Frobenius-type Eulerian and Poly-Bernoulli Mixed-type Polynomials

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1 Iteratioa Joura of Matheatica Aaysis Vo. 9, 2015, o. 15, HIKARI Ltd, Frobeius-type Eueria ad Poy-Beroui Mixed-type Poyoias Dae Sa Ki Departet of Matheatics, Sogag Uiversity Seou , Repubic of Korea Taekyu Ki Departet of Matheatics, Kwagwoo Uiversity Seou , Repubic of Korea Hyuck I Kwo Departet of Matheatics, Kwagwoo Uiversity Seou , Repubic of Korea Jogi Seo Departet of Appied Matheatics Pukyog Natioa Uiversity Pusa, Repubic of Korea Copyright c 2015 Dae Sa Ki, Taekyu Ki, Hyuck I Kwo ad Jogi Seo. This is a ope access artice distributed uder the Creative Coos Attributio Licese, which perits urestricted use, distributio, ad reproductio i ay ediu, provided the origia work is propery cited. Abstract I this paper, we cosider Frobeius-type Eueria ad poy-beroui ixed-type poyoias ad give various idetities which are derived fro ubra cacuus. Matheatics Subect Cassificatio: 05A15, 05A40, 11B68, 11B83

2 712 Dae Sa Ki, Taekyu Ki, Hyuck I Kwo ad Jogi Seo Keywords: poy-beroui poyoia, Frobeius-type Eueria poyoia, ubra cacuus 1. Itroductio ad preiiaries For k Z, the cassica poyogarith fuctio Li k (x is x Li k (x, (see [9, 13]. (1 k 1 The poy-beroui poyoias are defied by the geeratig fuctio to be Whe x 0, B (k we easiy get Li k (1 e t 1 e t e xt if 0 B (k (x t!. (2 B (k (0 are caed the poy-beroui ubers. By (2, B (k (x 0 ( B (k x 0 ( B (k x. (3 Aso, the Frobeius-type Eueria poyoias A (α (x λ of order α are give by the geeratig fuctio to be ( e (λ 1t λ α e xt where λ C with λ 1. Whe x 0, A (α (λ A (α ubers. By (4, we easiy get A (α (x λ 0 ( 0 A (α (x λ t, (see [10 13]. (4! (0 λ are caed the Frobeius-type Eueria A (α (λ x, (see [10 13, 15]. (5 For α C with α 1 ad s N, we ote that the Frobeius-Euer poyoias of order s are give by ( s 1 α e xt H e t (s (x α t, (see [1 21] (6 α! 0 ad that the higher-order Beroui poyoias are defied by the geeratig fuctio to be ( s t e xt B (s e t (x t, (s N, (see [10 21]. (7 1! 0 The Stirig ubers of the first kid are defied by the faig factoria fuctio to be (x x (x 1 (x + 1 S 1 (, x, ( 0, (8 0

3 By (11, we easiy get t k x k!δ,k, (, k 0, (12 Frobeius-type Eueria ad poy-beroui ixed-type poyoias 713 ad the Stirig ubers of the secod kid are give by x S 2 (, (x, ( 0, (see [15, 20]. (9 0 Let C be the copex uber fied ad et F be the set of a fora power series i the variabe t : { } t k F f (t a k k! a k C. (10 k0 Let P C [x] ad et P be the vector space of a iear fuctioas o P. The actio of the iear fuctioa L o the poyoia p (x is deoted by L p (x. Let f (t k0 a k tk F. The we defie the iear fuctioa k! f (t o P by f (t x a, ( 0, (see [4, 14, 20]. (11 ad δ,k is the Kroecker s sybo. For f L (t k0 L x k k! t k, we ote that f L (t x L x. The ap L f L (t is a vector space isoorphis fro P oto F. Heceforth, F is thought of as both a fora power series ad a iear fuctioa. We ca F the ubra agebra. The ubra cacuus is the study of ubra agebra. The order o (f (t of the o-zero power series f (t is the saest iteger k for which the coefficiet of t k does ot vaish. If o (f (t 1, the f (t is caed a deta series ad if o (f (t 0, the f (t is caed a ivertibe series. For f (t, g (t F with o (f (t 1 ad o (g (t 0, there exists a uique sequece s (x of poyoias such that g (t f (t k s (x!δ,k, (, k 0. The sequece s (x is caed the Sheffer sequece for (g (t, f (t, which is deoted by s (x (g (t, f (t. If s (x (1, f (t, the s (x is caed the associated sequece for f (t ad if s (x (g (t, t, the s (x is said to be the Appe sequece for g (t (see [15, 20]. Fro (12, we ote that e yt p (x p (y ad e yt p (x p (x + y. For f (t, g (t F ad p (x P, we have ad f (t f (t g (t p (x g (t f (t p (x f (t g (t p (x, (13 f (t x k t k k!, p (x t k x k p (x, (see [20]. (14 k! k0 Thus, by (14, we get k0 t k p (x p (k (x dk p (x dx k, (k 0. (15

