Research Article On a Class of q-bernoulli, q-euler, and q-genocchi Polynomials

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1 Absrac ad Applied Aalysis Volume 04, Aricle ID , 0 pages hp://dx.doi.org/0.55/04/ Research Aricle O a Class of -Beroulli, -Euler, ad -Geocchi Polyomials N. I. Mahmudov ad M. Momezadeh Easer Medierraea Uiversiy, Gazimagusa, Norher Cyprus, Turey Correspodece should be addressed o M. Momezadeh; mmomezadeh@gmail.com Received 0 Jauary 04; Revised 3 April 04; Acceped 3 April 04; Published 8 April 04 Academic Edior: Sofiya Osrovsa Copyrigh 04 N. I. Mahmudov ad M. Momezadeh. This is a ope access aricle disribued uder he Creaive Commos Aribuio Licese, which permis uresriced use, disribuio, ad reproducio i ay medium, provided he origial wor is properly cied. The mai purpose of his paper is o iroduce ad ivesigae a class of -Beroulli, -Euler, ad -Geocchi polyomials. The -aalogues of well-ow formulas are derived. I addiio, he -aalogue of he Srivasava-Piér heorem is obaied. Some ew ideiies, ivolvig -polyomials, are proved.. Iroducio Throughou his paper, we always mae use of he classical defiiio of uaum coceps as follows. The -shifed facorial is defied by (a; ) 0, (a;) (a; ) I is ow ha j0 j0 ( j a), ( j a), N, <, a C. (a; ) 0[ ] (/)( ) ( ) a. () The -umbers ad -facorial are defied by [a] a, (, a C); [0]!, []![] [ ]!. The -polyomial coefficie is defied by () (3) [ ] (; ), (,, N). (4) (; ) (; ) I he sadard approach o he -calculus wo expoeial fucios are used, hese -expoeial fucios ad improved ype -expoeial fucio (see []) are defied as follows: z e (z) []! ( ( ) z), 0 E (z) e / (z) 0 0 0< <, z <, 0 (/)( ) z []! ( + ( ) z), 0 < <, z C, E (z) e ( z )E ( z ) (, ) z []! 0 ( + ( ) (z/)), 0<<, 0 ( ( ) z < (z/)). (5)

2 Absrac ad Applied Aalysis The form of improved ype of -expoeial fucio E (z) moivaed us o defie a ew -addiio ad -subracio as (x y) : (x y) : I follows ha 0[ ] (, ) (, ) x y, 0,,,..., 0[ ] (, ) (, ) x ( y), 0,,,.... E (x) E (y) 0 (6) (x y) []!. (7) The Beroulli umbers {B m } m 0 are raioal umbers i a seuece defied by he biomial recursio formula: m 0 ( m )B, m, B m { 0, m >, or euivalely, he geeraig fucio 0 B! (8) e. (9) -Aalogues of he Beroulli umbers were firs sudied by Carliz [] i he middle of he las ceury whe he iroduced a ew seuece {β m } m 0 : m 0 ( m )β +, m, β m { 0, m >. (0) Here ad i he remaider of he paper, for he parameer we mae he assumpio ha <. Clearlywerecover(8) if we le i (0). The -biomial formula is ow as ( a) (a;) ( j a) j0 0[ ] (/)( ) ( ) a. () The above -sadard oaio ca be foud i [3]. Carliz has iroduced he-beroulli umbers ad polyomials i []. Srivasava ad Piér proved some relaios ad heorems bewee he Beroulli polyomials ad Euler polyomials i [4]. They also gave some geeralizaios of hese polyomials. I [4 6], he auhors ivesigaed some properies of he -Euler polyomials ad -Geocchi polyomials. They gave some recurrece relaios. I [7], Ceci e al. gave he -exesio of Geocchi umbers i a differe maer. I [8], Kim gave a ew cocep for he -Geocchi umbers ad polyomials. I [9], Simse e al. ivesigaed he -Geocchi zea fucio ad l-fucio by usig geeraig fucios ad Melli rasformaio. There are umerous rece sudies o his subjec by, amog may oher auhors, Cigler [0], Ceci e al. [7, ], Choi e al. [], Cheo [3], Luo ad Srivasava [8 0], Srivasava e al. [4, 4], Nalci ad Pashaev [5]GabouryadKur,[6], Kim e al. [7], ad Kur [8]. We firs give he defiiios of he -umbers ad polyomials. I should be meioed ha he defiiio of - Beroulli umbers i Defiiio ca be foud i [5]. Defiiio. Le C,0< <.The -Beroulli umbers b, ad polyomials B, (x, y) are defied by meas of he geeraig fucios: B () : e ( /) e (/) e ( /) E () b, 0 []!, <π, E () E (x) E (y) 0 B, (x, y) []!, <π. () Defiiio. Le C,0 < <.The -Euler umbers e, ad polyomials E, (x, y) are defied by meas of he geeraig fucios: Ê () : e ( /) e (/) +e ( /) E () + e, 0 []!, <π, E () + E (x) E (y) 0 E, (x, y) []!, <π. (3) Defiiio 3. Le C,0< <.The-Geocchi umbers g, ad polyomials G, (x, y) are defied by meas of he geeraig fucios: Ĝ () : e ( /) e (/) +e ( /) E () + g, 0 []!, <π, E () + E (x) E (y) 0 G, (x, y) []!, <π. (4)

