Multifarious Implicit Summation Formulae of Hermite-Based Poly-Daehee Ploynomials

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1 Appl. Mah. If. Sci. 12, No. 2, ( Applied Maheaics & Iforaio Scieces A Ieraioal Joural hp://dx.doi.org/ /ais/ Mulifarious Iplici Suaio Forulae of Herie-Based Poly-Daehee Ployoials Wasee A. Kha 1, K. S. Nisar 2, Ugur Dura 3, Mehe Acikgoz 4 ad Serka Araci 5, 1 Depare of Maheaics, Faculy of Sciece, Iegral Uiversiy, Luckow , Idia 2 Depare of Maheaics, College of Ars ad Sciece-Wadi Al dawaser, Price Saa bi Abdulaziz Uiversiy, Riyadh regio 11991, Saudi Arabia 3 Depare of he Basic Coceps of Egieerig, Faculy of Egieerig ad Naural Scieces, İskederu Techical Uiversiy, TR Haay, Turkey 4 Depare of Maheaics, Faculy of Sciece ad Ars, Gaziaep Uiversiy, TR Gaziaep, Turkey 5 Depare of Ecooics, Faculy of Ecooics, Adiisraive ad Social Scieces, Hasa Kalyocu Uiversiy, TR Gaziaep, Turkey Received: 13 Ja. 2018, Revised: 21 Feb. 2018, Acceped: 28 Feb Published olie: 1 Mar Absrac: I his paper, we iroduce he geeraig fucio of Herie-based poly-daehee ubers ad polyoials. By akig use of his geeraig fucio, we ivesigae soe ew ad ieresig ideiies for he Herie-based poly-daehee ubers ad polyoials icludig recurrece relaios, addiio propery ad correlaios wih poly-beroulli polyoials of secod kid. We he derive diverse iplici suaio forula for Herie-based poly-daehee ubers ad polyoials by applyig he series aipulaio ehods. Keywords: Herie polyoials, Beroulli polyoials, Daehee polyoials, Herie-based poly-daehee polyoials, Geeraig fucio, Cauchy produc, Suaio forulas. 1 Iroducio I rece years, he Daehee polyoials ad Beroulli polyoials (kow closely relaed each oher i cojucio wih heir diverse geeralizaios have bee sudied by ay auhors (cf. [1, 2, 4-22]. For exaple, Daoli e al. [4] iroduced ew fors of Beroulli ubers ad polyoials, which are exploied o derive furher classes of parial sus ivolvig geeralized several idex ay variable polyoials. Haroo e al. [5] perfored o classify fully degeerae Herie-Beroulli polyoials wih forulaio i ers of p-adic ferioic iegrals o Z p ad also illusraed ovel properies wih Daehee polyoials i a cosolidaed ad geeralized for. Kha e al. [7] iroduced a ew class of Herie uliple-poly-beroulli ubers ad polyoials of he secod kid ad ivesigae soe properies for hese polyoials, ad he derived several iplici suaio forulae ad geeral syery ideiies by usig differe aalyical eas. Ki e al. [8] sudied λ -Daehee polyoials ad ivesigaed heir properies arisig fro he p-adic iegral equaios. Ki e al. [9] cosidered he Wi-ype forula for Daehee ubers ad polyoials ad derived assored relaioships for hese polyoials ad ubers icludig close relaios wih higher-order Beroulli ubers ad hose of he secod kid. Ki e al. [10] acquired ulifarious forulas for expressig ay polyoial as liear cobiaios of wo kids of higher order Daehee polyoial basis ad he used hese forulas i order o cerai polyoials o obai ovel ad quirky ideiies ivolvig higher-order Daehee polyoials of he firs ad he secod kids. Ki e al. [11], by cosiderig Bares-ype Daehee polyoials of he firs kid as well as poly-cauchy polyoials of he firs kid, iroduced ixed-ype polyoials of hese polyoials ad exaied heir soe properies arisig fro ubral calculus. Ki e al. [12] cosidered he Daehee ubers ad polyoials of order k ad gave various relaioship Correspodig auhor e-ail: srk@hoail.co Naural Scieces Publishig Cor.