4 714 Dae Sa Ki, Taekyu Ki, Hyuck I Kwo ad Jogi Seo Let s (x (g (t, f (t. The we have f (t s (x s 1 (x, ( 0, ds (x dx 1 ( f (t x s (x, 0 (16 s (x ad 0 s (x + y 1 g ( f (t 1 f (t x x, f (t xp (x t f (t p (x,! (17 ( s (x p (y, where p (y g (t s (y, ( g ( f (t exf(t 0 s (x t, for a x C, (see [20]. (19! For s (x (g (t, f (t, r (x (h (t, (t, we have s (x C, r (x, (see [4, 6, 7, 16, 20], (20 where 0 C, 1 ( h f (t! g ( f (t ( ( f (t x. (21 I this paper, we study Frobeius-type Eueria ad poy-beroui ixedtype poyoias ad give various idetities of those poyoias which are derived fro ubra cacuus. 2. Frobeius-type Eueria ad poy-beroui ixed-type poyoias Now, we cosider the poyoias AB (α,k (x λ whose geeratig fuctio is give by ( α Li k (1 e t e xt e (λ 1t λ 1 e t 0 AB (α,k (x λ t!, (22 where k Z, α, λ C with λ 1. AB (α,k (x λ wi be caed the Frobeius-type Eueria ad poy-beroui ixed-type poyoias. Whe x 0, AB (α,k (λ AB (α,k (0 λ are caed the Frobeius-type Eueria ad poy-beroui ixed-type ubers. We ote that ( B (k (x, A (α (x λ ad AB (α,k (x λ are the Appe sequeces 1 e for t, e (λ 1t λ α, ( α Li k (1 e t 1 λ ad e (λ 1t λ 1 e t, respectivey. 1 λ Li k (1 e t

5 ad That is, Frobeius-type Eueria ad poy-beroui ixed-type poyoias 715 ( 1 e B (k t (x (( e (λ 1t λ A (α (x λ AB (α,k (x λ Fro (16, we have Li k (1 e t, t, (23 α, t, (( e (λ 1t λ tb (k (x B (k 1 (x, tab (α,k (x λ 1 (x, ( 1. By (1, we easiy see that d dx Li k (x d dx ( 1 α 1 e t Li k (1 e t, t. ta (α (x λ A (α 1 (x λ, (24 x 1 k x Li k 1 (x. (25 Fro (12, we ca derive the foowig equatio : AB (α,k (y λ i (y λ ti i! x i0 ( α Li k (1 e t e yt e (λ 1t λ 1 e t x ( α Li k (1 e t e yt x e (λ 1t λ 1 e t ( α B (k e (λ 1t (y t λ! x 0 ( ( α B (k (y x e (λ 1t λ 0 ( B (k (y A (α (λ. Thus, by (26, we get 0 (x λ O the other had, (y λ 0 (26 ( A (α (λ B(k (x, ( 0. (27 ( α Li k (1 e t e yt e (λ 1t λ 1 e t x (28

