l b B k x; a, b, c k0 k0 l a+l bt y l c+ e l a l a+l bu e t + e u e t+u. 4.5 x, y; a, b, the left-had de ca be wrtte a k t y l c y l c l b l a + l b l a + l b B k x; a, b, c t u k k! t e xt B k l a+l b a, b, c k0 u k! k! l b l t u k u l l a + l b k! l! uk k! e xt l c L k e t e t e t l b l a+l b u k k! x l c l b l a+l bt k0 B k t u k k! So, by applyg the double geeratg fucto 4.4, we obta the dered reult. A a drect coequece of Theore 4.2, we have the followg corollary cotag the well kow dualty property. Corollary 4.. Dualty Property For 0, we have C x, y; a, b, c C y, x; b, a, c. 4.6 Now, we are ready to how a cloed forula for C x, y; a, b, c whch portat ad fudaetal. Theore 4.3. Cloed Forula For 0, we have C x, y; a, b, c! 2 x l c + 0 l0 p0 y l c + l b l a + l b l a l a + l b l l p p { l { } p 4.7 }

13 Soe Idette o Geeralzed Poly-Euler ad Proof: By applyg Theore 4.2, we have C x, y; a, b, c t u! ex l c+ l a l a+l bt y l c+ e l b l a+l bu e t + e u e t+u ex l c+ l a l a+l bt y l c+ e l e t e u l a x l c+ e l a+l bt e e 0 x l c+ l a b l a+l bu y l c+ l b l a+l bu l a+l bt e t e e t e u 0 y l c+ l b l a+l bu e u By applyg the geeratg fucto for Strlg uber of ecod kd 2.4, the rghthad de of the lat expreo becoe x l c + l a l a+l b t { } t!! 0 y l c + l b l a+l b u { } u!! l l { } l a l t l! x l c + l a + l b l! 0 l0 p p r { } l b p r u p! y l c + l a + l b r p! l0 p0 p0 r0 p r0 t l u p l! p! y l c + whch yeld the dered detty. 5 Suary ad Cocluo l l { } 0! 2 l a l x l c + l a + l b p r { } l b p r l a + l b r Prelary vetgato for geeralzed poly-euler ad poly-beroull polyoal ha bee doe the paper [9, 6], whch provde oe eceary properte for the two polyoal cludg explct forula ad recurrece relato. However, there are oe teretg properte ad forula that are ot codered, epecally, for the geeralzed poly-euler ad poly-beroull polyoal wth three paraeter. I Secto 2 ad the frt part of Secto 3, the dcuo focue o oe forula that expreed the geeralzed poly-euler ad poly-beroull polyoal wth three paraeter ter of Strlg uber of the ecod kd, rg ad fallg factoral, ad certa geeralzato of Beroull polyoal. Thee would be of great help vualzg the

14 56 R. Corco, C. Corco, J. Otola, H. Jolay tructure of the ad polyoal, partcularly, drawg ther cobatoral terpretato a the Strlg uber of the ecod kd a well a the rg ad fallg factoral are kow to have cobatoral eag. The lat part of Secto 3 devote t dcuo o the expreo of B k y; a, b, c a polyoal x ad ther clafcato a Appell polyoal. The ad polyoal expreo gve by B k x; a, b, c 0 Note that th polyoal ca further be wrtte a B k x; a, b, c 0 0 l c B k a, bx. l c B k a, bx l c B k a, bx! l c B k a, bx!. 0 Clearly, the uber l c B k a, b are coeffcet of the Taylor ere expao of B k x; a, b, c. Th ple that d dx Bk x; a, b, c l c B k a, b. 5. x0 Equato 5. ca alo be obtaed by applyg the recurve forula Theore 3.6 repeatedly. That, d 2 dx 2 Bk d 3 dx 3 Bk x; a, b, c d [ l cb k dx x; a, b, c d dx ] x; a, b, c [ ] l c 2 B k 2 x; a, b, c 3 l c 3 B k 3 x; a, b, c l c 2 B k 2 x; a, b, c d dx Bk. x; a, b, c l c B k x; a, b, c. 5.2 Evaluatg the th dervatve 5.2 at x 0 ad ug the fact that B k 0; a, b, c a, b, we obta 5.. The dervatve forula Theore 3.6 ha bee ued to how B k that the geeralzed poly-beroull polyoal wth two paraeter ca be clafed a Appell polyoal. Moreover, a addto forula for the geeralzed poly-beroull polyoal wth two paraeter ha bee obtaed a a coequece of ther beg Appell polyoal. However, for three-paraeter cae, a addto forula derved ug dfferet ethod, the geeratg fucto ethod.

15 Soe Idette o Geeralzed Poly-Euler ad B k Fally, th paper ha bee cocluded by troducg the yetrzed geeralzato of x; a, b, c, whch gve by C x, y; a, b, c l a + l b k0 k B k l b x; a, b, c y l c. k l a + l b Th yetrzed geeralzato atfe the followg double geeratg fucto C x, y; a, b, c t u! ex l c+ l a whch, coequetly, yeld the followg dualty relato ad cloed forula C x, y; a, b, c C x, y; a, b, c C y, x; b, a, c! 2 0 l0 p0 y l c + x l c + l b l a + l b l a+l bt y l c+ e l a l a+l bu e t + e u e t+u. l a l a + l b l l p p { l { } p The above dualty relato ay be codered a couterpart of the dualty relato etablhed by Kaeko [3] for poly-beroull uber. Ackowledgeet. The author wh to thak the two aoyou referee for ther coet ad uggeto, whch are valuable ad very helpful revg ad provg our aucrpt. The reearch of the frt author wa upported by the Natoal Reearch Coucl of the Phlppe NRCP Proect No. P-004 Referece [] S. Arac, M. Ackgoz ad E. Se, O the exteded K p-adc q-defored feroc tegral the p-adc teger rg, J. Nuber Theory, , [2] A. Bayad ad Y. Haahata, ArakawaKaeko L-fucto ad geeralzed poly-beroull polyoal, J. Nuber Theory, 3 20, [3] B. Beńy, Advace Bectve Cobatorc, Ph.D. The, 204. [4] C. Brewbaker, A Cobatoral Iterpretato of the Poly-Beroull Nuber ad Two Ferat Aalogue, Iteger, , #A02. }. [5] L. Cotet, Advaced Cobatorc, D. Redel Publhg Copay, 974. [6] M-A. Coppo ad B. Cadelpergher, The Arakawa-Kaeko Zeta Fucto, Raaua J., , [7] R.B. Corco, C.B. Corco ad R. Aldea, Ayptotc Noralty of the r, β-strlg Nuber, Ar Cob., 8, 2006, 8-96.

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