Una nueva clase de polinomios q-apostol-bernoulli de orden α. A new class of q-apostol-bernoulli polynomials of order α.

Transcription

1 Revist Tecocietífic URU Uiversidd Rfel Urdet Fcultd de Igeierí Nº 6 Eero - Juio 24 Depósito legl: ppi 242ZU4464 ISSN: U uev clse de poliomios -Apostol-eroulli de orde α Mridul Grg d Subhsh Alh Deprtmet of Mthemtics Uiversity of Rsth Jipur 324 Rsth Idi grgmridul@gmilcom d subhshlh@gmilcom Recibido: Aceptdo: Resume E este trbo e primer lugr se d u itroducció de los úmeros y poliomios de eroulli y sus - geerlicioes Luego se defie u uev clse de poliomios -Apostol-eroulli de orde α y sus úmeros correspodietes Se obtiee represetcioes eplicits teorem de dició y fórmul difereciles de est uev clse de poliomios Plbrs clve: Poliomios -Apostol-eroulli A ew clss of -Apostol-eroulli polyomils of order α Abstrct I this pper we first give itroductio of eroulli polyomils d umbers d their -geerlitios We the defie ew clss of -Apostol-eroulli polyomils of order α d correspodig umbers We obti eplicit represettios dditio theorem d differetil formul for these ewly defied clss of polyomils Key words: -Apostol-eroulli polyomils Nottios d Defiitios We shll use the followig ottios d defiitios of -theory (Gsper & Rhm [8] The -umber [ ] d the -umber fctoril [ ]! [ ] = d [ ] [ ] N re defied by! = ( The -shifted fctoril (-logue of Pochhmmer symbol is defied s ; = N with = ; = (2 67

2 68 U uev clse de poliomios -Apostol-eroulli de orde α Revist Tecocietífic URU Nº 6 Eero - Juio 24 (67-76 If we cosider ( ; wheever ( ; the s the ifiite product diverges whe d therefore ppers i formul we shll ssume tht < Further for y comple umber we hve ( ( ; ; = (3 ( ; There is oe more defiitio of -umber shifted fctorilwhich is ofte used i the defiitios of -etesio of eroulli polyomils This is s follows [ ] = [ + ] ; with [ ] ( ( The -biomil theorem is give by ( ( ( ( = = ; (4 ; ; ; = < < < (5 ; ; For ; ( C = the result c be writte s follows ; ; = = < < < ; ; ; = (6 Itroductio d umbers d their fmilir geerlitios d ( eroulli polyomils d umbers of order c be see i the tets (Erdélyi et l [6]; Olver et l[2] Some iterestig logues of the clssicl eroulli polyomils were ivestigted by Apostol [] so-clled Apostol eroulli polyomils ( ; λ Further Luo d ( Srivstv [8] itroduced d ivestigted the Apostol-eroulli polyomils of order ( ; λ Some more geerlitios d logues of these polyomils hve bee studied by reserchers mely Ntlii d erdii[2] Luo et l[4] reeti[2] Srivstv et l [23]Kurt [2] Trembly [25] The defiitios of clssicl eroulli polyomils ( -logues of eroulli umbers were first studied by Crlit[3]Therefter vrious other - logues of eroulli umbers d polyomils hve bee studied risig from vryig motivtios My uthors hve further studied d developed this subect mog which few to metio re Koblit [] Tsumur [26] Srivstv et l [24] Cecid C[4]Erst [7]Ryoo [22] Choi et l [5] Kim et l [] Luo [7] Luo d Srivstv [5] Mhmudov [9] d Lee d Ryoo[3] We recll here some of these defiitios -etesios of eroulli polyomils d umbers of order C followig geertig fuctios (see Luo d Srivstv [5] re defied by mes of the [ ] ; + [ + ] ( ; [ ] (7 =! =! e =

