A NEW q-analogue FOR BERNOULLI NUMBERS

Transcription

1 A NEW -ANALOGUE FOR BERNOULLI NUMBERS O-YEAT CHAN AND DANTE MANNA Absrac Ispired by, we defie a ew seuece of -aalogues for he Beroulli umbers uder he framewor of Srod operaors We show ha hey o oly saisfy may ideiies similar o hose of he -aalogue proposed by Carliz 3, bu also ieresig aalyical properies as fucios of I paricular, we give a simple aalyic proof of a geeralizaio of a explici formula for he Beroulli umbers give by Woo 5 We also defie a se of -aalogues for he Sirlig umbers of he secod id wihi our framewor ad prove a -exesio of a relaed, well-ow closed form relaig Beroulli ad Sirlig umbers 00 AMS Classificaio Numbers: B68, 30C5, 05A30 Keywords: Beroulli umber, Beroulli polyomial, -aalogue, Srod operaor, Sirlig umber Iroducio The Beroulli umbers {B } are raioal umbers i a seuece defied by he biomial recursio formula m { m, m, B B m 0, m >, 0 or euivalely, he geeraig fucio m B m m! e This seuece appears i a muliude of diverse coexs, icludig umber heory, combiaorics, ad fiie approximaios of iegrals A shor lis of applicaios ca be foud a 8 -aalogues of he Beroulli umbers were firs sudied by Carliz 3, 4 i he middle of he las ceury whe he iroduced a power of o, givig a ew seuece {β m, m 0}: m { m β +, m, 3 β m 0, m > 0 Here, ad i he remaider of he paper, he parameer we mae he assumpio ha < Clearly we recover if we le i 3 Sice Carliz, here have bee may disic -aalogues of Beroulli umbers arisig from varyig moivaios Tsumura s 4 defiiio comes from a geeralizaio of a Dirichle s series, while he -Beroulli umbers due o Suslov 3 come from a geeralized Fourier series Improvemes o he laer wor were proposed by Ismail ad Rahma i, who produced a bivariae exesio of Beroulli polyomials ha comes from he iverse of a Asey-Wilso operaor Because of heir relaio o Fourier series, hese are also associaed wih a -geeralizaio of he Riema zea fucio I his paper we propose a -aalogue of Beroulli umbers which is differe from hose lised above Our defiiio is moivaed by a developme of Beroulli umbers ha appears i, which uifies he defiiio of Beroulli ad Euler umbers uder he heory of Srod Operaors The resulig seuece { β } has aural -aalogues of ideiies ad, which we cosruc

2 O-YEAT CHAN AND DANTE MANNA i Secio 3, alog wih a closed form i Secio 4 We go o o prove a umber of aalyic properies of he β as raioal fucios of, icludig iformaio abou is zeroes ad poles Nex, we defie -Beroulli polyomials usig a -aalogue of x i Secio 5, ad use his o geeralize he classical ideiy relaig Beroulli polyomials o power sums This culmiaes i aoher closed form which relaes our -Beroulli umbers o a -aalogue of Sirlig umbers of he secod id, which we discuss i Secio 6 Throughou our wor we will compare he properies of our seuece β o hose of Carliz seuece β Because of he way i is defied, our seuece is primarily combiaorial i aure, ad so, of he geeralizaios lised above, i is mos comparable o he β This o oly provides a poi of referece o he suiabiliy of β as a good -aalogue, bu also highlighs he differeces bewee i ad is closes relaive amog he myriad of -aalogues i exisece Before we coiue, le us iroduce some oaio ha is used i he remaider of he paper The -aalogue of a ieger is deoed usig he -braceoaio x : x, for all x R ad > 0 We also defie he -risig facorial or he-pochhammer symbol i he usual way: wih a; : a, 0 a; lim a; ad a; 0 : We will also mae sigifica use of he well-ow -biomial heorem, p 7, Thm : Theorem For, z < ad ay complex a we have 4 ad he limiig case az; z; a; z, a; a / 5 We also defie he -biomial coefficies by : ; I geeral, a -biomial coefficie is a polyomial i also called a Gaussia polyomial whose j -coefficie cous he umber of pariios of j wih a mos pars each less ha or eual o, p 33 Fially, we iroduce he followig coveie shorhad aalogue o he facorial:! : ;