3 Absrac ad Applied Aalysis 3 Noe ha Cigler [0] defied -Geocchi umbers as e () +e ( ) e () +e ( ) ( ) ( ; ) g,. (5) []! 0 The comparig g, wih g,,weseeha ( ) + g +, ( ;) + g +,. (6) Defiiio 4. Le C,0 < <.The-age umbers T, are defied by meas of he geeraig fucios: ah ia (i) e () e ( ) e () +e ( ) E () E () + ( ) + T +, [ + ]!. (7) I is obvious ha, by leig ed o from he lef side, we lead o he classic defiiio of hese polyomials: b, : B, (0), B, (x, y) B (x + y), e, : E, (0), E, (x, y) E (x + y), g, : G, (0), G, (x, y) G (x+y) B, (x) B (x), b, B, E, (x) E (x), e, E, G, (x) G (x), g, G. (8) Here B (x), E (x),adg (x) deoe he classical Beroulli, Euler, ad Geocchi polyomials, respecively, which are defied by e ex 0 B (x)!, e + ex e + ex 0 0 E (x)!, G (x)!. (9) The aim of he prese paper is o obai some resuls for he above ewly defied -polyomials. I should be meioed ha -Beroulli ad -Euler polyomials i our defiiios are polyomials of x ad y ad whe y 0, hey are polyomials of x. Firs advaage of his approach is ha for, B, (x, y) (E, (x, y), G, (x, y))becomes he classical Beroulli B (x + y) (Euler E (x + y), Geocchi G, (x, y))polyomialadwemayobaihe-aalogues of well-ow resuls, for example, Srivasava ad Piér [], Cheo [3], ad so forh. Secod advaage is ha, similar o he classical case, odd umbered erms of he Beroulli umbers b, ad he Geocchi umbers g, are zero, ad eve umbered erms of he Euler umbers e, are zero.. Preiary Resuls I his secio we will provide some basic formulae for he -Beroulli, -Euler, ad -Geocchi umbers ad polyomials i order o obai he mai resuls of his paper i he ex secio. Lemma 5. The -Beroulli umbers b, saisfy he followig -biomial recurrece: 0[ ] (, ),, b, b, { 0, >. Proof. By a simple muliplicaio of (8)weseeha So (0) B () E () + B (). () 0 0[ ] (, ) b, []! + 0 b, The saeme follows by comparig m coefficies. We use his formula o calculae he firs few b, : b 0,, b,, b, (+) 4 ++ [], 4[3] b 3, 0. []!. () (3) The similar propery ca be proved for -Euler umbers m 0[ ] (, ), m 0, e, + e m, { 0, m > 0. ad -Geocchi umbers m 0[ ] (, ), m, g, + g m, { 0, m >. (4) (5) Usigheaboverecurreceformulaewecalculaehefirs few e, ad g, ermsaswell: e 0,, g 0, 0, e,, g,, e, 0, e 3, [3] [] [4] 8 g, [] +, (+), g 8 3, 0. (6)

