On Generalized Multi Poly-Euler and Multi Poly-Bernoulli Polynomials

Transcription

1 O Geeralized Multi Poly-Euler ad Multi Poly-Beroulli Polyoials arxiv: v [ath.nt] 6 Dec 205 Roberto B. Corcio, Hassa Jolay, Cristia B. Corcio ad Takao Koatsu Abstract I this paper, we establish ore idetities of geeralized ulti poly-euler polyoials with three paraeters ad obtai a kid of syetrized geeralizatio of the polyoials. Moreover, geeralized ulti poly-beroulli polyoials are defied usig ultiple polylogarith ad derive soe properties parallel to those of poly-beroulli polyoials. These are geeralized further usig the cocept of Hurwitz-Lerch ultiple zeta values. Matheatics Subect Classificatio 200. B68, B73, 05A5. Keywords: ulti poly-euler polyoials, Appell polyoials, ultiple polylogarith, poly-beroulli polyoial, Hurwitz-Lerch ultiple zeta value, geeratig fuctio. Itroductio The Euler ubers, deoted by E, are usually itroduced as the coefficiets of the geeratig fuctio cosht 2et e 2t + t E. Evetually, these ubers have bee geeralized i polyoial for as 2e xt e t + E x t where E x deote the so-called Euler polyoials. Euler ubers ad polyoials have a rich literature i the history of atheatics where buch of idetities ad properties have bee established icludig their atheatical ad physical applicatios. These ubers ad polyoials have close coectios with Beroulli ubers ad polyoials, particularly, i the structures of their properties ad geeralizatios. I fact, i alost every property ad geeralizatio of Beroulli ubers

2 ad polyoials there correspods property ad geeralizatio for Euler ubers ad polyoials. For istace, Kaeko [8] itroduced the poly-beroulli ubers B k by eas of the followig expoetial geeratig fuctio Li k e x B k x e x where Li k z z k, while Oho ad Sasaki [2] defied poly-euler ubers as Li k e 4t E k t 4tcosht which have bee recetly exteded by H. Jolay et al. [7] i polyoial for as 2Li k e t e xt E k +e t xt. It is worth-etioig that the above geeralizatio of Kaeko has bee geeralized further by Cekci ad Youg [7] usig the cocept of Hurwitz-Lerch zeta fuctio Φz, s, a as follows Φ e t,k,a B,a k t 2 where Φz,k,a z +a k. 3 The ubers B,a k are called the Hurwitz type poly Beroulli ubers. These ubers have bee show to have explicit forula B k,a 0!S, +a k 4 where S, deotes the Stirlig ubers of the secod kid. A parallel versio of geeralizatio for Euler ubers is still to be doe. However, it is expected that the structure is quite coplicated. For istace, oe ay defie it as 2 e t Φ e t,k,a +e t E k t,a. 5 The ubers E,a k ay becalled the Hurwitz type poly Euler ubers. Oeca easily derive the explicit forula for E,a k as follows E k +,a r r r++q+ + q ++, r q, r r q!s r,. 6 +a k r q 0 2

