Some Identities on the Generalized Changhee-Genocchi Polynomials and Numbers

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1 28 Joura of Advaces i Appied Matheatics, Vo. 4, No. 1, Jauary Soe Idetities o the Geeraized Chaghee-Geocchi Poyoias ad Nubers Da-Da Zhao * ad Wuyugaowa Departet of Matheatics, Coege of Scieces ad Techoogy, Ier Mogoia Uiversity, Huhhot, Chia Eai: zhaodada1231@163.co Abstract I this paper, we geeraize the geeratig fuctio of the Chaghee-Geocchi poyoias. I particuar, by eas of the ethod of geeratig fuctios ad Riorda arrays, we study soe properties of the geeraized Chaghee-Geocchi poyoias. At the sae tie, we estabish soe idetities betwee the geeraized Chaghee-Geocchi poyoias ad other cobiatoria sequeces. Keywords: Geeraized Chaghee-Geocchi poyoias, geeraized Chaghee-Geocchi ubers, geeratig fuctios, Riorda arrays. 1 Itroductio I 2016, Byug-Moo Ki first itroduced the cocept of Chaghee-Geocchi poyoias, the Chaghee- Geocchi poyoias are defied by the geeratig fuctio (see[1] CG (x t 2og(1 + t (1 + t x. (1 2 + t Whe x 0, CG CG (0 are caed the Chaghee-Geocchi ubers. I additio, Byug-Moo Ki aso gived the Chaghee-Geocchi poyoias of the order r by the geeratig fuctio (see[1] CG (r (x t + t (2og(1 r (1 + t x. (2 2 + t For coveiece, et us reca soe defiitios ad otatios. Here, the geeraized Haroic ubers are defied by the geeratig fuctio (see[2] H,k,r (, βt ( og(1 tr (1 βt k. (3 As is we kow, the higher-order Chaghee ubers are defied by the geeratig fuctio (see[3] Ch (k t k ( t. (4 We cosider the -th twisted Daehee poyoias of order k, which are defied by the geeratig fuctio (see[4] D (k,ξ (xt + ξt (og(1 k (1 + ξt x. (5 ξt I specia case, whe x 0, D (k,ξ (0 D(k,ξ are caed twisted Daehee ubers of order r. Whe ξ 1, D (k,1 D(k are higher-order Daehee ubers. Copyright 2019 Isaac Scietific Pubishig

2 Joura of Advaces i Appied Matheatics, Vo. 4, No. 1, Jauary Next, we give severa kids of geeratig fuctios which we eed i this paper(see[5,6,7,8,9] G (r t ( 2t e t + 1 r. (6 B (r t ( t e t 1 r. (7 b (r t ( t og(1 + t r. (8 G (x 2 t ( 2 e t + 1 x. (9 Let I R[[t]] be the rig of the fora power series with rea coefficiets i soe ideteriate t, if f(t I ad f(t f t, et [t ]f(t be the coefficiet of [t ] i the fora power series of f(t. If f(t ad g(t are fora power series, the we get the foowig reatios: [t ](f(t + βg(t [t ]f(t + β[t ]g(t. (10 [t j ]f(t[t j ]g(t [t ]f(tg(t. (11 j0 A Riorda array is a coupe D (d(t, h(t i which d(t, h(t I ad h 0 h(0 0. It defies a ifiite ower triaguar array (d,k,k N accordig to the rue d,k [t ]d(th(t k. So we set {d,k } (d(t, h(t. Let D (d(t, h(t be a Riorda array ad f(t be the geeratig fuctio of the sequece {f i } i N, we have (see[10] d,k f k [t ]d(tf(h(t [t ]d(t[f(y yh(t ]. (12 k0 Recety, ay papers have bee devoted to the study of the Chaghee-Geocchi poyoias ad ubers by various ethods. I this paper, we geeraize the geeratig fuctio of the Chaghee- Geocchi poyoias o the basis of these papers, ad ivestigate soe iterestig idetities reated to the geeraized Chaghee-Geocchi poyoias ad ubers. 2 Soe Properties of the Geeraized Chaghee-Geocchi Poyoias I this paper, we cosider the geeraized Chaghee-Geocchi poyoias which are defied by the geeratig fuctio (, β x t 2k og r (1 + t (2 + βt k (1 + βt x. (13 where k 1, r 1 are itergers, ad β are rea ubers, ad β 0. Whe x 0, (, β 0 (, β are caed the geeraized Chaghee-Geocchi ubers. Particuary, whe β 1 ad k r, CG r,r (1, 1 x CG (r (x, 0. Copyright 2019 Isaac Scietific Pubishig