2 306 WASEEM A. KHAN e al.: Mulifarious iplici suaio forulae of... bewee Daehee polyoials of order k ad soe special polyoials. Ki e al. [13] sudied q-exesio of he Daehee polyoials ad ubers. Ki e al. [14] cosidered he poly-beroulli ubers ad polyoials of he secod kid ad preseed ew ad explici forulas for calculaig he poly-beroulli ubers of he secod kid ad he Sirlig ubers of he secod kid. Kwo e al. [15] cosidered Appell-ype Daehee polyoials ad derived ay ideiies ad forulas. Li e al. [16] defied he poly-daehee ubers ad aaied explici ideiies for hose ubers ad polyoials relaed o poly-beroulli ubers, polyoials ad hose of he secod kid. Moo e al. [17] cosidered he geeralized q-daehee ubers ad polyoials of higher order ad saed diverse ieresig forulas ad a represeaio for he as he sus of producs of he geeralized q-daehee polyoials ad ubers. Park [18] provided a p-adic iegral represeaio of he wised Daehee polyoials wih a q-paraeer ad developed soe ieresig properies. Park e al. [19] preseed Wi-ype forula for he wised Daehee polyoials ad ivesigaed heir various properies. Paha e al. [20] iroduced a ew class of geeralized Herie-Beroulli polyoials ad derived ay iplici suaio forulae ad syeric ideiies. Seo e al. [21] defied geeralized Daehee ubers of higher order ad represeed he as he sus of producs of geeralized Daehee ubers. There are various applicaios of he aforeeioed polyoials ad ubers i ay braches of o oly i aheaics ad aheaical physics, bu also i copuer ad egieerig sciece wih real world probles icludig he cobiaorial sus, cobiaorial ubers such as he Beroulli ubers ad polyoials, he Euler ubers ad polyoials, he Sirlig ubers of firs ad secod kids, he Chaghee ubers ad polyoials, ec. (see [1-22]. We ow begi wih recallig soe kow ubers ad polyoials as follows. Assuig ha N deoes he se of aural ubers wih he associaed se N 0 :N {0}. Le H (x,y be he 2-variable Kapé de Férie geeralizaio of he Herie polyoials give by eas of he followig geeraig fucio (cf. [3], [4]: saisfyig he followig propery H (x,y! ex+y2 (1 H (2x, 1H (x, where H (x are called he ordiary Herie polyoials (cf. [1]. For k N wih k > 1, he k-h polylogarih fucio is defied by Li k (z 1 z k (z Cwih z <1. (2 Noice ha if k 1, he Li 1 (z log(1 z, cf. [6], [7], [11], [14], [16]. The Daehee polyoials D (x are defied by eas of he followig geeraig fucio (cf. [8-14]: D (x! log(1+ (1+ x. (3 I case whe x0, D : D (0 sads for he Daehee ubers. The firs few Daehee ubers D are as follows. D 0 1,D 1 1 2,D 2 1 3,D 3 1 4,D 4 1 5,. The Beroulli polyoials B (x are defied via he followig expoeial geeraig fucio (cf. [5],[7]: B (x! e 1 ex ( <2π (4 where x0, B B (0 are called he Beroulli ubers. The Beroulli polyoials of he secod kid b (x are defied by he followig geeraig fucio o be (see [5],[7]: b (x! log(1+ (1+ x. (5 The poly-beroulli ubers B (k ad polyoials (x are respecively defied by (cf. [6], [7], [14], [16]: B (k Li k (1 e e 1 B (k! ad Li k(1 e e e x 1 fro which if we leig k1 i Eq. (6, i he yields B (1 : B ad B (1 (x : B (x. Recely, Kha e al. [7] iroduced he 3-variable Herie uli poly-beroulli polyoials of he secod kid via he followig geeraig fucio: where Hb (k 1,,k r (x,y,z Li k1,,k r (z! r!li k 1,,k r (1 e (log(1+ r (1+ x e y+z2 z r 0< 1 < 2 < < r r i1 k i i is he uliple polylogarih. I his paper, we cosider he Herie-based poly-daehee ubers ad polyoials. We he derive explici ideiies for hose ubers ad polyoials which are relaed o poly-beroulli ubers ad polyoials. We also ivesigae soe iplici suaio forula for he foregoig ubers ad polyoials by usig he series aipulaio ehods. B (k (x!. (6 (7 Naural Scieces Publishig Cor.