3 We otice that the explicit forula of E,a k is ore coplicated tha that of B,a. k Aother for of geeralizatio of Beroulli ubers has bee defied by Iatoi et al. [3] i ters of ultiple polylogarith as follows where Li k,k 2,...,k r e t e t Li k,k 2,...,k rz B k,k 2,...,k r t zr 0< < 2 <...< r k k kr r These ubers possess respectively the followig recurrece relatio ad explicit forula B k,k 2,...,k r B k,k 2,...,k r B k,k 2,...,k r 9 + B k,k 2,...,k r + > 2 >...> r>0!s,. 0 k k 2... kr r Parallel to the above geeralizatio is the geeralized ulti poly-euler polyoials which are deoted by E k,k 2,...,k r x;a,b,c. These polyoials have bee itroduced i [6] by eas of the above ultiple poly-logarith, also kow as ultiple zeta values. More precisely, we have 2Li k,k 2,...,k r ab t c rxt E k,k 2,...,k r a t +b t r x;a,b,c t. Whe r, boils dow to the geeralized poly-euler polyoials with three paraeters a,b,c. Moreover, whe c e, reduces to the ulti poly-euler polyoials with two paraeters a, b. These special cases have bee discussed itesively i [6, ]. Soe properties of geeralized ulti poly-euler polyoials E k,k 2,...,k r x;a,b,c are i0 established i [6] which iclude the followig idetities E k,k 2,...,k r x;a,b,c i E k,k 2,...,k r x;a,b,c la+lb E k,k 2,...,k r d dx Ek,k 2,...,k r i0 rlc i E k,k 2,...,k r i a,bx i 2 rxlc+la la+lb + x;a,b,c +rlce k,k 2,...,k r x;a,b,c E k,k 2,...,k r x+y;a,b,c i rlc i E k,k 2,...,k r i x;a,b,cy i. Furtherore, a explicit forula for E k,k 2,...,k r x;a,b,c is give by E k,k 2,...,k r x;a,b,c J, 2,..., r 4 i0 0< < 2 <...<r c +c r 3 3

4 where J, 2,..., r r 0 2rxlc lab i r! +s slab+rla i r i c!c 2!... k k kr r with s c +2c Whe r, these idetities reduce to d dx Ek E k x;a,b,c i0 E k x;a,b,c la+lb E k lc i E k i a,bx i i xlc+la la+lb +x;a,b,c +lce k x;a,b,c E k x+y;a,b,c i0 lc i E k i x;a,b,cy i i which are properties of geeralized poly-euler polyoials see [6]. I this paper, soe idetities of E k,k 2,...,k r x;a,b,c related to Stirlig ubers of the secod kid are established ad certai syetrized geeralizatio for E k,k 2,...,k r x;a,b,c is obtaied. O the other had, geeralized ulti poly-beroulli polyoials are defied ad soe properties of these polyoials are established parallel to those of the poly-beroulli polyoials. Moreover, certai geeralizatio of ulti poly-beroulli ubers is defied i ters of geeralized Hurwitz-Lerch ultiple zeta values. 2 Geeralized Multi Poly-Euler Polyoials ad Stirlig Nubers The geeralized poly-euler polyoials with three paraeters a, b ad c are defied i [6] as follows 2Li k ab t c xt E k a t +b t x;a,b,c t. 5 Soe idetities o geeralized poly-euler polyoials are expressed i ters of Stirlig ubers of the secod kid. Such idetities have appeared i Theore 2.6 of [6] but with c e. More precisely, we have the followig theore. 4

5 Theore 2.. [6] The geeralized poly-euler polyoials E k x;a,b satisfy the followig explicit forulas E k x;a,b 0 l E k x;a,b 0 l E k x;a,b 0 E k x;a,b 0 } l l } l l l0 λ s E k l ;a,bx 6 E k l 0;a,bx 7 l l+s l s 0 } l +s E k s λ s E k s l 0;a,bBs x 8 ;a,bh s x;λ, 9 where x xx+...x+, x xx...x +, s t s λ e xt B s e t xt ad e xt H e t s λ x;λt. Here, we derive soe idetities for E k,k 2,...,k r x;a,b,c which are parallel to those i Theore 2.. The first such idetity is give i the followig theore. Theore 2.2. E k,k 2,...,k r x;a,b,c 0 l } l rlogc l l E k,k 2,...,k r l logc;a,bx. 20 Proof. Note that ca be writte as E k,k 2,...,k r x;a,b,c t 2Li k,k 2,...,k r ab t e rtlogc x a t +b t r Usig Newto s Bioial Theore, we have E k,k 2,...,k r x;a,b t 2Li k,k 2,...,k r ab t x+ e rtlogc a t +b t r l 0 x ertlogc 2Li k,k 2,...,k r ab t e rtlogc! a t +b t r } rtlogc x } } l rlogc l E k,k 2,...,k r l rlogc;a,bx l Coparig coefficiets copletes the proof of E k,k 2,...,k r rlogc;a,b t t