3 A NEW -ANALOGUE FOR BERNOULLI NUMBERS 3 Basic Properies of he Carliz -aalogue We begi by collecig some basic facs abou he Carliz seuece β I is obvious from 3 ha he β will always be raioal i, ad we easily compue he firs few values: β 0, β, β 3, β β β A aalogue of also exiss for he β Defie B : m β m m! Muliplyig boh sides of 3 by m m! ad summig over m, we ge m m β 0 + β + β + m m! m β m m! + β 0 + β 0 + β 0 We observe ha he double sum is a produc of sigle sums ad use his o rewrie his formula as B + e B Ieraig, we obai B m e m m e m m m e m, which coverges as log as < I is clear ha leig recovers 3 New -Beroulli umbers I, he auhors defie he Srodumbers correspodig o a cerai probabiliy disribuio {P } via he expoeial geeraig fucio 0 P! Q, where Q is he mome geeraig fucio of he disribuio They showed ha he Srod umbers of he uiform disribuio o 0,, are acually Beroulli umbers, while he average of wo Dirac dela fucios a 0 ad yield he Euler umbers

4 4 O-YEAT CHAN AND DANTE MANNA We defie our -aalogue usig his framewor, replacig he iegral o 0, wih is correspodig -iegral The Jacso -iegral of a fucio f is defied by a fxd x : fa a a + 0 Usig his defiiio, he h -mome of he uiform disribuio o 0, is 0 µ : 0 0 x d x + Coiuig wih he momes µ, we use a -expoeial geeraig fucio wih < o obai µ + ; 0 The secod eualiy follows from 4, wih z ad a 0 As i, we ow obai he geeraig fucio for our -aalogue of he Beroulli umbers β by iverig he mome geeraig fucio: F, : β : 3 ; 0 Upo replacig by, ad usig 4 agai, i is easy o see ha we regai as eds o I is also simple o see from here ha a aalogue of holds Proposiio 3 The -Beroulli umbers β defied by 3 saisfy he -biomial recurrece m { m β β, m, 3 m 0, m > 0 Proof We see ha F, F, ;, ad by muliplyig power series o he lef usig 4, ad machig m -coefficies, he saeme follows I is i 3 where a major differece o Carliz s -aalogue is see Isead of a power of appearig i he biomial recursio, he biomial iself is replaced by is -aalogue We use his formula o calculae he firs few β, cofirmig ha his seuece is i fac subsaially differe from he β β 0, β, β 3, β β

5 A NEW -ANALOGUE FOR BERNOULLI NUMBERS 5 β Properies ad Closed Formulas I his secio we collec some of he properies of he β ad derive a closed-form formula i erms of -muliomial coefficies, which gives us a ew proof of he correspodig combiaorial closed form for B We begi by provig a srucural propery Proposiio 4 For all 0, he scaled -Beroulli umbers + β are ieger liear combiaios of raioal fucios of of he form +!!! m! + κ!, where m 0 ad he i are posiive iegers such ha κ : + + m m + We briefly remar ha he form of he erms i he previous proposiio i -Pochhammer oaio is +++ +m m ; + + m m m ; ; m ad he erm is eual o whe m 0 Proof We use srog iducio o The claim is saisfied i he base case β 0 usig m 0 Now assume ha he claim holds for all 0 From 3 we ca see ha + + β β 0 A arbirary raioal erm from a arbirary β, where 0, by assumpio, ca be wrie as a ieger imes +! +!! m! + κ! + This erm is muliplied by, leavig a ieger imes +!! +!!!! m! + κ! +! +!!! m! + κ!, wih κ m m + κ by he iducive hypohesis Therefore, he srucure of he erm i he ieger liear expasio of β is as claimed Example 4 As illusraios of he previous proposiio, we offer ad 4! 4 β3 +!3! 4!!! 6 β5 + 6! 5! 3 6!! 4! + 4 6! 3 3! 6!! 4! + 6! 3!4! 3 6!!3!3! The sig ad value of he ieger coefficie o which each erm is muliplied have combiaorial ierpreaios which are explaied i Theorem 4