3 Appl. Mah. If. Sci. 12, No. 2, (2018 / O he properies of Herie-based poly-daehee polyoials I his par, we sar by defiig Herie-based poly-daehee polyoials H D (k (x, y, z as follows. Defiiio 1.Le N 0. The, (x,y,z log(1+! Li k (1 e (1+ x e y+z2, (8 where if we ake xyz0, he D (k sads for he poly-daehee ubers. : H D (k (0,0,0 Reark.Upo seig k 1 i Eq. (8, oe ca easily derive HD (1 (x,y,z : H D (x,y,z. (9 Reark.O seig y z 0 i Eq. (8, i reduces o he poly-daehee polyoials give by Li ad Kwo i [16, p. 220]. Reark.Takig z0 i Eq. (8, we have H D (k (x,y,0 : (x,y ha will be used i Theore 8. Theore 1.The followig resul holds rue for N 0 : ( (x,y,z D (k (xh (y,z. 0 Proof.Usig (1 ad (8, we have (x,y,z! log(1+ Li k (1 e (1+x e y+z2 ( D (k (x!( H (y,z 0! Replacig by i above equaio ad coparig he coefficies of! i boh sides, we arrive a he desired resul. Theore 2.Le N 0. Herie-based poly-daehee polyoials have he followig relaio: (x,y,z H D (k +1 (x+1,y,z +1 (x,y,z. +1 Proof.I follows fro Eq. (8 ha (x+1,y,z! (x,y,z! log(1+ Li k (1 e (1+x+1 e y+z2 log(1+ Li k (1 e (1+x e y+z2 log(1+ Li k (1 e (1+x e y+z2 (x,y,z +1!. Now coparig he coefficies of o he boh sides, we coplee he proof.. Theore 3.Herie based poly-daehee polyoials saisfy he followig addiio ideiy for N 0 ; (x+w,y,z 0 ( (x,y,z(w, where (w is well kow as fallig facorial defied as w(w 1 (w +1. Proof.By Defiiio (8, we have (x+w,y,z! log(1+ Li k (1 e (1+x+w e y+z2 log(1+ Li k (1 e (1+x e y+z2 (1+ w ( (x,y,z (w!(. 0! Replacig by i above equaio ad coparig he coefficies of! i boh sides, we ge he required resul. Theore 4.The followig correlaios holds rue for N 0 ; ( ( B H D (x,y,z B (k (x, y, z. 0 0 Proof.Cobiig Eq. (8 wih Eq. (6, i becoes log(1+ e 1 (1+x e y+z2 Li k(1 e log(1+ e 1 Li k (1 e (1+x e y+z2 (10 ( Lik (1 e ( log(1+ e 1 Li k (1 e (1+x e y+z2 ( B (k (x,y,z!. (11 ( 0 By he lef-had side of Eq. (10, usig Eq. (4 ad Eq. (7, we have log(1+ e 1 (1+x e y+z2 e 1 log(1+ (1+ x e y+z2 ( B HD (x,y,z 0!(! ( 0 ( B H D (x,y,z!. (12 Therefore, by Eq. (11 ad Eq. (12, we arrive a he desired resul. Upo seig r 1 ad y z 0 i Eq. (7, we he obai poly-beroulli polyoials of he secod kid give below: b (k (x! Li k(1 e log(1+ (1+ x. We here give a correlaio icludig classical Herie polyoials, Herie-based poly-daehee polyoials ad poly-beroulli polyoials of he secod kid. Naural Scieces Publishig Cor.