6 I particular, whe c e ad r, 20 yields 6. For the geeralizatio of 7, we have the followig theore. Theore 2.3. E k,k 2,...,k r x;a,b,c 0 l } l rlogc l l E k,k 2,...,k r l 0;a,bx. 2 Proof. Note that ca be writte as E k,k 2,...,k r x;a,b,c t 2Li k,k 2,...,k r ab t e rtlogc + x a t +b t r Usig Newto s Bioial Theore, we have E k,k 2,...,k r x;a,b t 2Li k,k 2,...,k r ab t a t +b t r 0 e rtlogc 2Li k,k x 2,...,k r ab t! a t +b t r } rtlogc x } l rlogc l l l Coparig coefficiets copletes the proof of 2. Theore 2.4. E k,k 2,...,k r x;a,b,c 0 l0 l l+s l x e rtlogc E k,k 2,...,k r 0;a,b t } E k,k 2,...,k r l 0;a,bx } l+s s t E k,k 2,...,k r l 0;a,bB s xrlogc. 22 Proof. Note that ca be writte as E k,k 2,...,k r x;a,b,c t e t s t s e xrtlogc 2Lik,k 2,...,k r ab t s! s! e t s a t +b t r t s } +s t +s B s s +s! xrlogct E k,k 2,...,k r 0;a,b t s!! t s 0 } +s t +s B s s +s! xrlogct! Ek,k 2,...,k r t s! 0;a,b! t s 0 } } l+s t l+s s l+s! Bs xrlogcek,k 2,...,k r t l t l 0; a, b l!!s! t s 0 l0 } } l l +s l+s E k,k 2,...,k r l 0;a,bB s t xrlogc s 0 l0 s! 6

7 Coparig coefficiets copletes the proof of 22. Theore 2.5. E k,k 2,...,k r x;a,b,c 0 λ s s 0 s λ s E k,k 2,...,k r ;a,bh s xrlogc;λ. Proof. Note that ca be writte as E k,k 2,...,k r x;a,b,c t λ s e t λ s 2Lik,k 2,...,k r ab t e t λ sexrtlogc λ s a t +b t r H s xrlogc;λ t s 0 s s λ s λ s 0 s s λ s λ s 0 0 s s λ s 0 0 s λ s 2Li k,k 2,...,k r ab t e t a t +b t r H s xrlogc;λt H s Coparig coefficiets copletes the proof of 23. E k,k 2,...,k r ;a,b t xrlogc;λe k,k 2,...,k r ;a,b t t. λ s E k,k 2,...,k r ;a,bh s xrlogc;λ The ext theore cotais a idetity which is obtaied by akig use of the followig differetial forula for the geeralized poly-logarith Haahata ad Masubuchi, Itegers 23 d dz Li k,...,k r z z Li k,...,k r,k r z if k r > ; z Li k,...,k r z if k r. 24 Theore 2.6. If k r >, the E k,...,k r + x rlogac x E k,...,k r x r ν If k r, the E k,...,k r ν0 logab ν0 0 ν νe logab k,...,k r ν x 0 ν logab E k,...,k r,k r ν x. + x rlogac x E k,...,k r x ν r logab E k,...,k r ν x+logabe k,...,k r x. ν ν0 0 7

8 Proof. We differetiate both sides of 2ac x rt Li k,...,k r ab t +ab t r with respect to t usig 24. If k, the 2rac x rt logac x Li k,...,k r ab t E k,...,k r x t +2ac x rt ab t logab Li ab t k,...,k r,k r ab t r +ab t r ab t logab E k,...,k r x t + +ab t r Dividig both sides by +ab tr, we have rlogac x 2ac x rt +ab t rli k,...,k r ab t E k,...,k r t x!. + logab 2ac x rt ab t +ab t rli k,...,k r,k r ab t rabt +ab logab t E k,...,k r x t + E + t. Sice ad logab ab logab ab t logab t logab 0 0 ν0 rab t +ab r ab t t r 0 0 ν0 ν logab ν! 0 t ν ν logab t ν, ν! e tlogab 8