6 6 O-YEAT CHAN AND DANTE MANNA 4 Closed form for β We ow derive a closed form for he -Beroulli umbers β i erms of -muliomial coefficies summed over pariios of Our heorem will imply he resul of Proposiio 4, bu firs we eed a few defiiios Defie he -muliomial for posiive iegers, ad oegaive iegers,,, m, where m ad i i, by he formula!,,, m!! m! m m Noe ha: As eds o ad we ge lim muliomial coefficie,,, m The -muliomials also geeralize -biomial coefficies, sice, he classical,,, m, We also have he followig resul due o P A MacMaho, p 4, Thm 36: Theorem 4 I a o-commuaive zero characer rig wih ideermiaes {x i } i 0, ad, such ha x j x i x i x j for all i < j, ad x i x i for all i, x + x + x x i i, i,, i xi xi for all iegers, i 0 Tha is, he coefficie of m i i +i + +i i, i,, i cous he umber of permuaios ξ ξ of he -eleme mulise { i,, i } where he mulipliciy of j i he mulise is i j, wih i + + i such ha exacly m pairs of he ξ i, ξ j appear wih i < j bu ξ i > ξ j Le C deoe he se of ieger composiios ordered pariios of Hece a eleme of C is a ordered m-uple of iegers p,,, m, for some m, where each i > 0 ad i i We also defie he legh of he composiio as he legh of he uple: lp m We defie almos similarly he elemes of P, he se of uordered pariios of, excep ha he iegers i each uple are arraged i o-icreasig order so ha all pariios wih he same pars are euivale Thus he se C is bigger ha, ad coais, he se P Le NCp be he umber of composiios ha correspod o he pariio p By coveioal couig echiues, i is clear ha lp! NCp j!j! j!, where j i is he umber of is i he pariio p, for all i Example 4 Here is a eumeraio of C ad P, icludig leghs ad umber of composiios for each pariio, for 4 C4 p lp 4 3,, 3,,, 3,, 3,, 3,,, 4 P4 p lp NCp 4 3,,,, 3 3,,, 4

7 A NEW -ANALOGUE FOR BERNOULLI NUMBERS 7 We are ow ready o sae he mai heorem of his secio Theorem 4 For all, β! lp 4! +, +,, lp +,,, p C! lp 4 NCp! +, +,, lp +,,, p P where i he sum we have,,, lp p i our muliomial, followed by lp s Proof We prove he firs of he wo saemes by srog iducio o The secod follows from he firs by reidexig Begi by oicig ha β!! which proves he saeme for sice is he oly ieger composiio of Now assume ha he saeme is rue for all < where > By he recursive defiiio of β, β i0 i β i β 0 i + + +!! +! i i! i +! i! i! i lp p Ci,,! i! i! i + βi i i +, i +,, i lp +,,, where he muliomial is i, i,, i lp p, followed by i lp s, by he iducio hypohesis Now rearrage facors o produce β +! lp +! i +, i +, i +,, i lp +,,, i p Ci The -muliomial coefficie wihi he sum is ideed well-defied sice our hypohesis implies hece i + + i i lp + + i lp i, i + + i + + i i lp + + lp We rewrie he erm ha precedes he summaio o obai!! +! +!!! +,,, where he srig of oes i he muliomial is of legh Hece his erm correspods o he excepioal sigleo composiio All oher composiios of have muliple erms, ad hus ca be wrie uiuely i he form i, i, i,, i m, where i, i,, i m is a composiio of i ad i We see ha he legh of he composiio of is oe more ha he legh of he correspodig composiio of i The saeme for hus follows by reidexig he double sum as a sigle sum over composiios of Noe 43 I Example 4 we oed he exisece of cerai ieger coefficies muliplyig he raioal fucios The secod of hese wo formulas clearly shows ha he values of hese coefficies are he N Cp hey cou he umber of ordered composiios for each pariio; ad he sigs of hese coefficies are lp hey express he pariy of he legh of each pariio,,

8 8 O-YEAT CHAN AND DANTE MANNA The previous heorem has he followig immediae coseuece, from aig he limi as Corollary 4 For all,! B lp 43! +, +,, lp +,,, p C! lp 44 NCp,! +, +,, lp +,,, p P where i he sum we have,,, lp p i our muliomial, followed by lp s Remar 43 This id of closed form for he Beroulli umbers has bee described i deail by Woo 5, who carefully cosrucs a ree ad he sums is odes o ge a similar expressio However, o such resul for he β appears i he lieraure 4 Properies of F, A his poi we ur o he properies of he geeraig fucio F,, sarig wih a recursio From he represeaio 3, we have ha Afer replacig by, we have F, F, ; F, ; ; ; ; F, ; ; To obai he las eualiy we use ; ; Bu ; F ; + ; ; Therefore, 45 F, F, F, F, Euaig coefficies of we ge he followig ideiy Proposiio 4 For all, 46 β β 0 β 0 β β ; Proposiio 4 is a -aalogue of he uadraic recurrece 7, E 44 B B B B I is worh oig ha he recursio 45 does o lead o a coiued fracio expasio for F Ideed, isolaig F, gives F, F, F, + which, whe ieraed, recovers he defiiio of F, + F, We may also derive a differeial euaio for F, If we differeiae boh sides of 3 wih respec o, we fid ha F, F, ; ;