3 Mridul Grg d Subhsh Alh Revist Tecocietífic URU Nº 6 Eero - Juio 24 ( [ ] ; [ ] ( ; [ ] (8 =! =! e = The followig formul for ( ; i terms of c esily be obtied from (7 d (8 ; ( ( + ( ; = {[ ] } (9 Remr From the reltio (9 it is obvious tht the degree of which mes tht for o itegrl vlues of it is ot polyomil i ( is ( + i Ceci d C [4] itroduced -etesios of Apostol-eroulli polyomils d umbers re defied by mes of the followig geertig fuctios e ; ; (! + [ + ] ( λ = ( λ = = e (! [ ] ( λ = ( λ; = = Choi et l [5] gve the followig defiitios for -etesios of Apostol-eroulli polyomils d umbers of order Î N [ ] ; + [ + ] ( λ ( ; λ; (2 = [ ]! =! e = [ ] ; [ ] ( λ ( λ; (3 = [ ]! =! e = The followig formul c esily be obtied from (2 d (3 = ( λ ( + ( λ ; ; {[ ] } ; (4 Remr 2 It is observed from (4 tht the degree of ; λ; ( the ottio ( ; λ; idictes tht it should be of degree ( is ( + i wheres We would lie to metio here tht i the sme pper the uthors hve lso defied -etesio of Apostol-Euler polyomils d umbers of order Î N by the followig geertig fuctios [ ] ; + [ + ] ( λ e = E ( λ (5 = [ ]! =! 2 ; ; [ ] ; [ ] ( λ e = E ( λ (6 = [ ]! =! 2 ; From eutios (38 d (332 of the sme pper (Choi et l [5] it is esy to derive the followig reltio betwee λ; d E λ; which shows tht these re ot idepedet

4 7 U uev clse de poliomios -Apostol-eroulli de orde α Revist Tecocietífic URU Nº 6 Eero - Juio 24 (67-76 ( ( λ = ( ( + ( ( λ ; E ; ( Î N Î N (7 I the preset pper we further eted this study d defie -etesio of Apostol-eroulli ( λ polyomils of order λ C We shll lso prove i Theorem tht these re polyomils of degree i This property overcomes the shortcomig poited out i Remr 2 of erlier defiitio ( ( ; λ; A ew clss of -Apostol-eroulli polyomils d umbers of order Defiitio For λ Î C < < we defie ew clss of -Apostol-eroulli polyomils of ( λ order by mes of the followig geertig fuctio [ ] ; [ + ] ( λ λ e = [ ] (8 =! =! d correspodig umbers re give by Obviously [ ] ; [ ] ( λ λ e = [ ] (9 =! =! Specil Cses ( λ ( λ = ( If we set = Î N i (8 d (9 we get [ ] ; [ + ] ( λ λ e [ ] (2 =! =! = [ ] ; [ ] ( λ λ e [ ] (22 =! =! = give by (2 ccordig with the fol- We observe tht lowig reltio ( λ is ssocited with ( λ! = ( ; ; +! ( + ( ( ; λ; (2 ( λ (23 Further o tig = i (2 d (22 we rrive t the followig -etesio of Apostol-eroulli polyomils d umbers e =! [ + ] ( λ λ (24 = = =! [ ] ( λ λ e (25 = = Here ( λ is ssocited with ( ; ; λ give by ( ccordig with the followig reltio

5 Mridul Grg d Subhsh Alh Revist Tecocietífic URU Nº 6 Eero - Juio 24 (67-76 ( λ = ( ; λ; + ( + 7 (26 2 If we te i (8 d (9 we get the followig clss of eroulli polyomils ( d umbers λ ( λe ( λe = ( λ ( λ e = ( < lλ! (27 ( λ = ( < l λ (28! = λ We observe tht is ot comprble with the defiitio of Apostol-eroulli polyomils ( of order ( ; λ defied by Luo d Srivstv [8] s follows e = λe =! ( ( ; λ (29 [8] through the fol- Rther it is relted with Apostol Euler polyomils of order lowig reltio ( λ ( 2 ( ; λ ( ( λ E ; = E (3 3 If we set λ = i (8 d (9 we get the followig -etesio of eroulli polyomils d umbers of order [ ] ; [ + ] ( e = [ ] (3 =! =! [ ] ; [ ] ( e = [ ] =! =! (32 ( ( Here is ot comprble with the defiitios give by (7 but we hve the reltio betwee ( ( d -Euler polyomils of order E (see [Luo d Srivstv [5] by the followig reltio ( = ( 2 ( E ( ( We would lie to remr here tht give by (3 hs improvemet over give by (7 i the sese tht for o itegrl vlues of it is polyomil of degree i s obvious from the reltio (34 (33