3 30 Joura of Advaces i Appied Matheatics, Vo. 4, No. 1, Jauary 2019 Whe β 1 ad k r 1, CG 1,1 (1, 1 x CG (x, 0. I this sectio, we study soe properties of the geeraized Chaghee-Geocchi poyoias by the geeratio fuctio ethod. For equatio (13, we aso get Hece, we have 2 k og r (1 + t [1 + (1 + βt] k 0 (, β x t 2k og r (1 + t [1 + (1 + βt] k (1 + βtx. ( k i O the other had, we aso ca get (1 + βt x ( x i β i ti i! 2 k og r (1 + t [1 + (1 + βt] k 0 ( k 0 (, β x t (1 + βt x (, β x t ( x β (, β x t (! CGk,r (, β xt. (14 (, β x t (1 + βt x ( (1 + βt (, β x t x ( β t ( ( ( k x ( 1 β t p p0 0 ( ( ( k x p (, β x t p(, β x ( 1p β p t. (15 ( p! Theore 1. Let be oegative itegers, k 1 ad r 1 are itegers, we have p p ( ( ( k x p p(, β x ( 1p β p ( p! ( x β (, β x (! 2 k r! B,r(1, 1, 2!, 3!,.... (16 By coparig the coefficiets of t o both sides of the equatio (14 ad (15, theore 1 is proved. Theore 2. Let be oegative itegers, k, r, p, q 1 are itegers, we have j j0 0 0 ( x CG k,r j (, β x (j! CG p,q j (, β y ( j! β 2 k CG p,q+r (, β y. (17 Copyright 2019 Isaac Scietific Pubishig

4 Joura of Advaces i Appied Matheatics, Vo. 4, No. 1, Jauary By equatio (9 ad equatio (13, we get j j0 0 0 ( x β (j! CGk,r j j (, β y (, β xcgp,q ( j! [t j ]2 k og r (1 + t[t j ] 2p og q (1 + t (2 + βt p (1 + βt y j0 [t ]2 k 2p og r+q (1 + t (2 + βt p (1 + βt y 2k CGp,r+q (, β y. Theore 3. Let be oegative itegers, k, r 1 are itegers, we have ( k ( β 2 (, β (! r! S 1 (, r. (18 0 By equatio (13, whe x 0, we have Hece, we ca get (1 + β 2 tk (, β t 2k og r (1 + t (2 + βt k. Here, we sipify the eft side of this equatio (1 + β 2 tk 0 For the right side, we have (see[11] CG k,r (, β t CG k,r (, β t ogr (1 + t. ( k ( β 2 t 0 ( β 2 t (, β (!. og r (1 + t r By coparig the coefficiets of t, theore 3 is proved. Coroary 1. I theore 3, whe k 1 ad 1, we get r! S 1 (, r t. CG 1,r (, β + β 2 CG1,r 1 (, β r! S 1 (, r. (, β t Coroary 2. I theore 3, whe β 1, ad k r 1, we get theoer 11 of the referece [1]. Theore 4. Let be oegative itegers, k 1, r 2 are itegers, we have +1 (, β x + kβ 2 CGk+1,r (, β x ( (! [ rcgk,r 1 ( (, β x + xβ +1 ( +1 (, β x 1]. (19 0 Copyright 2019 Isaac Scietific Pubishig

5 32 Joura of Advaces i Appied Matheatics, Vo. 4, No. 1, Jauary 2019 Let s take the derivative about t, o both sides of equatio (13, we get 1 t 1 (, β x ( 1! [ kβ 2 k+1 og r (1 + t 2 (2 + βt k+1 + 2k og r 1 (1 + t r (2 + βt k 1 + t ](1 + βtx + 2k og r (1 + t (2 + βt k βx(1 + βt x 1 kβ CG k+1,r (, β x t 2 + r 1 (, β x t ( t + βx (, β x 1 t ( 1 β t kβ CG k+1,r (, β x t 2 + ( [(!( 1 +1 r 1 (, β x 0 + xβ +1 ( +1 (, β x 1] t. By coparig the coefficiets of t o both sides of this equatio, theore 4 is proved. Theore 5. Let be oegative itegers, k, r 1 ad, 1 are itegers, we have p0 p0 ( p (, β xch ( p p(yβ p CG k+,r (, β x + y, (20 ( p (, β xcg, p p(, β y CG k+,r+ (, β x + y. (21 (, β x t Ch ( (y (βt p0 ( p p (, βch ( p(yβ 2k og r (1 + t 2 (2 + βt k ( 2 + βt (1 + βt (x+y CG k+,r (, β x + y t. p t By coparig the coefficiets of t o both sides of this equatio, equatio (20 is proved. The proof of (21 is siiar to that of (20. 3 Idetities Ivovig the Geeraized Chaghee-Geocchi Poyoias ad Nubers I this sectio, we estabish soe idetities which are reated to the geeraized Chaghee-Geocchi poyoias. The we fid the Riorda array of the geeraized Chaghee-Geocchi ubers, ad give severa idetities by eas of the Riorda arrays. Theore 6. Let be oegative itegers, k, r, p 1 are itegers, we have 0 Ch (k (x + pβ ( 1 +r H,p,r (, β (, β x. (22 Copyright 2019 Isaac Scietific Pubishig