4 308 WASEEM A. KHAN e al.: Mulifarious iplici suaio forulae of... Theore 5.The followig relaio is valid for N 0 ; ( H (y,z (x,y,zb(k ( x. 0 Proof.Fro Eq. (8 ad Eq. (1, we have he followig applicaios: H (y,z! ey+z2 Li k(1 e log(1+ (1+ x ( b (k ( x 0!( ( 0 (x,y,z! (x,y,z! ( (x,y,zb (k ( x!. Coparig he coefficies of! of boh sides above, we ge he required resul. 3 Iplici suaio forulae for Herie-based poly-daehee polyoials I his secio, we ivesigae various iplici suaio forulae of Herie-based poly-daehee polyoials. Theore 6.The followig iplici suaio forula for Herie-based poly-daehee polyoials H D (k (x,y,z holds rue; q,l ( ( q l q+l (x,w,z,p0 p (w y +p q+l p (x,y,z. Proof.We firs eed he followig series aipulaio forula: f(n (x+yn N0 N!,0 y f(+ x!! (13 which ca be foud i [22, p.52 (2]. We ow cosider he followig geeraig fucio which is obaied by chagig o + u ad fro (13 i (8: log(1+ + u Li k (1 e +u (1+(+ ux e z(+u2 e y(+u q u l q+l (x,y,z q,l0 q! l!. Afer replacig y by w i Eq. (14, we equae obaied resul wih Eq. (14. I he becoes e (w y(+u q,l0 H D (k q q+l (x,y,z q! ul l! q,l0 H D (k q q+l (x,w,z q! ul l!. O expadig expoeial fucio i Eq. (14 gives (14 [(w y(+u] N N! N0 q,l0 H D (k q q+l (x,y,z q! ul l! q,l0 H D (k q q+l (x,w,z q! ul l!. (15 Fro (13 ad (15, we see (w y +p u p!p!,p0 q,l0 H D (k q q+l (x,y,z q! ul l! q,l0 H D (k q q+l (x,w,z q! ul l!. (16 Now replacig q by q, l by l p i he lef had side of Eq. (16, we ge q,l q,l0,p0 (w y +p!p! q+l p (x,y,z q u l (q! (l p! q u l q+l (x,w,z q,l0 q! l!. Fially, o equaig he coefficies of he like powers of q ad u l i he above equaio, we ge he claied resul. By subsiuig l 0 i Theore 6, we iediaely obai he followig corollary. Corollary 1.The followig forula is valid; ( q q (x,w,z q (w y q (x,y,z. Corollary 2.O replacig w by w + y ad seig x 0 i Theore 6, we ge he followig resul ivolvig Heriebased poly-daehee polyoials of oe variable; q+l (w+y,z q,l,p0 ( q ( l p w +p q+l p (z. Theore 7.Herie-based poly-daehee polyoials saisfy he followig iplici suaio forula; ( (x,y+u,z+w s s(x,y,zh s (u,w. s0 Proof.Replacig y by y+u ad z by z+w i Eq. (8 ad usig Eq. (3, we he have (x,y+u,z+w! log(1+ Li k (1 e (1+x e (y+u+(z+w2 ( (x,y,z!( H (u,w.! Now chagig by s i lef-had side ad coparig he coefficies of, we acquire he required ideiy. Theore 8.The followig correlaios holds rue; ( (x,y,z D (k s s(x,y wh s (w,z. s0 Proof.By Eq. (8, we have log(1+ Li k (1 e (1+ x e (y w e ( w+z2 D (k (x,y w!( H (w,z!. (17 By applyig Cauchy produc o righ-had side of (17, we ge H D (k ( (x,y,z! s0 ( (k s D s (x,y wh s(w,z!. Equaig he coefficies of o he boh sides above, we coplee he proof of heore. Naural Scieces Publishig Cor.