6 72 U uev clse de poliomios -Apostol-eroulli de orde α Revist Tecocietífic URU Nº 6 Eero - Juio 24 (67-76 Eplicit Represettios Theorem For λ Î C < < we hve ( ( λ = ( λ {[ ] } (34 (b ( λ ; = λ ([ ] (35! [ ] [ ] ( λ ( Where d λ ( λ re defied by eutio (8 d (9 It is cler from (34 tht is polyomil of degree i Proof ( Usig the reltio[ + ] = [ ] + [ ] we c write (8 s e [ ] ; [ ] ( λ λ e = [ ] (36 =! =! [ ] Usig (9 i LHS of (36 we get [ ] e!! ( λ ( λ = (37 = = Writig series for [ ] e we hve the followig [ ] +!!! ( λ ( ( λ = (38 = = Usig series mipultio d eutig coefficiets of we get the desired result (34! (b Result (35 c esily be obtied from (9 o usig series of epoetil fuctio d eutig coefficiets of! Form eutio (34 it is esy to see tht ( λ We ow clculte the vlues of -Apostol-eroulli umbers for differet vlues of with the help of (35 d (34 Few -Apostol-eroulli umbers s clculted from (35 re is polyomil of degree i ( λ d polyomils ( λ

7 Mridul Grg d Subhsh Alh Revist Tecocietífic URU Nº 6 Eero - Juio 24 ( ( λ ( λ ( λ = ( λ; λ [ ] = ( λ; λ [ ] [ ] λ [ ] + = + ( λ 2 2 ; 2 ( λ; + + (39 Further usig bove vlues i (34 we get few -Apostol-eroulli polyomils s follows ( λ ( λ = ( λ ( λ ( λ ( λ = ( 2 ( λ ( λ 2 ( λ ( λ ( λ ( λ ( λ = ( 2 (4 Theorem 2 For λ Î C < < we hve ( λ = Φ ( λ (4 where Φ µ ( s * Φ ( s µ µ (see Choi et l[5] is -etesio of geerlied Hurwit-Lerch et fuctio defied by Goyl d Lddh[9] d is defied s follows [ µ ] + ; = [ ]! [ + ] ( ( µ Φ s = s C;Re > Proof I (8 if we write epoetil fuctio [ esily rrive t (4 s ] e + If we let i (4 we get the followig eplicit represettio for (42 i series form d compre it with (42 we ( λ ( λ = Φ ( λ ( λ < * (43 Additio Theorem Theorem 3 For λ Î C < < we hve ( λ y ( λ ( + y = {[ y] } (44

8 74 U uev clse de poliomios -Apostol-eroulli de orde α Revist Tecocietífic URU Nº 6 Eero - Juio 24 (67-76 Proof It follows from (8 tht [ ] ; [ + + y] ( λ λ e = ( y [ ] + (45 =! =! Usig the reltio [ + + y] = [ y] + y [ + ] (45 c be writte s e [ ] y ; [ + ] ( λ λ e = ( y [ ] + (46 =! =! [ y] Usig (8 the bove result ssumes the followig form [ y] y e y!! Epdig e the result (44 ( λ ( λ = ( + (47 = = [ y] i series form usig series mipultio d eutig coefficiets of If we let i (44 we get the followig dditio formul for ( λ defied by (27 we get! ( λ ( λ + = = y y (48 If we te λ = i (44 we get the followig result for -etesio of eroulli polyomils of order defied by (3 ( y ( ( + = {[ ] } (49 y y Differetil Formul Thoerem 4 For λ Î C < < we hve d ( λ l ( λ = d ( (5 Proof Differetitig the geertig fuctio (8 wrt d usig the followig result + [ ] l [ ] { e } = e d d (5 we esily rrive t (5 If we te the limit i (5 it gives the followig differetil formul for by (27 ( λ defied d ( λ ( λ = ( (52 d