4 4 Absrac ad Applied Aalysis Remar 6. The firs advaage of he ew -umbers b,, e,,adg, is ha similar o classical case odd umbered erms of he Beroulli umbers b, ad he Geocchi umbers g, are zero, ad eve umbered erms of he Euler umbers e, are zero. Nex lemma gives he relaioship bewee -Geocchi umbers ad -Tage umbers. Lemma 7. For ay N,wehave T +, g +, ( ) + [ + ]. (7) Proof. Firs we recall he defiiio of -rigoomeric fucios: cos e (i) +e ( i) i a e (i) e ( i) e (i) +e ( i), Now by choosig zii B(z),wege B (i) I follows ha B (i) e ( i) si i E (i) e ( i) si, si e (i) e ( i), i co i e (i) +e ( i) e (i) e ( i). 0 (8) (i) b, []!. (9) si (cos isi ) co i (i) b 0, +ib, + b, []! By choosig zii Ĝ(z),wege Ĝ (i) 4i E (i) + ie ( i) cos 4i Ĝ (i) E (i) + ie ( i) cos i cos (cos isi ) I follows ha Thus 0 (i) g, []!, (i) i+a g 0, +ig, + g, []! (i) i+ g, []!. a a (i) g, []!, ( ) () g,, []! ( ) (i) ah ia (i) ig, []! () g, []! () + g +, [ + ]!, ah ia (i) e () e ( ) e () +e ( ) E () E () + ( ) + T +, [ + ]!, T +, g +, ( ) + [ + ]. (3) (33) (34) (i) i+ b, []!. (30) Sice he fucio co is eve i he above sum odd coefficies b +,,,,..., are zero, ad we ge (i) (i) co + b, + b []!, []!. (3) The followig resul is a -aalogue of he addiio heorem, for he classical Beroulli, Euler, ad Geocchi polyomials. Lemma 8 (addiio heorems). For all x, y C we have B, (x, y) B, (x, y) 0[ ] b, (x y), 0[ ] (, ) B, (x) y,

5 Absrac ad Applied Aalysis 5 E, (x, y) E, (x, y) G, (x, y) 0[ ] e, (x y), 0[ ] (, ) E, (x) y, 0[ ] g, (x y), G, (x, y) 00[ ] (, ) G, (x) y. (35) Proof. We prove oly he firs formula. I is a coseuece of he followig ideiy: 0 B, (x, y) []! E () E (x) E (y) b, 0 (x []! y) []! 0 0 0[ ] b, (x y) []!. (36) I paricular, seig y0i (35), we ge he followig formulae for -Beroulli, -Euler ad -Geocchi polyomials, respecively: B, (x) E, (x) G, (x) Seig yi (35), we ge B, (x, ) E, (x, ) G, (x, ) Clearly (39)is -aalogues of B (x+) 0[ ] (, ) b, x, 0[ ] (, ) e, x, (37) 0[ ] (, ) g, x. (38) 0[ ] (, ) B, (x), 0[ ] (, ) E, (x), 0[ ] (, ) G, (x). 0 ( )B (x), (39) respecively. E (x+) G (x+) 0 0 ( )E (x), ( )G (x), (40) Lemma 9. The odd coefficies of he -Beroulli umbers, excephefirsoe,arezero.thameasb, 0 where r + (r N). Proof. I follows from he fac ha he fucio f () b, 0 []! b, E () + (E () + E () ), By usig -derivaive we obai he ex lemma. Lemma 0. Oe has B, (x) + B, (x) D,x B, (x) [], E, (x) + E, (x) D,x E, (x) [], G, (x) + G, (x) D,x G, (x) []. Lemma (differece euaios). Oe has (4) (4) B, (x, ) B, (x) ( ; ) [] x,, (43) E, (x, ) + E, (x) ( ; ) x, 0, (44) G, (x, ) + G, (x) ( ; ) [] x,. (45) Proof. We prove he ideiy for he -Beroulli polyomials. From he ideiy E () E () E (x) E (x) + E () E (x), (46) i follows ha 0 0[ ] 0 (, ) B, (x) (, ) x + []! []! + B, (x) 0 []!. (47)