9 we obtai Hece, rlogac x rlogac x +logab E k,...,k r 0 ν0 r x t ν t ν logab 0 ν0 E k,...,k r x t + r ν ν0 0 logab ν! µ0 ν t ν logab ν0 E k,...,k r,k r µ x tµ µ! ν! t µ E µ µ! + µ0 E k,...,k r + x t ν logab E k,...,k r ν x ν 0 Coparig the coefficiets o both sides, we have E k,...,k r + x rlogac x E k,...,k r x r ν ν0 logab 0 ν0 E k,...,k r t +. t ν logab E k,...,k r,k r ν x νe logab k,...,k r ν x ν 0 ν logab E k,...,k r,k r ν x. If k r, the the secod ter o the left-had side becoes 2ac x rt ab t logab ab t Li k,...,k r. After dividig of +ab t r, this secod ter becoes Fially, we get E k,...,k r 2ac x rt logab +ab t rli k,...,k r logab E k,...,k r x t. + x rlogac x E k,...,k r x νe r logab k,...,k r ν x+logabe k,...,k r x. ν ν0 0 t. 9

10 3 Syetrized Geeralizatio of E k,k 2,...,k r x;a,b,c The poly-euler polyoials E k x; a, b, c have bee give syetrized geeralizatio [6] of the for D x,y;a,b,c la+lb k0 k E k x;a,b,c k ylc+la. la+lb This syetrized geeralizatio possesses the followig double geeratig fuctio 0 D u x,y;a,b,ct! 2ey lc+la la+lb u e xlc+la la+lb t e t+u e t e t +e t +e u e t+u 25 ad explicit forula D x,y;a,b,c 2 0! 2 l0 i0 ylc+2la+lb la+lb r0 Defiitio 3.. For, 0, we defie D x,y;a,b,c k +k k r ilcx a i+2 b i+ l lc x a i+ b i l la+lb l k,k 2,...k r r r r k E, k 2,... k r x;a,b,c la+lb l l } }. 26 kr r ylc+la. la+lb The followig theore cotais the double geeratig fuctio for D x,y;a,b,c. 27 Theore 3.2. For, 0, we have 0 Proof. 0 D u x,y;a,b,ct! 2e D u x,y;a,b,ct! r y lc+la la+lb u e r r xlc+la la+lb t e r2u+r t e t r +e t r r i et +e iu e t+iu. 28 0

11 0 k +k k r u k!k 2!...k r! k +k k r 0 uk +k k r k!k 2!...k r! k +k k r 0 t u k +k k r k!k 2!...k r! lc+la r y e la+lb u Usig idetity 3, we obtai 0 D u x,y;a,b,ct! E k, k 2,... k r x;a,b,c la+lb E k, k 2,... k r x;a,b,c la+lb E k, k 2,... k r x;a,b,c la+lb k r 0 k +k k r 0 E k, k 2,... k r kr r ylc+la t la+lb kr r ylc+la t la+lb kr r ylc+la u kr la+lb k r! t u k +k k r x;a,b,c la+lb k!k 2!...k r! lc+la r y e la+lb u e uk +k k r k!k 2!...k r! k +k k r 0 lc+la r y la+lb u e r r xlc+la la+lb t uk +k k r k!k 2!...k r! 2er y lc+la la+lb u e r r xlc+la la+lb t +e t r where Su,, 2,..., r k +k k r 0 E k, k 2,... k r k +k k r 0 r xlc+la t la+lb 2Li k, k 2,..., k r e t +e t r 0< < 2 <...< r e t r Su,, 2,..., r u k u 2 k 2...u r k r k!k 2!...k r!