9 Noig ha we have F, F, A NEW -ANALOGUE FOR BERNOULLI NUMBERS 9 F, F, F, ; F, ;, 0 0 F, F, ;, F, If we differeiae wih respec o, we isead obai F, F, F, ; ; Combiig his wih he derivaive we ge he parial differeial euaio Proposiio F F F, + 43 Aalyic Properies of β ad is Zeroes ad Poles We ow ur our aeio o he sudy of β as raioal fucios of The degree of a raioal fucio rx ax bx is commoly defied as he maximum of he polyomial degrees of umeraor ad deomiaor; ha is, degr : max{dega, degb} O he oher had, we defie he oal degree Tdegr as he differece i degrees: Tdegr dega degb The oal degree of a raioal fucio gives a asympoic as x, sice rx x Tdegr Proposiio 44 Properies of Tdegr For ay raioal fucios r, r, a Tdegr r Tdegr + Tdegr, b Tdegr + r max{tdegr, Tdegr }, provided ha Tdegr Tdegr, c For all 0, Tdeg as raioal fucios i he variable Proof a ad b are sraighforward c Usig a, we see ha Tdeg Tdeg! Tdeg! Tdeg! + Proposiio 45 For all 0, + as a raioal fucio i he variable Tdeg β + 3

10 0 O-YEAT CHAN AND DANTE MANNA Proof We use srog iducio o Sice β 0, Tdeg β 0 0, so he base case holds For he iducive sep, suppose ha Tdeg β 3 for all 0 The by he recurrece relaio 3 we have Tdeg β + Tdeg + β Tdeg β By he iducive hypohesis, + Tdeg β which is a icreasig fucio of for 0 Thus Proposiio 44b applies ad we fid ha Tdeg β max Sice i is well-ow ha he odd idexed Beroulli umbers B + 0 for, we expec ha β + has a zero a for Looig a he lis of he β for >, we observe ha he zero a is simple, ad ha here is also a zero of order a 0 for all > We ow prove hese facs by aig limis wih he geeraig fucio Theorem 44 The β have a zero of order exacly a 0 excep a Proof Recall he geeraig fucio Replacig by / we fid ha F, β β 0 + β + β ; ; + β 48 Thus β ; ; ; + ; ; + ; + ; ; ; + + ;

11 A NEW -ANALOGUE FOR BERNOULLI NUMBERS To prove ha he order is exacly, use he fac ha o obai lim ; 0 + β + + lim 0 + We have hus prove a slighly sroger saeme ha reuired, ha is, lim 0 β /, for all To prove ha he β + have a simple zero a, we will also prove a sroger saeme ha reuired I fac, he saeme we will prove will provide a coecio o eve Beroulli umbers To isolae he β + i he geeraig fucio, we wrie β + F, F, β Therefore, 49 wih 40 Noe ha lim β+ ; + + +! lim + ; + N, N : + + ; ; + ad so by l Hôpial s Rule 4 lim N e + e + 0, lim N N, provided N exiss ad is fiie as a lef-sided derivaive I evaluaig N, i is coveie o separaely evaluae he derivaive of a relaed objec as a lemma Lemma 4 4 d ; d ; + + Proof The lef-had side is he logarihmic derivaive d d d log ; d