6 716 Dae Sa Ki, Taekyu Ki, Hyuck I Kwo ad Jogi Seo Li k (1 e t 1 e t A (α (y λ t! x 0 ( A (α Lik (1 e t (y λ 1 e t x 0 ( A (α (y λ B (k. By (28, we get 0 (x λ 0 ( A (α (x λ B (k. (29 Therefore, by (27 ad (29, we obtai the foowig theore. Theore 1. For 0, we have ( A (α (λ B(k (x 0 Fro (19, we have Note that (x λ Li k (1 e t 1 e t x 0 ( A (α (x λ B (k (x λ. ( α Li k (1 e t x, ( 0. (30 e (λ 1t λ 1 e t (1 e t 1 x k 1 ( ( + 1 k 0 1 ( ( + 1 k Thus, by (30, ad (31, we get 0 (1 e t 0 ( + 1 k x (31 ( 1 e t x ( 1 (x. (x λ (32 1 ( ( α ( 1 e t x ( + 1 k e (λ 1t λ 0 1 ( ( 1 A (α ( + 1 k (x λ 0 0 0

7 Frobeius-type Eueria ad poy-beroui ixed-type poyoias 717 By (1, we get Li k (1 e t 1 e t x ( 1 ( + 1 k ( 1 ( ( ( + 1 k ( e t 1 x ( 1 ( + 1 k! { 0 Thus, by (30 ad (33, we get ( 1 + A (α (λ (x. S 2 (, ( 1 t ( + 1 k!s 2 (, ( 1 ++ ( + 1 k! x ( x (!S 2 (, } x. (33 (x λ (34 { ( 1 ( } ( α!s 0 ( + 1 k 2 (, x e (λ 1t λ ( 1 (!S 2 (, A (α (x λ ( + 1 k ( 1 ( + 1 k { ( ( (!S 2 (, 0 ( A (α (λ x ( }! ( + 1 S k 2 (, A (α (λ x. It is ot difficut to show that ( α A (α (x λ x (35 e (λ 1t λ ( ( α + 1!S 2 (, (λ 1 x. 0 By (30 ad (35, we get (x λ (36 Li ( k (1 e t α x 1 e t e (λ 1t λ

8 718 Dae Sa Ki, Taekyu Ki, Hyuck I Kwo ad Jogi Seo ( ( α + 1!S 2 (, (λ 1 B (k (x { 0 0 ( ( ( α + 1! (λ 1 S 2 (, B (k x ( ( ( } α + 1! (λ 1 S 2 (, B (k x. Therefore, by (32, (34 ad (36, we obtai the foowig theore. Theore 2. For 0, we have ( 1 ( (x λ. ( ( + 1 k ( (! ( 1 ( ( ( α + 1 A (α (λ (x ( + 1 S k 2 (, A (α (λ x! (λ 1 S 2 (, B (k x We observe that ( α Li k (1 e t t e (λ 1t λ 1 e t x ( α Li k (1 e t e (λ 1t λ 1 e t t x ( α Li k (1 e t ( e (λ 1t λ 1 e t x ( (λ ti i0 i ( (λ. Thus, by (17, (23 ad (37, we get (x λ 0 i! x Therefore, by (38, we obtai the foowig theore. (37 ( (λ x. (38

9 Frobeius-type Eueria ad poy-beroui ixed-type poyoias 719 Theore 3. For 0, we have Fro AB (α,k (x λ ( e (λ 1t λ p (x (x λ (( By (18 ad (39, we get 0 α e (λ 1t λ 1 e t 1 λ α 1 e t (x + y λ ( (λ x., t Li k, we have (1 e t Li k (1 e t AB(α,k (x λ x (1, t. (39 0 ( (x λ y. (40 For s (x (g (t, f (t, we ote that ( s +1 (x x g (t 1 g (t f (t s (x. (41 Thus, by (41, we get +1 (x λ x (x λ g (t g (t AB(α,k (x λ, (42 where ( e (λ 1t α λ 1 e t g (t Li k (1 e t. (43 Fro (43, we have g (t g (t (og g (t (44 ( α og ( e (λ 1t λ α og ( + og ( 1 e t ( og Li k 1 e t α (λ 1 e(λ 1t + t ( Lik (1 e t Li k 1 (1 e t. e (λ 1t λ e t 1 tli k (1 e t Thus, by (44, we get g (t g (t AB(α,k (x λ (45 ( α+1 αe (λ 1t Li k (1 e t x e (λ 1t λ 1 e t + t e t 1 ( Lik (1 e t Li k 1 (1 e t t (1 e t ( α x e (λ 1t λ αab (α+1,k (x + λ 1 λ + t ( ( Lik (1 e t Li k 1 (1 e t α x. e t 1 t (1 e t e (λ 1t λ