6 Joura of Advaces i Appied Matheatics, Vo. 4, No. 1, Jauary (, β x t ( βt k (1 + βt (x+p ogr (1 + t (1 + βt p Ch (k (x + pβ t 0 ( 1 +r H,p,r (, βt Ch (k (x + pβ ( 1 +r H,p,r (, β t!. By coparig the coefficiets of t o both sides of this equatio, theore 6 is proved. Coroary 3. I theore 6, whe β, ad p 1, the foowig reatios hod: 0 ( ( 1 +r Ch (k (x + 1H(, r 1 (, x. (23 (! Theore 7. Let r, k 1 ad r 1 are itegers, the r 0 ( r Ch (k (xβ r D (r r, Accordig to the equatio (4 ad (5, we have (, β x t 2k og r (1 + t (2 + βt k (1 + βt x ( r! CGk,r (, β x. (24 2 ( 2 + βt k (1 + βt x ogr (1 + t (t r (t r Ch (k (x (βt D (r t, (tr ( Ch (k (xβ r D (r r t +r,. 0 By coparig the coefficiets of t o both sides of this equatio, theore 7 is proved. Coroary 4. I theore 7, whe 1, the foowig reatio hods: r 0 ( r Ch (k (xβ D (r r ( r! CGk,r (1, β x. (25 Coroary 5. I theore 7, whe β 1, ad k r, the foowig reatio hods: r 0 ( r Ch (r (xd (r r ( r! CG(r (x. (26 Theore 8. Let i{r, s}, k, r, s 1 are itegers, we have b (s (, β x s s (, β x s Ch (k!(! s(x s β s 1 0 r β 0 ( r ( s!, ( s!, ( r! Ch (k (xb (s r r. r > s r s r < s (27 Copyright 2019 Isaac Scietific Pubishig

7 34 Joura of Advaces i Appied Matheatics, Vo. 4, No. 1, Jauary 2019 By equatio (8,(11,(13, we have b (s (, β x ( 1 0 [t ] s t s 2k og r s (1 + t (2 + βt k s s (, β s ( s!, r > s Ch (k s(x s β s 1 ( s!, r s r β Ch (k (xb (s r r. r < s j0 p0 0 0 ( r ( r! Theore 9. Let k, r 1 ad p, 0 be itegers, we have r j r j ( ( r k j where δ,r is the Kroecker deta sybo. Repacig t by et 1 Hece, we have β s 2(j, ps 2 (r j, 2 p+ p (, β! r! δ,r, (28 i equatio (13, we have (, β (et 1 (, β (et 1 (et 1 k 2 k t r [2 + β(et 1 ]. k As is we kow, k s 2(, k t k! (see[12]. Now, we cosider the eft-had side of the equatio (29, we have (, β (et 1 [2 + β(et 1 ] k 0 j0 p0 0 (, β s 2 (, t! j j ( j By coparig the coefficiets of t r, theore 9 is proved. [2 + β(et 1 ] k 2 k t r. (29 i i0 0 2 k ( β!s 2 (i, ti i! β s 2(j, ps 2 ( j, 2 k p+ p (, β! t. By the cocept of Riorda arrays ad equatio (13, we get { CGk,r } (2 k (2 + βt k, (1 + t, we ca get the fowig resuts: Theore 10. Let, k 1 be itegers, we have (, β 1 j! β 1 Ch (k 1. (30 j1 (,β j1 (, β 1 1 j! [t ]2 k (2 + βt k [e y 1 yog(1+t ] [t 1 ]2 k (2 + βt k [t 1 ] Ch (k β t 1 Ch(k 1 β ( 1!. Hece, the idetity (30 ca be obtaied iediatey. Copyright 2019 Isaac Scietific Pubishig