12 O-YEAT CHAN AND DANTE MANNA Now we sae ad prove he heorem o zeroes a Theorem 45 For all, Proof Usig 40 ad 4, lim N lim + lim β+ B 4 ; ; ; ; I each of he ifiie sums above, because of he facor of i he umeraor, all bu he erms will vaish i he limi Therefore, our oe-sided derivaive exiss: N e e 4 + e e 4 e 43 4 e Usig 43, alog wih 49 ad 4, we ge: β+ + lim +! e 4 e 3 4 e e e e + e 3 4 d d + e d B d! B ! I he sep where we expad i a series, we appeal o fac ha he expaded fucio is eve Aside from beig ieresig i is ow righ, because B 0 for all, his heorem has our desired resul as a immediae corollary Corollary 4 For all, β + has a zero of order a This also cosiues a poi of similariy wih Carliz s -Beroulli umbers Boh he previous heorem ad is corollary are similar o properies of he β 3, Sec 7, Thm iii We close his secio wih a heorem o he poles of β

13 A NEW -ANALOGUE FOR BERNOULLI NUMBERS 3 Theorem 46 For all, β has simple poles a he primiive + s roos of uiy Proof Fix We apply he explici formula 4 o fid ha he erm correspodig o he sigleo pariio gives rise o a erm / + i he sum I he remaiig pariios all he pars are less ha ad so i + i he res of he summads Thus, cacellig he! we see ha he summads are all of he form lp! NCp +! lp +! which are aalyic a primiive + s roos of uiy Thus we have simple poles a ζ + arisig from he / + erm 5 -powers ad -Beroulli polyomials Oe of he mos impora classical applicaios of he Beroulli umbers is he power sum formula, which ivolves he Beroulli polyomials B x give by 5 B x B x, ad saisfyig he geeraig fucio 5 0 B x! ex e The power sum formula ca he be expressed as 5, E S m m + B +m B +, where S m :, valid for all posiive iegers m ad I his secio we show ha our seuece β saisfy a aalogue of 53, wih a appropriae -aalogue of he Beroulli polyomials as well as ha of a h power Le us firs sae he resul Theorem 5 For all iegers, x, we have where ad σ x x σ x :, β x : : 0 + β + x β +, i,,i 0 i +i + +i β x, i, i,, i Remar 5 I is clear from he muliomial heorem ha lim, ad herefore Theorem 5 ideed specializes o 53 i he limi for all posiive iegers Before we prove Theorem 5, we eed o firs derive he geeraig fucios for ad β x

14 4 O-YEAT CHAN AND DANTE MANNA Proposiio 5 For all iegers, 54 ; Proof Expadig he righ-had side by he -biomial heorem 4 we fid ha i i ; ; i ; i i 0 i 0 ; i ; i i,,i 0 i + +i Usig 54 we may i fac exed he defiiio of o all real umbers This observaio jusifies he ame -Beroulli polyomial for β x The ex proposiio gives us he geeraig fucio for β x Proposiio 5 For x R we have 55 β x ; x ; Proof We apply he defiiio of β x o he geeraig fucio ad reidex replace by + o fid β x β x, 0 x β ; 0 x ;, ; as desired We oe ha, as usual, upo replacig by ad aig he limi as eds o, we recover he classical geeraig fucio 5 We are ow ready o prove Theorem 5 Proof of Theorem 5 Le x N recall ad simplify We cosruc he geeraig fucio for he righ-had side β+ x + β + β+ x β β x β

15 A NEW -ANALOGUE FOR BERNOULLI NUMBERS 5 where we used he fac ha β 0 x β 0 i he las eualiy Applyig 55, 3, ad 54 we obai x β+ x + β ; x + ; ; ; as reuired x 0 0 σ x, 6 -Sirlig umbers The -aalogue of h powers defied i he previous secio opes he door o may exesios of classical ideiies o our world of β I his secio, we prove a geeralizaio of 6, E 469: 6 B j0 j j! S, j j + j0 j + j j, where S, j are he Sirlig umbers of he secod id They saisfy he geeraig fucio 6 ad closed form x 0 xx x + S,, 0 S,! j j j j0 To prove our geeralizaio of 6, we defie S,, a -aalogue of S,, based o x he explici formula above Le 63 S, :! ad S0, 0 : j j, j j0 63 may be compared wih he aural aalogue wihi he Carliz framewor i 3, give by a, :! j i j0 j j The a,j appear i a closed form for he β, replacig S, j i Carliz s exesio of 6 They were furher geeralized by Gould 0 A overview is preseed i 9, where ideiies cocerig -exesios of boh Beroulli ad Sirlig umbers are geeraed via -differece euaios Our -Sirlig umbers S, saisfy a aalogue of 6: Proposiio 6 For all oegaive iegers ad fixed real x, 64 x xx x + S, 0 via