6 6 Absrac ad Applied Aalysis From (43) ad(37), (44) ad(38), we obai he followig formulae. Lemma. Oe has x [ + ( ; ) [] 0 ] ( ; ) + + B, (x), x ( ( ; ) 0[ ] ( ; ) E, (x) + E, (x)), x ( ; ) [+] + ( 0 [ + ] ( ; ) + + G, (x) + G +, (x)). (48) The above formulae are -aalogues of he followig familiar expasios: x + x [ ( + )B (x), ( )E (x) +E (x)], x + (+) [ ( + )E (x) +E + (x)], respecively. Lemma 3. The followig ideiies hold rue: (49) i follows ha 00[ ] (, ) B, (x, y) []! 00[ ] (, ) (, ) x y + []! + B, (x, y) []! Some New Formulae (5) The classical Cayley rasformaio z Cay(z, a) : ( + az)/( az) moivaed us o derive he formula for E ().I addiio, by subsiuig Cay(z, ( )/) i he geeraig formula we have B () B () ( B () B () ( + ( ) )) (53) E () +. The righ had side ca be preseed by -Euler umbers. Now euaig coefficies of we ge he followig ideiy. Ihecaseha 0,wefidhefirsimproved-Euler umber which is exacly. Proposiio 4. For all, 0[ ] B, B, 0[ ] (, ) B, (x, y) B, (x, y) 0[ ] (, ) B, E, [ ] (54) [] (x y), 0[ ] (, ) E, (x, y) + E, (x, y) (x y), 0[ ] (, ) G, (x, y) + G, (x, y) [] (x y). (50) Proof. We prove he ideiy for he -Beroulli polyomials. From he ideiy E () E () E (x) E (y) E (x) E (y) + E () E (x) E (y), (5) 0 [ ] (, ) B, E, []. Le us ae a -derivaive from he geeraig fucio, afer simplifyig he euaio, by owig he uoie rule for uaum derivaive, ad also usig oe has B () B () E () ( )(/) +( )(/) E (), D (E ()) E () + E (), (55) +( ) E () ( ) ( B () B ()). (56) I is clear ha E () E ( ). Now,byeuaig coefficies of we obai he followig ideiy.

7 Absrac ad Applied Aalysis 7 Proposiio 5. For all, 0[ ] B, B, + 0 0[ ] (, ) B, [ ] ( ) [ ( ) 0 ] 0 [ B, B +, [ + ( ) ] ] (, ) B, [ ] ( ), (, ) + + B, [ ] ( ) 0[ ] (, ) B, [ ] ( ). (57) We may also derive a differeial euaio for B (). If we differeiae boh sides of he geeraig fucio wih respec o, afer a lile calculaio we fid ha B () B () ( ( ) E () E () ( ( ) )). (58) If we differeiae B () wih respec o, weobai, isead, B () B () E 4 ( (+) ) () 4 ( ). (59) 0 4. Explici Relaioship bewee he -Beroulli ad -Euler Polyomials I his secio, we give some explici relaioship bewee he -Beroulli ad -Euler polyomials. We also obai ew formulae ad some special cases for hem. These formulae are exesios of he formulae of Srivasava ad Piér, Cheo, ad ohers. We prese aural -exesios of he mai resuls i he papers [9, ]; see Theorems 7 ad 9. Theorem 7. For N 0, he followig relaioships hold rue: B, (x, y) 0[ ] m [ B, (x) + [ j0[ j ] E, (my) (, ) j B j, (x) ] j m j ] 0[ ] m [B, (x) + B, (x, m )] E, (my). (6) Proof. Usig he followig ideiy E () E (x) E (y) we have 0 B, (x, y) 0 E () E (x) E (/m) + []! (, ) E, (my) m []! m E (/m) + E ( m my) 0 B []!, (x) 0 (6) []! Agai, usig he geeraig fucio ad combiig his wih he derivaive we ge he parial differeial euaio. Proposiio 6. Cosider he followig: B () B () + E, (my) m B []!, (x) []! 0 : I +I. I is clear ha 0 (63) B () B + () E () 0 4 ( (+) ) ( ) 4 ( ). (60) I E, (my) m B []!, (x) []! [ j ] m B, (x) E, (my) []!. (64)