8 Joura of Advaces i Appied Matheatics, Vo. 4, No. 1, Jauary Theore 11. Let, k, j, 1 be itegers, we set up the foowig equatio: j1 j1 j1 (, β 1 (, β G( j j! ( + 1 ( 1 2 CG k, (, β, (31! 0 (, β G(x j 2 j j! G ( j j! 0 [t ] 2k og (1 + t 2 (2 + βt k (2 + t [t ] ( + 1 ( 0 1 ( + k 1 k 1 [t ]2 k (2 + t k 2y [( e y + 1 yog(1+t ] 2! CG k, (, β. ( x ( 1 2 β. (32 CG k, (, β t ( + 1 ( 1 2 t Hece, the equatio (31 is proved. The proof of (32 is siiar to that of (31, ad it is oitted here. Coroary 6. I theore 11, whe β, the foowig reatios hod: j1 j1 (, G( j j! CG k+, (,, (, β G(x j 2 j j! ( 2 ( + k + x 1 Theore 12. Let, k, j, 1 be itegers, we set the foowig equatios:. ( ( +!B j (, β j! j1 CG k, +(, β. (33 j1 (, β 1 B ( j j! [t ]2 k (2 + t k y [( e y 1 yog(1+t ] [t ] 2k og (1 + t (2 + βt k (t CG k, +(, β ( +!. Hece, theoer 12 ca be obtaied iediatey. The secod ad third sectios are our ai resuts. We geeraize the geeratig fuctio of the Chaghee-Geocchi poyoias ad fid soe ew idetities by the ethod of geeratig fuctios ad Riorda arrays. Speciay, these idetities cotai soe reatios about cassica Chaghee-Geocchi poyoias. I additio, it is easy to see that cobiatios of specia sequeces ca be represeted by the geeraized Chaghee-Geocchi poyoias, such as the Chaghee poyoias ad the geeraized Haroic ubers, the Chaghee poyoias ad the Daehee ubers, etc. Ackowedgeets. The research is supported by the Natura Sciece Foudatio of Chia uder Grat ad Natura Sciece Foudatio of Ier Mogoia uder Grat 2016MS0104. Copyright 2019 Isaac Scietific Pubishig

9 36 Joura of Advaces i Appied Matheatics, Vo. 4, No. 1, Jauary 2019 Refereces 1. B. M. Ki, J. Jeog, ad S. H. Ri, Soe expicit idetities o chaghee-geocchi poyoias ad ubers, Adv. Differ. equ., vo. 2016, o. 1, p. 202, F. Z. Zhao ad Wuyugaowa, Soe resuts o a cass of geeraized haroic ubers, Utiitas Matheatica, vo. 87, o. 1, pp , D. S. Ki ad T. Ki, Higher-order chaghee ubers ad poyoias, Adv. Studies Theor. Phys., vo. 8, o. 8, pp , D. S. Ki, T. Ki, J. J. Seo, ad S. H. Lee, A ote o the twisted abda-daehee poyoias, Appied Mathetica Scieces, vo. 7, o. 141, pp , G. D. Liu, Arithetic idetities ivovig geocchi ad stirig ubers, Discrete Dyaics i Nature ad Society, vo. 2009, o. 1, pp , L. Caritz, A ote o beroui ad euer poyoias of the secod kid, Scripta Math, vo. 25, pp , G. D. Liu ad H. M. Srivastava, Expicit foruas for the örud poyoias of the first ad secod kid, Coputers ad Matheatics with Appicatios, vo. 51, o. 9-10, pp , D. S. Ki ad T. Ki, Soe idetities of higher order euer poyoias arisig fro euer basis, Itegra Trasfors ad Specia Fuctios, vo. 24, o. 9, pp , D. S. Ki, T. Ki, ad D. V. Dogy, Beroui poyoias of the secod kid ad their idetities arisig fro ubra cacuus, J. Noiear Sci. App., vo. 9, o. 3, pp , R. Sprugoi, Riorda arrays ad cobiatoria ubers, Discrete Math, vo. 132, pp , F. Qi, A recurrece forua for the first kid stirig ubers, Eprit Arxiv., vo. 10, B. N. Guo ad F. Qi, A expicit forua for beroui ubers i ters of stirig ubers of the secod kid, J. Aa. Nu. Theor., vo. 3, o. 1, pp , Copyright 2019 Isaac Scietific Pubishig