12 0 0! k +k k r u +u u r e u r.! u k u 2 k 2...u r k r k,k 2,...k r Thus, 0 D u x,y;a,b,ct! 2er y lc+la la+lb u e r r xlc+la la+lb t +e t r 2er y lc+la la+lb u e r r xlc+la la+lb t +e t r 0< < 2 <...< r e t r e u r eu e t e 2u e t e u e t e 2u e t... e r u e t e r u e t 2er y lc+la la+lb 2er y lc+la la+lb u e r r xlc+la la+lb t e r2u e t r +e t r r i eiu e t u e r r xlc+la la+lb t e r2u+r t e t r +e t r r i et +e iu e t+iu. Note that equatio 25 ca easily be deduced fro equatio 28 by takig r. It is the iterestig to establish a explicit forula for D x,y;a,b,c parallel to equatio 26. To do this, let us cosider first the followig expressio fro the right-had side of 2

13 equatio 28. That is, +e t r r i et +e iu e t+iu r e t r r e t e t e iu i r r e t e t e iu i 0 r r e t e t c i e iu c i k r q0 k r 0 k r 2 0 i c i 0 k r... k 2... k 0 q 2 q 0 q ++ q! q0 e qt q e qt r e t c i e iu c i i c i 0 0 c +c c r 0 c +c c r r e t e iu c i! i } t r } u c i! i c i r } u c i! i c i e r r xlc+la la+lb d r 2 0 d r 2 0 d r 3 0 d r 2... d 2... d 0 r 2 dr 2... d i+ i t e t r e r! t e r r xlc+la la+lb t! r e q0 k0! r d i } dr 2... d c! di c i+ } t c }c i+!i+ d i u! r q 2 k e kt q 0 q ++ e qt k q! k0 q0 } t r 2 xlc+q klb+q k+r la la+lb } t 3 k+q r k q 2 0 q ++ q!

14 r r 2 xlc+q klb+q k +r la la+lb q0 k0 k+q r q 2 k 0 q ++ t } t! q! r r 2 xlc+q klb+q k +r la p la+lb p0 q0 k0 k+q r q 2 } k 0 q ++ p t! q! lc+la r } r y u e la+lb u e r 2u c i! 0 0 i c i r ylc+ r 2 lb+ r } 2 + la u la+lb! d r 2... d 2 d 0 di dr 2... d c! c } r 2 d r 2 0 d r 2 0 d r 3 0 dr 2... d i+ }c i+!i+ d u i c i+! r ylc+ r 2 lb+ r } l 2 + la l la+lb 0 l0 l d r 2... d 2 d 0 di c i+ D u x,y;a,b,ct! 0 l d r 2 0 d r 3 0 } di c i+ l d r 2 l dr 2... d c! c } } c i+!i+ d i u! 0 c +c c r l0 l d r 2... d 2... d 0 c i+!i+ d i } r 2 i d i l d r 2 d r 2 0 d r 3 0 l dr 2... d i+ i d i r ylc+ r 2 lb+ r } 2 + la la+lb l dr 2... d c! r p0 q0 k0 p c } r 2 d i... l p... l dr 2... d i+ i q 2 k 0 k+q r q ++ p! q! 4 } F p q } t u!

15 where r Fq p 2 p xlc+q klb+q k +r la. la+lb Coparig coefficiets, we obtai the followig theore. Theore 3.3. For, 0, we have D x,y;a,b,c 0 c +c c r r ylc+ r 2 lb+ r } l 2 + la la+lb l0 where l l d r 2 d r 2 0d r 3 0 di c i+ l d r 2... d 2... } c i+!i+ d i d 0 p0 q0 k0 l dr 2... d c! r p c l dr 2... d i+ } r 2 i q 2 k 0 k+q r q ++ p! q! r Fq p 2 p xlc+q klb+q k +r la. la+lb 4 Geeralized Multi Poly-Beroulli Polyoials Parallel to the defiitio of geeralized ulti poly-euler polyoials i, we have the followig geeralizatio of poly-beroulli ubers. Defiitio 4.. The geeralized ulti poly-beroulli polyoials are defied by Li k,k 2,...,k r ab t c rxt B k,k 2,...,k r b t a t r x;a,b,c t. 29 Oe ca easily prove the followig theore usig the sae arguet i derivig the idetities i Theore Theore 4.2. The geeralized ulti poly-beroulli polyoials satisfy the followig idetities. } B k,k 2,...,k r l x;a,b,c rlogc l B k,k 2,...,k r l logc;a,bx l 0 l } B k,k 2,...,k r l x;a,b,c rlogc l B k,k 2,...,k r l 0;a,bx l 0 l B k,k 2,...,k r } l l +s x;a,b,c l+s B k,k 2,...,k r l 0;a,bB s s xrlogc 0 l0 l s B k,k 2,...,k r x;a,b,c s λ s B k,k 2,...,k r λ s ;a,bh s xrlogc;λ 0 0 d i } F p q 5