5 Appl. Mah. If. Sci. 12, No. 2, (2018 / Theore 9.Herie-based poly-daehee polyoials fulfill he followig iplici suaio forula; (x,y+1,z s0 ( s s(x, y, z. Proof.By chagig he variable y o y+1 i (8, ad by siple calculaios, our asserio follows iediaely. Therefore, we oi he proof. 4 Coclusio ad Observaio I his paper, we have cosidered he geeraig fucio of Daehee polyoials as D (x! log(1+ (1+ x (18 which was iroduced by Ki e al. [8-14]. Firsly, we have uliplied he righ-had side of (18 wih e y+z2, he i becae HD (x,y,z! log(1+ (1+ x e y+z2 (19 which was called Herie-based Daehee polyoials. Secodly, sice Li 1 ( 1 e we have cosidered (19 as (x,y,z log(1+! Li k (1 e (1+ x e y+z2 (20 saisfyig H D (1 (x,y,z : H D (x,y,z. Thus, by (20, we have iroduced he geeraig fucio of Herie-based poly-daehee polyoials ad derived heir ew properies. Also, by applyig he series aipulaio ehods o he geeraig fucio of Herie-based poly-daehee polyoials, we have obaied soe ieresig iplici suaio forulae. Refereces [1] Adrews, L. C. Special fucios for egieers ad aheaicias, Macilla Co., New York, [2] Araci, S., Acikgoz, M., Esi, A. A oe o he q- Dedekid-ype Daehee-Chaghee sus wih weigh α arisig fro odified q-geocchi polyoials wih weigh α, J. Assa Acad. Mah., 5 (2012, [3] Bell, E.T. Expoeial polyoials, A. Mah. 35 (1934, [4] Daoli, G., Lorezua, S., Cesarao, C. Fiie sus ad geeralized fors of Beroulli polyoials, Red. di Maheaica, 19, 1999, [5] Haroo, H., Kha, W. A. Degeerae Beroulli ubers ad polyoials associaed wih degeerae Herie polyoials, To be published i Co. Korea Mah. Soc. [6] Kha W. A. A oe o Herie-based poly-euler ad uli poly-euler polyoials, Pales. J. Mah. 5 (1, 17-26, [7] Kha, W. A., Ghayasuddi, M., Shadab, M. Muliple poly-beroulli polyoials of he secod kid associaed wih Herie polyoials, Fasc. Mah., 58, 2017, [8] Ki, D. S., Ki, T., Lee, S.-H., Seo, J.-J. A Noe o he labda-daehee polyoials, I. Joural of Mah. Aalysis, 7 (62, , [9] Ki, D. S., Ki, T. Daehee ubers ad polyoials, Appl. Mah. Sci., 7 (120, , [10] Ki, D. S., Ki, T. Ideiies arisig fro higherorder Daehee polyoial bases, Ope Mah. 2015; 13: [11] Ki, D. S. Ki, T., Koasu, T., Seo, J.-J. Baresype Daehee of he firs kid ad poly-cauchy of he firs kid ixed ype polyoials, Adv. Differece Equ., 2014, 2014:140. [12] Ki, D. S., Ki, T., Lee, S. H., Seo, J.-J. Higherorder Daehee ubers ad polyoials, I. Joural of Mah. Aalysis, 8 (6, 2014, [13] Ki, T., Lee, S.-H., Masour, T., Seo, J.-J. A oe o q-daehee polyoials ad ubers, Adv. Sud. Coep. Mah., 24 (2, 2014, [14] Ki, T., Kwo, H. I., Lee, S.-H., Seo, J.-J. A oe o poly-beroulli ubers ad polyoials of he secod kid, Adv. Differece Equ., 2014, 2014:219. [15] Kwo, J. K., Ri, S. H., Park, J.