8 8 Absrac ad Applied Aalysis O he oher had Therefore 0 I E, (my) m []! B, (x, y) 0 0 j0[ j ] B j, (x) (, ) j m j j []! 0 0[ ] E, (my) j0[ j ] B j, (x) (, ) j m m j j []!. []! 0 0[ ] m [ B, (x) + [ j0[ j ] (65) (, ) j B j, (x) ] j m j ] E, (my) []!. (66) I remais o euae he coefficies of. Nex we discuss some special cases of Theorem 7. Corollary 8. For N 0 he followig relaioship holds rue: B, (x, y) 0[ ] (B, (x) + ( ; ) [] x ) E, (y). (67) The formula (67) is a -exesio of he Cheo s mai resul [3]. Theorem 9. For N 0,hefollowigrelaioships E, (x, y) [+] hold rue bewee he -Beroulli polyomials ad -Euler polyomials. Proof. The proof is based o he followig ideiy: E () + E (x) E (y) Ideed 0 E () + E (y) E (/m) E, (x, y) I follows ha []! E, (y) []! 0 B, (mx) 0 E (/m) E ( m mx). 0 (, ) m m []! []! E, (y) B []!, (mx) 0 : I I m []! I E, (y) B []!, (mx) m []! 0 0[ ] m E, (y) B, (mx) []! [+] [ + ] I B, (mx) 0 (69) (70) m E +, (y) B +, (mx) []!, m []! (, ) m E, (y) 0 0[ ] 0 0[ ] m B, (mx) []! + 0 m [+ + ] ( j0[ j ] (, ) j m j E j j, (y) E, (y)) B +, (mx) (68) j0[ j ] (, ) j m j j E j, (y) []!. (7) Nex we give a ieresig relaioship bewee he - Geocchi polyomials ad he -Beroulli polyomials.

9 Absrac ad Applied Aalysis 9 Theorem 0. For N 0,hefollowigrelaioship G, (x, y) [+] + 0 B, (x, y) [+] + 0 [+ m ] ( j0[ j ] (, ) j m j G j j, (x) G, (x)) B +, (my), [+ m ] ( j0[ j ] (, ) j m j B j j, (x) + B, (x)) G +, (my) (7) holds rue bewee he -Geocchi ad he -Beroulli polyomials. Proof. Usig he followig ideiy m 0 ( 0[ ] B, (my) 0 m m [ ] 0 0 (, ) m G, (x) G, (x)) m []! []! ( j0[ j ] (, ) j m j G j j, (x) G, (x)) B, (my) []! [+ [+] m ] ( j0[ j ] (, ) j m j G j j, (x) G, (x)) B +, (my) []!. (74) Thesecodideiycabeprovedialiemaer. Coflic of Ieress The auhors declare ha here is o coflic of ieress wih ay commercial ideiies regardig he publicaio of his paper. we have 0 E () + E (x) E (y) m E () + E (x) (E ( m ) )m G, (x, y) 0 0 m /m E (/m) E ( m my) []! G, (x, y) (, ) m 0 []! B []!, (my) 0 m []! G, (x, y) B []!, (my) 0 m []! (73) Refereces [] J. L. Cieślińsi, Improved -expoeial ad -rigoomeric fucios, Applied Mahemaics Leers,vol.4,o.,pp.0 4, 0. [] L. Carliz, -Beroulli umbers ad polyomials, Due Mahemaical Joural,vol.5,pp ,948. [3] G.E.Adrews,R.Asey,adR.Roy,Special Fucios, vol.7 of Ecyclopedia of Mahemaics ad Is Applicaios, Cambridge Uiversiy Press, Cambridge, Mass, USA, 999. [4] H. M. Srivasava ad Á. Piér, Remars o some relaioships bewee he Beroulli ad Euler polyomials, Applied Mahemaics Leers,vol.7,o.4,pp ,004. [5] T. Kim, -geeralized Euler umbers ad polyomials, Russia Joural of Mahemaical Physics, vol. 3, o. 3, pp , 006. [6] D. S. Kim, T. Kim, S. H. Lee, ad J. J. Seo, A oe o - Frobeius-Euler umbers ad polyomials, Advaced Sudies i Theoreical Physics,vol.7,o.8,pp ,03. [7] B. A. Kupershmid, Reflecio symmeries of -Beroulli polyomials, Joural of Noliear Mahemaical Physics,vol., suppleme, pp. 4 4, 005. [8] Q.-M. Luo, Some resuls for he -Beroulli ad -Euler polyomials, Joural of Mahemaical Aalysis ad Applicaios, vol.363,o.,pp.7 8,00.