16 The ext theore cotais a explicit forula for B k,k 2,...,k r x;a,b,c. Theore 4.3. Explicit Forula For k Z, 0, we have B k,k 2,...,k r x;a,b,c Proof. r>...> >0 Li k,k 2,...,k r ab t b t a t r b rt So, we get k k kr r b rt r>...> >0 r>...> >0 r>...> >0 r r 0 r r ab t r r k k kr r k k kr r k k kr r 0 rx la +lb. 30 r r r r e tlab 0 r r r r e tla++lb. Li k,k 2,...,k r ab t b t a t r e xrtlc r>...> k >0 k 2 r>...> >0 2...kr r r r 0 k k kr r r r r r 0 r r e trx la +lb rx la +lb By coparig the coefficiets of t o both sides, the proof is copleted. t The ext theore cotais a expressio of B k,k 2,...,k r x;a,b,c as polyoial i x. Theore 4.4. The geeralized ulti poly-beroulli polyoials satisfy the followig relatio B k,k 2,...,k r x;a,b,c lc i B k,k 2,...,k r i a,bx i 3 i i0 6

17 Proof. Usig 29, we have B k,k 2,...,k r x;a,b,c t Li k,k 2,...,k r ab t c xt e xtlc b t a t r xtlc i B k,k 2,...,k r i a,b ti i! i! i0 lc i B k,k 2,...,k r i a,bx i i i0 B k,k 2,...,k r a,b t t. Coparig the coefficiets of t, we obtai the desired result. Note that, whe a c e ad b, Defiitio 4. reduces to Li k,k 2,...,k r e t e t r e rxt B k,k 2,...,k r x t. 32 The followig theore gives a relatio betwee B k,k 2,...,k r x;a,b,c ad B k,k 2,...,k r x. Theore 4.5. The geeralized ulti poly-beroulli polyoials satisfy the followig relatio B k,k 2,...,k r x;a,b,c la+lb B k,k 2,...,k r xlc rlb 33 la+lb Proof. Usig 29, we have B k,k 2,...,k r x;a,b,c t Li k,k 2,...,k r ab t e xtlc b rt ab t r e xlc rlb lab tlab Li k,k 2,...,k r e tlab +e tlab la+lb B k,k 2,...,k r xlc rlb la+lb t. Coparig the coefficiets of t, we obtai the desired result. Theore 4.6. The geeralized poly-beroulli polyoials satisfy the followig relatio d dx Bk,k 2,...,k r + x;a,b,c +lcb k,k 2,...,k r x;a,b,c 34 7

18 Proof. Usig 29, we have Hece, d dx Bk,k 2,...,k r d dx Bk,k 2,...,k r d + dx Bk,k 2,...,k r x;a,b,c t x;a,b,c t tlcli k,k 2,...,k r ab t e xrtlc b t a t r rlcb k,k 2,...,k r x;a,b,c t. + x;a,b,c t lcb k,k 2,...,k r x;a,b,c t. Coparig the coefficiets of t, we obtai the desired result. The followig corollary iediately follows fro Theore 4.6 by takig c e. For brevity, let us deote B k,k 2,...,k r x;a,b,e by B k,k 2,...,k r x;a,b. Corollary 4.7. The geeralized poly-beroulli polyoials are Appell polyoials i the sese that d dx Bk,k 2,...,k r + x;a,b +B k,k 2,...,k r x; a, b 35 Cosequetly, usig the characterizatio of Appell polyoials [20, 22, 23], the followig additio forula ca easily be obtaied. Corollary 4.8. The geeralized poly-beroulli polyoials satisfy the followig additio forula B k,k 2,...,k r x+y;a,b B k,k 2,...,k r i x;a,by i 36 i i0 However, we ca derive the additio forula for B k,k 2,...,k r x;a,b,c as follows B k,k 2,...,k r x+y;a,b,c t Li k,k 2,...,k r ab t b t a t r c x+yrt Li k,k 2,...,k r ab t c xrt c yrt b t a t r B k,k 2,...,k r x;a,b,c t yrlc t yrlc i B k,k 2,...,k r t i x;a,b,c i. i0 Coparig the coefficiets of t yields the followig result. 8

