Bernoulli, poly-bernoulli, and Cauchy polynomials in terms of Stirling and r-stirling numbers

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1 Novembe 4, 2016 Beoulli, oly-beoulli, ad Cauchy olyomials i tems of Stilig ad -Stilig umbes Khisto N. Boyadzhiev Deatmet of Mathematics ad Statistics, Ohio Nothe Uivesity, Ada, OH 45810, USA -boyadzhiev@ou.edu Abstact. We eview ad discuss hee some esults o the eesetatio of Beoulli olyomials i tems of Stilig umbes of the secod id ad i tems of -Stilig umbes. Mathematics subject classificatio: 05A15; 11B68; 11B73. Key wods ad hases: Beoulli umbes; Beoulli olyomials; Cauchy olyomials; Stilig umbes; -Stilig umbes; fiite diffeeces. 1. Itoductio Moe tha foty yeas ago Hey Gould ublished a vey ifomative ae [6] discussig the classical fomula (1) B 1 1 ( 1) j j j, 0 j0 whee B ae the Beoulli umbes. Amog othe thigs, Gould idicated that the fomula was vey old ad was ow at least i It aeas also i Hasse [10] The eesetatio ca be witte also as (2) whee B 0 ( 1)! S(, ) 1 ( 1) S(, ) ( 1)! j0 j j j 1

2 ae the Stilig umbes of the secod id, oigiatig i the wos of James Stilig ( ) (see [2] fo the histoy of these umbes ad fo a shot oof of (2)). It would be ice to eted this esult to the Beoulli olyomials Ideed, if we elace hee fid B( ) B 0. B by thei eesetatios (2) ad chage the ode of summatio we (3) ( 1)! B ( ) S(, ) which diectly eteds (2), as B (0) B. Aothe etesio of (2) was obtaied ecetly by Guo, Mezo, ad Qi [9] (see also Neto [15]). Namely, Guo et al foud the eesetatio (4) ( 1)! B( ) S(, ) 1 0 fo all iteges, 0, whee S (, m ) ae the -Stilig umbes, etedig S(, m ). I combiatoics, S (, m ) is the umbe of ways to atitio the set {1,2,..., } ito m oemty disjoit subsets so that the umbes 1,2,..., ae i diffeet subsets. Thus S (, m) S(, m), 0 ad whe 0, the eesetatio (4) tus ito (2). The -Stilig umbes wee studied i details by Bode i [4]. Equatio (32) i Bode s ae [4] gives the followig eesetatio of -Stilig umbes i tems of Stilig umbes of the secod id (5) S (, ) S(, ) 0. Hee i the ight had side we ecogize the sum i the baces i (3). Thus (4) follows immediately fom (3) ad Bode s equatio (5). I the et sectio we discuss aothe esult, ossibly oigiatig fom Niels Nielse [16], which eteds diectly eesetatio (1). I the thid sectio we eted the eesetatio to oly- Beoulli olyomials. I sectio 4 we give some bief otes o Cauchy umbes ad olyomials. 2

3 Todoov [17] obtaied the fomula B ( ) 0 ( 1) 1 2. Nielse s eesetatio whee f ( ) f ( 1) ( ) is the fiite diffeece. It is well ow that fo ay fuctio f( ), j f ( ) ( 1) f ( j) j0 j ad thus Todoov s esult ca be witte i the fom Poositio 1. Fo evey 0 ad evey the Beoulli olyomials have the eesetatio (6) 1 j B ( ) ( 1) ( j) 0 1 j0 j. This is a diect etesio of (1), as B (0) B. The eesetatio was also obtaied by Guillea ad Sodow i [8] by etedig the wo of Hasse [10]. A ecet ideedet oof was give i [3]. The esult, howeve, is much olde; it ca be ecogized i equatio (18) o age 232 i Nielse s boo [16]. Poositio 1 imlies the eesetatio (3) ad theefoe, eteds also the Guo-Mezo-Qi esult. We have: ad whe Lemma 2. Fo evey 0 ad 0, ( 1) ( j) ( 1)! S(, ) j j j0 0 is a o-egative itege, j ( 1) ( j) ( 1)! S (, ). j0 j The oof is tivial. By eadig the biomial ad the chagig the ode of summatio we fid j j ( 1) ( j) ( 1) j j j j0 j0 0 3

4 0 j0 j 0 j ( 1) j!( 1) S(, ). The secod equatio comes fom Bode s fomula (5). 3. Poly-Beoulli umbes ad olyomials The oly-beoulli umbes ad olyomials wee itoduced by Kaeo [11]. The oly- Beoulli olyomials ( q ) ( ), q 1, ca be defied by the geeatig fuctio m t Li (1 ) q e t t e ( ) t 1 e! 0 (see Bayad ad Hamahata [1]). Hee Li q( ) is the olylogaithmic fuctio. Whe q 1 q 1, we have Li 1(1 e t ) t ad (1) () ( 1) B ( ). The umbes ( q ) (0) ae the oly-beoulli umbes. Clealy (1) ( 1) Kaeo showed that the oly-beoulli umbes ca be witte i tems of the Stilig umbes of the secod id, (7) ( 1)! ( 1) (, ) q S. ( 1) 0 This eesetatio ca be eteded to oly-beoulli olyomials by the same method as above. Coveietly, Bayad ad Hamahata [1] showed that 1 j ( ) q ( 1) ( j) 0( 1) j0 j (fo aothe oof see [3]). Eadig the biomial ad chagig the ode of summatio yields j j ( 1) ( j) ( 1) ( 1) j j j j0 j0 0 4