16 6 O-YEAT CHAN AND DANTE MANNA Proof By 54 ad he biomial expasio 65 wih y ;, we have x y x y + x 0 x ; x y, 0 x, ; valid o sufficiely small discs 0 <, < ε for ay real x Now we expad he h power ad applyig 54 agai, we obai x j x j j ; 0 j0 x j j j ; 0 j0 However, he iermos sum acually begis a sice he lowes power of i he expasio of ; is Therefore, x x j j j ; 0 j0 x j j j ; 0 j0 xx x + S, 0 The resul follows by euaig coefficies of / ad aalyically coiuig he ideiy o he eire ui circle < Hidde i he above proof is he fac ha for fixed he geeraig fucio for S, may be expressed as S, j! j ;! ; j0 Sice he power series expasio of his fucio begis a, we have he followig resul Proposiio 6 For all iegers 0, if > he S, 0 Sice S, 0 for >, we would a leas expec our -aalogue o vaish uder he limi I is somewha surprisig ha as fucios of hey are ideically zero However, Carliz s -Sirlig umbers a, also vaish for > 3 To obai he desired geeralizaio of 6, we firs prove a evaluaio of he -aalogue of he middle expressio i 6 i erms of a sum ivolvig β This will eveually lead o he closed form for β we are afer

17 A NEW -ANALOGUE FOR BERNOULLI NUMBERS 7 Lemma 6 For all N, we have j 66 j! j + S, j j0 0 j0 0 β ; + + Proof We begi wih he geeraig fucio T x,, : S, j 67 xx x j j + By Proposiio 6, he ier sum ca be exeded o ifiiy Applyig 63 ad exchagig he order of summaio, we have j j T x,, j m m xx x j j + j! m 0 j0 j0 j j x j m m j + 0 m j x j j m j + m ; j x j + ; j0 j0 m ; j j x j ; Add ad subrac he j 0 erm for his sum, ad he use he biomial expasio 65 agai o obai T x,, ; x ; Nex we divide by x ad ae he limi x 0 By l Hôpial s Rule, 68 T x,, lim x 0 x Expadig he umeraor i a series, we ge l l ; j 0 Combie his wih 68 ad 3 o ge l ; ; 0 j j j j T x,, lim x 0 x j0 j j0 j j j j j j + j + j j + j + F, j The, we muliply power series o he righ, ad combie he resul wih 67 ad 3, o ge he desired formula We ow sysemaically solve 66 for β The firs sep is o ge rid of β Isolaig β ad separaig he β erm from he sum gives j j! 69 β j + S, ; j β ; β + + j0 0

18 8 O-YEAT CHAN AND DANTE MANNA Subsiue β by usig 69, wih replaced by, o ge β j j! j + S, ; j j! j j + S, j j0 j0 + j0 ; β ; + + β ; + + Reidex he sum o he secod lie ad combie i wih he oher sum over, compleig he firs sep of he maipulaio: β j j! j + S, ; j j! j + j + S, j + β ; j0,, ; ; Nex, we remove β from he formula, by usig 69 agai, his ime wih replaced by The resul of his sep is β j j! j + S, ; j j! j + j0 j + S, j j0 ; ; ; j j!,, j + S, j β,, ; ; ; 3 3 j0,,,,, Coiuig i his way, we elimiae he sum idexed by, which leaves m β j j! C m, j + S m, j where C m, : pi,i,,i s Cm s j0 i, i,, i s, m ; ; s l ; ; il i l + i l + ; The -muliomial i he coefficies C m, ca be cacelled wih par of he produc, givig us C m, s s m i l + i l + pi,i,,i s Cm Noice ha oe facor iside he sum does o deped o p, ad so ca be facored ouside he sum The remaiig par is idepede of Thus we obai he followig heorem Theorem 6 Closed form for β For all N, m β m+ j j! ; m c m j + S m, j j0 l