10 0 Absrac ad Applied Aalysis [9] Q.-M. Luo ad H. M. Srivasava, Some relaioships bewee he Aposol-Beroulli ad Aposol-Euler polyomials, Compuers & Mahemaics wih Applicaios,vol.5,o.3-4,pp.63 64, 006. [0] Q.-M. Luo ad H. M. Srivasava, -exesios of some relaioships bewee he Beroulli ad Euler polyomials, Taiwaese Joural of Mahemaics,vol.5,o.,pp.4 57,0. [] H. M. Srivasava ad Á. Piér, Remars o some relaioships bewee he Beroulli ad Euler polyomials, Applied Mahemaics Leers,vol.7,o.4,pp ,004. [] N. I. Mahmudov, -aalogues of he Beroulli ad Geocchi polyomials ad he Srivasava-Piér addiio heorems, Discree Dyamics i Naure ad Sociey, vol.0,aricleid 69348, 8 pages, 0. [3] N. I. Mahmudov, O a class of -Beroulli ad -Euler polyomials, Advaces i Differece Euaios,vol.03,aricle 08, 03. [4] N. I. Mahmudov ad M. E. Kelesheri, O a class of geeralized -Beroulli ad -Euler polyomials, Advaces i Differece Euaios,vol.03,aricle5,03. [5] T. Kim, O he -exesio of Euler ad Geocchi umbers, Joural of Mahemaical Aalysis ad Applicaios,vol.36,o., pp , 007. [6] T. Kim, Noe o -Geocchi umbers ad polyomials, Advaced Sudies i Coemporary Mahemaics (Kyugshag), vol.7,o.,pp.9 5,008. [7]M.Ceci,M.Ca,ad V.Kur, -exesios of Geocchi umbers, Joural of he Korea Mahemaical Sociey, vol. 43, o., pp , 006. [8] T. Kim, A oe o he -Geocchi umbers ad polyomials, Joural of Ieualiies ad Applicaios, vol.007,aricleid 0745, 8 pages, 007. [9]Y.Simse,I.N.Cagul,V.Kur,adD.Kim, -Geocchi umbers ad polyomials associaed wih -Geocchi-ype lfucios, Advaces i Differece Euaios, vol.008,aricle ID 85750, pages, 008. [0] J. Cigler, -Chebyshev polyomials, hp://arxiv.org/abs/ [] M. Ceci, V. Kur, S. H. Rim, ad Y. Simse, O (i, ) Beroulli ad Euler umbers, Applied Mahemaics Leers,vol.,o.7,pp.706 7,008. [] J. Choi, P. J. Aderso, ad H. M. Srivasava, Some exesios of he Aposol-Beroulli ad he Aposol-Euler polyomials of order, adhemuliplehurwizzeafucio, Applied Mahemaics ad Compuaio,vol.99,o.,pp , 008. [3] G.-S. Cheo, A oe o he Beroulli ad Euler polyomials, Applied Mahemaics Leers,vol.6,o.3,pp ,003. [4] H. M. Srivasava ad C. Viga, Probabilisic proofs of some relaioships bewee he Beroulli ad Euler polyomials, Europea Joural of Pure ad Applied Mahemaics, vol.5,o., pp , 0. [5] S. Nalci ad O. K. Pashaev, -Beroulli umbers ad zeros of -sie fucio, hp://arxiv.org/abs/0.65v. [6] S. Gaboury ad B. Kur, Some relaios ivolvig Hermiebased Aposol-Geocchi polyomials, Applied Mahemaical Scieces,vol.6,o.8 84,pp ,0. [7] D. Kim, B. Kur, ad V. Kur, Some ideiies o he geeralized -Beroulli, -Euler, ad -Geocchi polyomials, Absrac ad Applied Aalysis,vol.03,AricleID9353,6pages,03. [8] V. Kur, New ideiies ad relaios derived from he geeralized Beroulli polyomials, Euler polyomials ad Geocchi polyomials, Advaces i Differece Euaios,vol.04,aricle 5, 04.

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