5 0 j0 j 0 ad this bigs to the et esult: j ( 1) ( 1) j ( 1)!( 1) S(, ), Poositio 4. Fo ay 0, q 1,!( 1) ( ) q S(, )( 1) 0 ( 1) 0. This tus ito (7) fo 0 ad tus ito (3) fo q 1. Substitutig hee, 0 a itege, we obtai the oly-beoulli aalog of fomula (4), eessig oly-bemoulli olyomials i tems of -Stilig umbes. Coollay 5. Fo ay iteges 0, 0, q 1, we have!( 1) ( ) ( 1) q S(, ) 0 ( 1) 0!( 1) ( 1) q S (, ). ( 1) 0 4. Cauchy umbes ad olyomials We metio hee vey biefly some simila develomets with Cauchy umbes ad olyomials. The Cauchy umbes of the fist id c, 0,1,..., ae defied by the eoetial geeatig fuctio c l(1 )! 0 o diectly by the itegal 5

6 1 c ( 1)...( 1) d. 0 Recallig that the Stilig umbes of the fist id s(, ) ae defied by the geeatig fuctio (see [5], [7]) ( 1)...( 1) s(, ) 0 we fid immediately the eesetatio c 0 s(, ) 1 which somewhat esembles (2). Ifomatio about Cauchy umbes ca be foud i [5], [12], [13], ad [14]. Fo istace, the above eesetatio is give o. 294 of [5]. Komatsu [12] has defied oly-cauchy umbes of the fist id c, q 1, which have the eesetatio ( q ) c 0 s(, ) q ( 1) simila to (7). The Cauchy olyomials c () z of the fist id ae defied by the fuctio () z c z, (1 ) l(1 )! 0 ad ecetly Komatsu ad Mezo [13] obtaied a eesetatio of these olyomials i tems of -Stilig umbes of the fist id s (, ) s (, ) c (). 1 0 Hee (, ) ( 1) s, whee the usiged -Stiig umbes of the fist id ae descibed i [13]. This eesetatio esembles (4). Moe details ad othe simila esults ca be foud i [13]. 6

7 Refeeces [1] Abdelmejid Bayad, Yoshioi Hamahata, Polylogaithms ad oly-beoulli olyomials, Kyushu J. Math., 65 (2011) [2] Khisto N. Boyadzhiev, Close ecoutes with the Stilig umbes of the secod id, Math. Magazie, 85 (4) (2012), [3] Khisto N. Boyadzhiev, Powe seies with biomial sums ad asymtotic easios, It. J. Math. Aalysis, Vol. 8 (2014), o. 28, [4] Adei Z. Bode, The -Stilig umbes, Discete Math., 49 (1984), [5] Louis Comtet, Advaced Combiatoics, Klue, 1974 [6] Hey W. Gould, Elicit fomulas fo Beoulli umbes, Ame. Math. Mothly, 79 (1) (1972), [7] Roald I. Gaham, Doald E. Kuth, ad Oe Patashi, Cocete Mathematics, (Addiso-Wesley, New Yo) (1994) [8] Jesus Guillea, Joatha Sodow, Double itegals ad ifiite oducts fo some classical costats via aalytic cotiuatios of Lech's tascedet, Ramauja J. 16 (2008) [9] B.-N. Guo, I. Mez o, ad F. Qi, A elicit fomula fo Beoulli olyomials i tems of -Stilig umbes of the secod id, Rocy Moutai J. Math., 2016, to aea. Available at htt://ojecteuclid.og/euclid.mjm/ [10] Helmut Hasse, Ei Summieugsvefahe fu die Riemaische -Reihe, Math. Z, 32 (1930) [11] Masaobu Kaeo, Poly-Beoulli umbes, J. Theoie Nombes Bodeau, Vol. 9 (1997), [12] Taao Komatsu, Poly-Cauchy umbes, Kyushu J. Math., Vol. 67 (2013), [13] Taao Komatsu, Istva Mezo, Seveal elicit fomulae of Cauchy olyomials i tems of -Stilig umbes, Acta Math. Huga.,148 (2) (2016), [14] Doatella Melii, Rezo Sugoli, M. Cecilia Vei, The Cauchy umbes, Discete Math., 306 (2006), [15] Atoio F. Neto, A Note o a Theoem of Guo, Mezo, ad Qi, J. Itege Seq., 19 (2016),

8 [16] Niels Nielse, Taité élémetaie des ombes de Beoulli, Gauthie-Villas, Pais, [17] Pavel G. Todoov, O the theoy of the Beoulli olyomials ad umbes, J. Math. Aal. Al. 104 (1984)