19 Theore 4.9. The geeralized poly-beroulli polyoials satisfy the followig additio forula B k,k 2,...,k r x+y;a,b,c i0 rlc i B k,k 2,...,k r i x;a,b,cy i. i 5 Hurwitz-Lerch Type Multi Poly-Beroulli Polyoials Cosider the case i which x, a e ad b c for the paraeters i Defiitio 4.. The we have Li k,k 2,...,k r e t B k,k 2,...,k r t e t r. 37 This ca be geeralized usig the followig geeralizatio of Hurwitz-Lerch ultiple zeta values Φ k,k 2,...,k rz,a zr a r+ k 2 +a r+2 k 2...r +a kr. r 38 Note that Li k,k 2,...,k rz zr 0< < 2 <...< r k k 2 z r 2...kr r k 2 k 2...r k r 0< < 2 <...< r zr r z r r z r Φ k,k 2,...,k rz,r zr + k 2 +2 k 2... r +r kr Thus, we have Li k,k 2,...,k rz Φ z r k,k 2,...,k rz,r. 39 More precisely, oe ca geeralize 40 as follows Φ k,k 2,...,k r e t,a B k,k 2,...,k r t,a. 40 We call B k,k 2,...,k r,a as Hurwitz-Lerch Type Multi Poly-Beroulli Nubers. Furtherore, we cadefiethehurwitz-lerch TypeMulti Poly-BeroulliPolyoials,deotedbyB k,k 2,...,k r,a x, as follows Φ k,k 2,...,k r e t,ae rxt 9 B k,k 2,...,k r,a x t 4

20 where B k,k 2,...,k r,a 0 B k,k 2,...,k r,a. The ext theore cotais a explicit forula for B k,k 2,...,k r },a x expressed i ters of the r, β-stirlig ubers [0], which satisfy the followig expoetial geeratig β,r fuctio } β! ert e βt t. 42 β,r Theore 5.. The Hurwitz-Lerch type ulti poly-beroulli polyoials have the followig explicit forula } r! B k,k 2,...,k r r,xr,a x +a r + k 2 +a r +2 k 2...r +a kr r Proof. Usig 38 ad 42, we have B k,k 2,...,k r,a x t Φ k,k 2,...,k r e t,ae xrt r! e xrt e t r a r+ k 2 +a r+2 k 2...r +a kr r r! r r! } a r+ k 2 +a r+2 k 2...r +a kr r r r } r r! r,xr t +a r + k 2 +a r +2 k 2...r +a kr Coparig the coefficiets of t copletes the proof of the theore. Note that 42 iplies } r,0 +r r }. Hece, as a direct cosequece of Theore 5., we have the followig corollary. Corollary 5.2. The Hurwitz-Lerch type ulti poly-beroulli ubers equal } +r r! B k,k 2,...,k r r,a +a r + k 2 +a r +2 k 2...r +a kr r,xr 43 t 20