-W. A oe o he Appell-ype Daehee polyoials, Global J. Pure Appl. Mah., 11 (5, , [16] Li, D., Kwo, J. A oe o poly-daehee ubers ad polyoials, Proc. Jagjeo Mah. Soc., 19 (2, , [17] Moo, E. J., Park, J. W., Ri, S.-H. A oe o he geeralized q-daehee ubers of higher order, Proc. Jagjeo Mah. Soc., 17 (4, , [18] Park, J. W. O he wised Daehee polyoials wih q-paraeer, Adv. Differece Equ., 2014, 2014:304. [19] Park, J. W, Ri, S.-H, Kwo, J, The wised Daehee ubers ad polyoials, Adv. Differece Equ., 2014, 2014:1. [20] Paha, M. A., Kha, W. A. Soe iplici suaio forulas ad syeric ideiies for he geeralized Herie-Beroulli polyoials, Medierr. J. Mah., 12, 2015, [21] Seo, J.-J., Ri, S. H, Ki, T, Lee, S. H. Sus producs of geeralized Daehee ubers, Proc. Jageo Mah. Soc., 17 (1, 1-19, [22] Srivasava, H. M., Maocha, H. L. A reaise o geeraig fucios, Ellis Horwood Liied Co. New York, Naural Scieces Publishig Cor.

6 310 WASEEM A. KHAN e al.: Mulifarious iplici suaio forulae of... Wasee A. Kha is currely Assisa Professor, Depare of Maheaics, Iegral Uiversiy, Luckow, Idia. He copleed Ph.D. fro Aligarh Musli Uiversiy, Aligarh i He has ore ha seve years of acadeic ad research experiece. His research field is Special Fucios, Iegral Trasfors ad Voig fucio. He has published ore ha 50 research papers i he aioal ad ieraioal jourals of repue. He has also aeded ad delivered alks i ay Naioal ad Ieraioal Cofereces, Syposius. He is a life eber of Sociey for Special fucios ad heir Applicaios (SSFA. K. S. Nisar has bee workig as a Associae Professor, Depare of Maheaics, Price Saa bi Abdulaziz Uiversiy, Saudi Arabia. He awarded he Ph.D. degree for his research wih he Depare of Applied Maheaics, Faculy of Egieerig, Aligarh Musli Uiversiy, Idia i His curre research ieress are Special fucios, Fracioal Calculus, Machie Learig ad SAC-OCDMA. He has ore ha 100 research publicaios i various aioal ad ieraioal repued jourals. Mehe Acikgoz received M. Sc. Ad Ph. D. Fro Cukurova Uiversiy, Turkey. He is Full Professor a Uiversiy of Gaziaep. His research ieress are approxiaio heory, fucioal aalysis, p-adic aalysis ad aalyic ubers heory. Serka Araci was bor i Haay, Turkey, o Ocober 1, He has published over ha 100 papers i repued ieraioal jourals. His areas of specializaio iclude p-adic Aalysis, Theory of Aalyic Nubers, q-series ad q-polyoials, Theory of Ubral calculus, Fourier series ad Nevalia heory. Currely, he works as a lecurer a Hasa Kalyocu Uiversiy, Gaziaep, Turkey. O he oher had, Araci is a edior ad a referee for several ieraioal jourals. For deails, oe ca visi he lik: hp://ik.hku.edu.r/eng/acadeic-saff/serka- ARACI/9 Ugur Dura is a PhD sude a Uiversiy of Gaziaep, Turkey. His aser of sciece i aheaics was obaied i May 2016 fro Uiversiy of Gaziaep i Turkey. Recely, his ai research ieress are heory of (p,q-calculus, p-adic aalysis ad aalyic uber heory. Naural Scieces Publishig Cor.