10 720 Dae Sa Ki, Taekyu Ki, Hyuck I Kwo ad Jogi Seo We observe that ( ( t Lik (1 e t Li k 1 (1 e t α x (46 e t 1 t (1 e t e (λ 1t λ 1 ( α Li k (1 e t Li k 1 (1 e t t + 1 e (λ 1t λ 1 e t e t 1 x ( ( α + 1 Li k (1 e t Li k 1 (1 e t B +1 x + 1 e (λ 1t λ 1 e t ( + 1 ( B +1 (x λ AB (α,k 1 (x λ, where B B (1 are the ordiary Beroui ubers. Therefore, by (42, (45 ad (46, we obtai the foowig theore. Theore 4. For 0, we have +1 (x λ xab (α,k (x λ + αab (α+1,k 1 +1 ( + 1 ( B (x + λ 1 λ (x λ AB (α,k 1 (x λ. Fro (17, we have ( α Li k (1 e t (y λ e yt e (λ 1t λ 1 e t x (47 (( α Li k (1 e t t e yt x 1 e (λ 1t λ 1 e ( ( t α Lik (1 e t t e yt e (λ 1t λ 1 e t x 1 ( α ( Li k (1 e t + e (λ 1t t e yt x 1 λ 1 e ( t α Li k (1 e t ( + t e yt e (λ 1t λ 1 e t x 1, ( 1. It is easy to show that ( α ( α+1 t αe (λ 1t. (48 e (λ 1t λ e (λ 1t λ Thus, by (48, we get ( t ( α α Lik (1 e t e yt e (λ 1t λ 1 e t x 1 α+1 Li k (1 e t e (y+λ 1t 1 e t ( e (λ 1t λ x 1 (49

11 Frobeius-type Eueria ad poy-beroui ixed-type poyoias 721 Fro (1, we have αab (α+1,k 1 (y + λ 1 λ. ( ( Lik (1 e t (1 e t t 1 e t t 0 ( + 1 k ( (1 e t 1 ( + 1 k 1 1 (1 e t 1 e t 0 0 ( + 1 k { e t } (1 e t (1 e t (1 e t 2 k 1 k 1 t e t 1 Li k 1 (1 e t Li k (1 e t. t (1 e t (50 Thus, by (50, we get ( α ( Li k (1 e t e (λ 1t t e yt x 1 (51 λ 1 e ( t α t Li k 1 (1 e t Li k (1 e t e yt x e (λ 1t 1 e t 1 1 e t 1 ( ( α Li k 1 (1 e t Li k (1 e t B e yt e (λ 1t λ 1 e t x 0 1 ( ( B AB (α,k 1 (y λ (y λ. 0 It is easy to show that ( α Li k (1 e t ( t e yt e (λ 1t λ 1 e t x 1 ( α Li k (1 e t y e yt e (λ 1t λ 1 e t x 1 y 1 (y λ. Therefore, by (47, (49, (51 ad (52, we obtai the foowig theore. Theore 5. For 1, we have (x λ αab (α+1,k 1 (x + λ 1 λ + x 1 (x λ + 1 ( ( B AB (α,k 1 0 (x λ (x λ. (52

12 722 Dae Sa Ki, Taekyu Ki, Hyuck I Kwo ad Jogi Seo Here, we copute the foowig expressio i two differet ways : ( α ( Li e (λ 1t k 1 e t x +1. (53 λ O the oe had, it is equa to ( α Li k (1 e t ( e (λ 1t λ 1 e t 1 e t x +1 ( α Li k (1 e t e (λ 1t λ 1 e t x+1 (x 1 +1 ( ( α + 1 ( 1 Li k (1 e t e (λ 1t λ 1 e t x 0 ( + 1 ( 1 AB (α,k (λ. 0 O the other had, it is equa to ( Li ( α k 1 e t x +1 e (λ 1t λ ( Li k 1 e t (α A +1 (x λ ˆ t ( ( Lik 1 e s ds A (α +1 (x λ 0 ˆ t e s Li k (1 e s ds 0 1 e s A(α +1 (x λ ˆ ( t ( ( s B (k 1 s ds!! ( ( 1 B (k 1 1 ( + 1! ( ( + 1 ( A(α +1 (x λ t +1 A (α +1 (x λ B (k 1 A (α (λ. Thus, by (53, (54 ad (55, we obtai the foowig theore. Theore 6. For 0, we have ( + 1 ( 1 AB (α,k (λ 0 ( ( + 1 ( 1 B (k 1 A (α + 1 (λ. 0 0 (54 (55