21 Whe r, equatio 44 gives B k,a 0 +! +a k which is exactly the explicit forula for Hurwitz type poly-beroulli ubers i Theore 2. of [7]. Refereces [] S. Araci, M. Acikgoz ad E. Se, O the exteded Ki s p-adic q-defored ferioic itegrals i the p-adic iteger rig, J. Nuber Theory, , [2] A. Bayad ad Y. Haahata, Arakawa-Kaeko L-fuctios ad geeralized poly-beroulli polyoials, J. Nuber Theory, 3 20, [3] A. Bayad ad Y. Haahata, Multiple polylogariths ad ulti-poly-beroulli polyoials, Fuct. Approx. Coet. Math., , [4] B. Beńyi, Advaces i Biective Cobiatorics, Ph.D. Thesis, 204. [5] C. Brewbaker, A Cobiatorial Iterpretatio of the Poly-Beroulli Nubers ad Two Ferat Aalogues, Itegers, , #A02. [6] B. Cadelpergher ad M. A. Coppo, A ew class of idetities ivolvig Cauchy ubers, haroic ubers ad zeta values, Raaua J., , [7] M. Cekci ad P. T. Youg, Geeralizatios of Poly-Beroulli ad Poly-Cauchy Nubers, Eur. J. Math., Published Olie o 0 Septeber 205, DOI 0.007/s [8] L. Cotet, Advaced Cobiatorics, D. Reidel Publishig Copay, 974. [9] M-A. Coppo ad B. Cadelpergher, The Arakawa-Kaeko Zeta Fuctio, Raaua J., , [0] R. B. Corcio, C. B. Corcio ad R. Aldea, Asyptotic Norality of ther, β-stirlig Nubers, Ars Cobi., , [] R. B. Corcio, H. Jolay, M. Aliabadi ad M. R. Darafsheh, A Note o Multi Poly- Euler Nubers ad Beroulli Polyoials, Geeral Matheatics, , ROMANIA. [2] Y. Haahata, Poly-Euler Polyoials ad Arakawa-Kaeko Type Zeta Fuctios, Fuct. Approx. Coet. Math., 5 204, [3] K. Iatoi, M. Kaeko ad E. Takeda, Multi-Poly-Beroulli Nubers ad Fiite Multiple Zeta Values, J. Iteger Seq., 7 204, Article }

22 [4] L. Jag, T. Ki, ad H. K. Pak, A ote o q-euler ad Geocchi ubers, Proc. Japa Acad. Ser. A Math. Sci., , [5] H. Jolay, R. E. Alikelaye ad S. S. Mohaad, Soe Results o the Geeralizatio of Beroulli, Euler ad Geocchi Polyoials, Acta Uiv. Apulesis Math. Ifor., 27 20, [6] H. Jolay, R. B. Corcio ad T. Koatsu, More Properties of Multi Poly-Euler Polyoials, Bol. Soc. Mat. Mex., 2 205, [7] H. Jolay, M.R. Darafsheh, R.E. Alikelaye, Geeralizatios of Poly-Beroulli Nubers ad Polyoials, It. J. Math. Cob., 2 200, 7 4. [8] M. Kaeko, Poly-Beroulli ubers, J. Théor. Nobres Bordeaux, 9 997, [9] T. Ki, q-geeralized Euler ubers ad polyoials, Russ. J. Math. Phys , [20] D. W. Lee, O Multiple Appell Polyoials, Proc. Aer. Math. Soc., 39 20, [2] Y. Oho ad Y. Sasaki, O the parity of poly-euler ubers, RIMS Kokyuroku Bessatsu, B32 202, [22] J. Shohat, The Relatio of the Classical Orthogoal Polyoials to the Polyoials of Appell, Aer. J. Math., , [23] L. Toscao, Polioi Ortogoali o Reciproci di Ortogoali Nella classe di Appell, Le Mateatiche, 956, Roberto B. Corcio Cebu Noral Uiversity Cebu City, Philippies e-ail: rcorcio@yahoo.co Hassa Jolay Uiversité des Scieces et Techologies de Lille UFR de Mathéatiques Laboratoire Paul Pailevé CNRS-UMR Villeeuve d Ascq Cedex/Frace e-ail: hassa.olay@ath.uiv-lille.fr Cristia B. Corcio Cebu Noral Uiversity Cebu City, Philippies e-ail: cristiacorcio@yahoo.co 22

23 Takao Koatsu School of Matheatics ad Statistics Wuha Uiversity Wuha Chia e-ail: 23