9 Mridul Grg d Subhsh Alh Revist Tecocietífic URU Nº 6 Eero - Juio 24 ( If we te λ = i (5 we get the followig result for -etesio of eroulli polyomils of order defied by (3 d ( l ( = d ( Acowledgemet The support provided by the Uiversity Grts Commissio New Delhi through Riv Gdhi Ntiol Fellowship to oe of the uthors Subhsh Alh is grtefully cowledged Refereces Apostol TMO the Lerch Zet fuctio Pcific J Mth ( retti G Ntlii P Ricci PE Geerlitios of the eroulli d Appell polyomils Abstr Appl Al 7( Crlit L -eroulli umbers d polyomils Due Mth J 5 ( Ceci M C MSome results o -logue of the Lerch et fuctio Adv Stud Cotemp Mth 2 ( Choi J Aderso PJ Srivstv HMSome -etesios of the Apostol-eroulli d the Apostol Euler polyomils of order d the multiple Hurwit et fuctio Appl Mth Comput 99 ( Erdélyi A Mgus W Oberhitteger F Tricomi F G Higher Trscedetl Fuctios Vol McGrw-Hill New Yor Toroto d Lodo (953 7 Erst T -eroulli d -Stirlig polyomils umbrl pproch It J Differece Eu ( Gsper G Rhm M sic Hypergeometric Series Cmbridge Uiversity Press Cmbridge (24 9 Goyl SP Lddh RKO the geerlied Riem Zet fuctios d the geerlied Lmbert trsform Git Sdesh ( Kim T Kim Y-H d Hwg K-W O the -etesios of the eroulli d Euler umbers relted idetities d Lerch et fuctio Proc Jgeo Mth Soc 2 ( Koblit N O Crlit s -eroulli umbers J Number Theory 4 ( Kurt A further geerlitio of the eroulli polyomils d o the 2D-eroulli polyomils 2 y Appl Mth Sci 4(47 ( Lee HY Ryoo CS A ote o the geerlied higher-order -eroulli umbers d polyomils with weight Tiwese J Mth 7(3 ( Luo Q M Guo N Qi F Debth LGeerlitios of eroulli umbers d polyomils It J Mth Mth Sci 59 ( Luo Q-M Srivstv HM -Etesios of some reltioship betwee the eroulli d Euler polyomils Tiwese J Mth 5( ( Luo Q-M Apostol-Euler polyomils of higher order d Gussi hypergeometric fuctiois Tiwese J Mth ( (53

10 76 U uev clse de poliomios -Apostol-eroulli de orde α Revist Tecocietífic URU Nº 6 Eero - Juio 24 ( Luo Q-M Some results for the -eroulli polyomils d -Euler polyomils J Mth Al Appl 363 ( Luo Q-M Srivstv H M Some geerlitios of the Apostol-eroulli d Apostol-Euler Polyomils J Mth Al Appl 38( Mhmudov N I O clss of -eroulli d -Euler polyomils Adv Differ Eu 23:8 (23doi:86/ Ntlii P errdii A A geerlitio of the eroulli Polyomils J Appl Mth 3 ( Olver FWJ Loier DW oisvert WF Clr CW: NIST Hdboo of Mthemticl Fuctios NIST d Cmbridge Uiversity Press New Yor (2 22 Ryoo CS A ote o -eroulli umbers d polyomils Appl Mth Lett 2 ( Srivstv H M Grg M Choudhry S: A ew Geerlitio of the eroulli d Relted Polyomils Russ J Mth Phys 7(2 ( Srivstv HM Kim T Simse Y -eroulli umbers d polyomils ssocited with multiple -Zet fuctios d bsic L-series Russi J Mth Phys 2 ( Trembly R Gboury S Fugère J: A ew clss of geerlied Apostol-eroulli polyomils d some logues of the Srivstv-Pitér dditio theorem Appl Mth Lett 24 ( Tsumur H A ote o -logues of the Dirichlet series d -eroulli umbers Jour Num Theory 39(