13 Frobeius-type Eueria ad poy-beroui ixed-type poyoias 723 Now, we cosider the foowig two Sheffer sequeces : (( e AB (α,k α λ 1 e t (x λ Li k (1 e t, t (56 ad (x ( 1, e t 1. (57 Let us assue that (x λ C, (x. (58 0 The, by (20, (21, (56, (57 ad (58, we get C, 1 ( α Li k (1 e t ( e t 1! e (λ 1t λ 1 e t x 1 ( α Li k (1 e t (! e (λ 1t 1 e t e t 1 x ( 1 α Li k (1 e t! e (λ 1 e t! S 2 (, t! x ( ( α Li k (1 e t S 2 (, e (λ 1t λ 1 e t x ( S 2 (, (λ. Therefore, by (58 ad (59, we obtai the foowig theore. Theore 7. For 0, we have ( { x AB (α,k (x λ! 0 ( } S 2 (, (λ. Let φ (x (1, og (1 + t. The we see that φ (x t! 1 x ( ex(et e t 1! 0 0 x!! S 2 (, t! Thus, by (60, we get φ (x 0 ( S 2 (, x 0 0 S 2 (, x. 0 t!. (59 (60

14 724 Dae Sa Ki, Taekyu Ki, Hyuck I Kwo ad Jogi Seo (( α For AB (α,k e (x λ (λ 1t λ 1 e t, t 1 λ Li k, φ (1 e t (x k0 S 2 (, k x k (1, og (1 + t, we have AB (α,k (x λ C, φ (x, (61 where 0 C, 1 ( α Li k (1 e t (og (1 + t! e (λ 1t λ 1 e t x ( ( α Li k (1 e t S 1 (, e (λ 1t λ 1 e t x ( S 1 (, (λ. Therefore, by (61 ad (62, we obtai the foowig theore. Theore 8. For 0, we have { AB (α,k (x λ 0 ( S 1 (, (λ } φ (x. (62 Let us cosider the foowig two Sheffer sequeces : (( e AB (α,k (λ 1t α λ 1 e t (x λ Li k (1 e t, t, (63 (( e H (s t s α (x α, t. 1 α The we have (x λ 0 C, H (s (x α, (64 where C, 1 ( e t s ( α α Li k (1 e t t! 1 α e (λ 1t λ 1 e t x (65 1 s ( ( α s! (1 α s ( α s e t Li k (1 e t e (λ 1t 1 e t t x 0 ( s ( ( s α (1 α s ( α s e t Li k (1 e t x e (λ 1t λ 1 e t 0 ( s ( s (1 α s ( α s e t (x λ 0

15 Frobeius-type Eueria ad poy-beroui ixed-type poyoias 725 ( s ( s (1 α s ( α s ( λ. 0 Therefore, by (64 ad (65, we obtai the foowig theore. Theore 9. For 0, we have (x λ 1 (1 α s For AB (α,k (x λ have (( 0 { ( e (λ 1t λ 1 λ s (x λ 0 α 1 e t ( } s ( α s ( λ H (s (x α. (( s, t Li k, B (s e (1 e t (x t 1 t, t, we 0 C, B (s (x, (66 where C, 1 ( e t s ( α 1 Li k (1 e t t! t e (λ 1t λ 1 e t x ( ( α ( ( Lik (1 e t e t s 1 x e (λ 1t λ 1 e t t ( s! ( + s! S 2 ( + s, s ( 0 ( α Li k (1 e t e (λ 1t λ 1 e t x ( s! ( + s! S 2 ( + s, s ( (λ 0 ( ( ( +s S 2 ( + s, s (λ. 0 s Therefore, by (66 ad (67, we obtai the foowig theore. Theore 10. For 0, we have ( ( AB (α,k (x λ ( +s S 2 ( + s, s 0 0 s (λ (67 B (s (x. Ackowedgeets. This paper was supported by Kwagwoo Uiversity i The correspodig author of this paper is Jog-Ji Seo.

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