19 A NEW -ANALOGUE FOR BERNOULLI NUMBERS 9 where c m : m m+ ; m c m s pi,i,,i s Cm l j0 j + j j m, 0 i l + i l + for m 0 Remar 6 We remar ha he facor m+ ; m 0 as, excep whe m 0 Hece, his closed form is ideed a -aalogue 6 This is aoher poi of similariy o Carliz s -aalogue; he β also admi a formula relaig hem o he a,, leadig o closed form which is a double sum 3, 63 Eve hough our closed form is a riple sum, he addiioal sum c m reveals ye aoher combiaorial coecio o ieger composiios disic from ha i 4, our aalogue of Woo s Theorem 7 Cocludig Remars Our moivaio here was o apply he uified framewor of Srod operaors from o develop a aural se of -aalogues o he Beroulli umbers More geerally, we may defie he -mome geeraig fucio of a disribuio fucio gx will be give as Q : 0 µ gx x; d x The iegral is a defiie iegral over he eire real lie, where he bouds o he ierval of iegraio may be effecively limied by he suppor of he disribuio fucio Uder his framewor, he geeraig fucio for he closely relaed -Euler umbers would hus be E : E Q, 0 where Q is he -mome geeraig fucio for he disribuio correspodig o he Euler umbers: he average of a pair of Dirac dela fucios a ad b: I, he auhors used 0 ad However, his geeraed scaled Euler umbers, whereas usig he values ad produces he umbers hemselves so ha Q E δ x + δ x d x + x, ; ; + 0 ; ; + ; This defiiio for -Euler umbers is similar o ha used i, where he auhors provide a combiaorial ierpreaio for heir Taylor coefficies as power series i, i erms of aleraig permuaios I would be ieresig o provide a similar ierpreaio for he coefficies of β as Taylor series i he variable We have also show ha he β admi similar properies ad closed forms o he major properies ad closed forms of he Carliz seuece β The oly major propery of he β ha appears i 3 which we have ye o prove is a weaer -versio of he vo Saud-Clause Theorem I is possible ha, usig oe of our closed forms, a similar heorem ca be prove for he β

20 0 O-YEAT CHAN AND DANTE MANNA We have provided oly a very brief ivesigaio io he aalyic properies of β Below is a plo of he zeros of β 0 / 0 o he complex plae, whose shape is ypical of geeral plos for he zeroes of β From his plo ad ohers, as well as explici expressios for β, we see here is a lo of srucure sill o be prove For example, he complex zeroes seem o cluser aroud he ui circle, wih gaps repulsio ear he + s roos of uiy We close wih a collecio of our observaios i he followig cojecure Cojecure 7 For all we have a β has o zeroes o he egaive real lie b β / is self-reciprocal for Tha is, / β / β This meas ha for every zero iside he ui circle here is a correspodig zero wih he same mulipliciy ouside c The zeroes of β / of larges ad smalles modulus are real d There is some small, posiive fucio f such ha he complex zeroes z of β saisfy f < z Refereces G Adrews, The Theory of Pariios, Cambridge Uiversiy Press, Cambridge, 984 J Borwei, N Cali, D Maa, Euler-Boole Summaio Revisied, Amer Mah Mohly Volume 6, Number L Carliz, -Beroulli umbers ad polyomials, Due Mah J Volume 5, Number L Carliz, -Beroulli ad Euleria umbers, Tras Amer Mah Soc Volume 76, Number The Digial Library of Mahemaical Fucios, Secio 44, release-dae , hp://dlmfisgov/44 6 The Digial Library of Mahemaical Fucios, Secio 46, release-dae , hp://dlmfisgov/46 7 The Digial Library of Mahemaical Fucios, Secio 44, release-dae , hp://dlmfisgov/44 8 The Digial Library of Mahemaical Fucios, Secio 47, release-dae , hp://dlmfisgov/47 9 T Ers, -Beroulli ad -Sirlig umbers, a umbral approach, I J Differece Eu Number H W Gould, The -Sirlig umbers of he firs ad secod ids, Due Mah J Volume 8, Number T Huber, A J Yee, Combiaorics of geeralized -Euler umbers, Jour Comb Theory Ser A, Volume 7, Issue M E H Ismail, M Rahma, Iverse operaors, -fracioal iegrals, ad -Beroulli polyomials, Jour Approx Theory Volume S K Suslov, Some expasios i basic Fourier series ad relaed opics, Jour Approx Theory Volume H Tsumura, A oe o -aalogues of he Dirichle series ad -Beroulli umbers, Jour Num Theory Volume S C Woo, A ree for geeraig Beroulli umbers, Mah Mag Volume 70, Number

21 A NEW -ANALOGUE FOR BERNOULLI NUMBERS address: Deparme of Mahemaics ad Compuer Sciece, Virgiia Wesleya College, Norfol, VA 350 USA address: