Some results on the Apostol-Bernoulli and Apostol-Euler polynomials

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1 Soe results o the Apostol-Beroulli ad Apostol-Euler polyoials Weipig Wag a, Cagzhi Jia a Tiaig Wag a, b a Departet of Applied Matheatics, Dalia Uiversity of Techology Dalia , P. R. Chia b Departet of Matheatics, Haia Noral Uiversity Haiou , P. R. Chia Abstract The ai object of this paper is to ivestigate the Apostol-Beroulli polyoials ad the Apostol-Euler polyoials. We first establish two relatioships betwee the geeralized Apostol-Beroulli ad Apostol-Euler polyoials. It ca be foud that ay results obtaied before are special cases of these two relatioships. Moreover, we have a study o the sus of products of the Apostol-Beroulli polyoials ad of the Apostol-Euler polyoials. Keywords : Apostol-Beroulli polyoials; Apostol-Euler polyoials; Geeralized Apostol-Beroulli polyoials; Geeralized Apostol-Euler polyoials; Cobiatorial idetities 1. Itroductio For a real or coplex paraeter α, the geeralized Beroulli polyoials B (α (x ad the geeralized Euler polyoials E (α (x, each of degree i x as well as i α, are defied by the followig geeratig fuctios (for details, see [1, Sectio 2.8] ad [2, Sectio 1.6]: t α e t e xt 1 2 α e t e xt B (α (x t, ( t < 2π, (1.1! E (α (x t, ( t < π. (1.2! Clearly, the classical Beroulli polyoials B (x ad the classical Euler polyoials E (x are give by B (x : B (1 (x ad E (x : E (1 (x, ( N 0, (1.3 respectively, where N 0 : N {0} ad N : {1, 2, 3, }. Moreover, the classical Beroulli ubers B ad the classical Euler ubers E are give by 1 B : B (0 ad E : 2 E, 2 ( N 0, (1.4 respectively. These polyoials ad ubers have uerous iportat applicatios i cobiatorics, uber theory ad uerical aalysis. They have therefore bee studied extesively over the last two ceturies. E-ail addresses : wpigwag@yahoo.co (Weipig Wag, cagzhijia@yahoo.co.c (Cagzhi Jia, wagt@dlut.edu.c (Tiaig Wag 1

2 It is the purpose of this paper to cosider the so called geeralized Apostol-Beroulli ad Apostol-Euler polyoials, which are atural geeralizatios of B (α (x ad E (α (x, respectively. These polyoials are defied as follows [3, 4, 5]. Defiitio 1.1. For arbitrary real or coplex paraeters α ad λ, the geeralized Apostol- Beroulli polyoials (x; λ ad the geeralized Apostol-Euler polyoials (x; λ are defied by the followig geeratig fuctios: t α λe t e xt (x; λ t, ( t + log λ < 2π, (1.5 1! 0 2 α λe t e xt (x; λ t, ( t + log λ < π. ( ! 0 The so called Apostol-Beroulli polyoials B (x; λ ad the so called Apostol-Euler polyoials E (x; λ are give by B (x; λ : B (1 (x; λ ad E (x; λ : E (1 (x; λ, ( N 0, (1.7 respectively. Furtherore, the Apostol-Beroulli ubers B (λ ad the Apostol-Euler ubers E (λ are give by B (λ : B (0; λ ad E (λ : 2 E ( 1 2 ; λ, ( N 0, (1.8 respectively. Obviously, whe λ 1 i (1.5 to (1.8, we obtai the correspodig well ow fors give by (1.1 to (1.4. The Apostol-Beroulli polyoials B (x; λ ad the Apostol-Beroulli ubers B (λ were first defied by Apostol [6] whe he studied the Lipschitz-Lerch Zeta fuctios. Recetly, Luo ad Srivastava itroduced the geeralized Apostol-Beroulli ad Apostol-Euler polyoials. They also studied these polyoials systeatically (see [3, 4, 5, 7, 8]. Fro the wors referred to above, we ca see that the (geeralized Apostol-Beroulli polyoials ad the (geeralized Apostol-Euler polyoials have ay iterestig ad useful properties, ad they deserve further study. This paper is orgaized as follows. Soe basic properties for (x; λ ad (x; λ will be listed below. Sectio 2 is devoted to the geeral relatioships ivolvig these polyoials. It ca be foud that ay results established before (see [5, 9, 10, 11] are special cases of these relatioships. Fially, i Sectio 3, we will preset soe idetities ad give the explicit expressios for the sus of the products of the Apostol-Beroulli polyoials ad of the Apostol- Euler polyoials. Now, let us give a brief review of the properties satisfied by (x; λ ad (x; λ. It is easily observed fro the geeratig fuctios (1.5 ad (1.6 that B (α+β (x + y; λ E (α+β (x + y; λ Fro (1.5 ad (1.6, it follows also that λ λ 0 0 (x; λb(β (x; λe(β (y; λ, (1.9 (y; λ. (1.10 (x + 1; λ (x; λ B (α 1 (x; λ, ( (x + 1; λ + (x; λ 2E (α 1 (x; λ. (1.12 2

3 Moreover, sice B (0 (x; λ E (0 (x; λ x, upo settig β 0 i (1.9 ad (1.10 ad iterchagig x ad y, we get (x + y; λ (x + y; λ ( ( 0 0 (y; λx, (1.13 (y; λx. (1.14 The properties (1.9 to (1.14 ad soe other oes ca be foud i [3, 4, 5]. Here we will oly preset two ore theores for (x; λ ad (x; λ which have ot appeared before. Theore 1.2. The geeralized Apostol-Beroulli polyoials satisfy B (α+1 (x; λ ( 1 α Theore 1.3. The geeralized Apostol-Euler polyoials satisfy αλ 2 E(α+1 (x; λ + (x α α B(α 1 (x; λ. (1.15 (x + 1; λ x (x; λ +1 (x; λ, (1.16 E (α+1 (x; λ 2 α E(α +1 (x; λ (x α 2 α E(α (x; λ. (1.17 These two theores ca be readily obtaied by coputig the geeratig fuctios, hece we chose ot to prove the here. It is worth oticig that all the properties give above could reduce to the correspodig oes for the geeralized Beroulli ad Euler polyoials by settig λ Relatios betwee (x; λ ad E(α (x; λ I 2003, Cheo [9] rederived several ow properties ad relatios ivolvig the classical Beroulli polyoials B (x ad the classical Euler polyoials E (x, by aig use of soe stadard techiques based upo series rearrageet as well as atrix represetatio. Srivastava ad Pitér [10] followed Cheo s wor [9] ad established two relatios ivolvig the geeralized Beroulli polyoials B (α (x ad the geeralized Euler polyoials E (α (x. More recetly, Luo ad Srivastava [5] exteded the results i [10] to the geeralized Apostol- Beroulli polyoials (x; λ ad the geeralized Apostol-Euler polyoials (x; λ. We also preseted two relatios betwee B (α (x ad E (α (x with atrix represetatio [11]. I this sectio, we will study further the relatios betwee (x; λ ad (x; λ with the ethods of geeratig fuctio ad series rearrageet. As a cosequece, it ca be foud that the relatioships deostrated here are i fact coo geeralizatios of the wors [5, 9, 10, 11]. Theore 2.1. For α, β, λ C ad N 0, we have the relatioship (x + y; λ 1 β 2 β λ (y + ; λ E (β (x; λ ( betwee the geeralized Apostol-Beroulli ad Apostol-Euler polyoials. 3

4 Proof. Let us copute the geeratig fuctios for both sides of (2.1. By Defiitio 1.1, the right side gives 1 β 2 β λ (y + ; λ E (β (x; λt! β 2 β λ E (β (x; λt t! (y + ; λ (! β λe t e xt t α + 1 λe t e yt 1 β 1 2 β λ e t t α λe t e (x+yt, 1 0 which coicides with the geeratig fuctio of the left side. Corollary 2.2. For α, β, λ C ad N 0, we have the relatioships: (x + y; λ 0 (x + y 1 2 β ( (y; λ + 2 B(α 1 1 (y; λ E (x; λ, (2.2 β (y + E (β (x. ( Proof. By settig β 1 i Theore 2.1, we have (x + y; λ (y; λ + λb(α (y + 1; λ E (x; λ, which, i light of the recurrece relatio (1.11, leads us at oce to (2.2. Next, by settig λ 1 i Theore 2.1, the relatioship (2.3 ca also be obtaied. Rear. (2.2 ad (2.3 are ai results of [5] ad [11], respectively (see [5, Sectio 3, Theore 1] ad [11, Sectio 3, Theore 1]. A coo special case of these two idetities is (x + y 0 ( (y + 2 B(α 1 1 (y E (x, which is oe of the ai results of [10] (see [10, Sectio 3, Theore 1]. It should be oticed that the idetity deostrated i [11, Sectio 3, Theore 1] is ot i the siplest for: oe of the ier su ca be coputed. Corollary 2.3. For β, λ C ad, j N 0, we have the relatioships: x 1 β 2 β λ E (β (x + ; λ, (2.4 0 ( j x j j ( 1 j λ B (j (x + ; λ, (2.5 0 where ( j ( 1 ( j

5 Proof. The special case of Theore 2.1 whe α 0 is (x + y 1 2 β β β 0 β β 0 λ β 0 λ (y + E (β λ E (β (x + y + ; λ, E (β (x; λ (x; λ(y + which, by settig y 0, yields (2.4. Equatio (2.5 is a aalogue of (2.4. To prove it, let us copute the geeratig fuctio agai: j ( 1 j λ B (j (x + ; λ t! 0 0 t j λe t e j xt ( 1 j λ e t t j e xt. 1 0 Now, it suffices to idetify the coefficiets of t /! i the first ad last ebers of the equatio above. Whe λ 1, Corollary 2.3 gives two idetities for the geeralized Beroulli polyoials ad the geeralized Euler polyoials: x 1 β 2 β E (β (x +, 0 ( j x j j ( 1 j B (j (x +. 0 Next, we will show that the geeralized Apostol-Euler polyoials ca be expressed by the geeralized Apostol-Beroulli polyoials. For coveiece, we itroduce a lea. Lea 2.4. Defie the ubers S(, j; λ by S(, j; λ 1 j! j j ( 1 j λ, (λ C,, j N 0. 0 The the S(, j; λ have the geeratig fuctio Proof. We have 0 S(, j; λ t! 1 j! 1 j! This copletes the proof S(, j; λ t! 1 j! (λet 1 j. (2.6 j ( j ( 1 λ j t! 1 j! j j ( 1 j λ t! 0 0 j j ( 1 j λ e t 1 j! (λet 1 j. 0 5

6 Accordig to the defiitio of the ubers S(, j; λ, we ca readily see that S(, j; 1 S(, j, where S(, j are the faous Stirlig ubers of the secod id (see, e.g., [12, p. 204]. Moreover, i view of the geeratig fuctio (2.6, we have t l+j j!s(l + j, j; λ (l + j! l 0 j j!s(, j; λ t! (λet 1 j which will be used i the proof of the theore below. 1 l j Theore 2.5. For α, λ C ad, j N 0, we have the relatioship (x + y; λ l l j 0!j! S(l + j, j; λe(α!(l + j!( l! l betwee the geeralized Apostol-Euler ad Apostol-Beroulli polyoials. t l+j j!s(l + j, j; λ (l + j!, (2.7 (y; λb(j (x; λ (2.8 Proof. Let F (,, l be the suad i relatio (2.8; the the double su of (2.8 ca be rewritte as l l 1 1 l F (,, l + + F (,, l. (2.9 l j 0 Let us further defie A : B : l0 0 l j 0 1 j 2 α 1 λe t 1 λe t e (x+yt + 1 l j ( t λe t 1 j e xt 1 l j With this otatio, we ow give the proof. l 1 i0 j! (l + j! l j +1 t l+j j!s(l + j, j; λ (l + j!, S(l + j, j; λe(α i (y; λ ti+l. i! We first cosider the case whe j N ad λ C \ {1}. I this case, by Defiitio 1.1 ad equatio (2.7, the first part of (2.9 gives l F (,, l t! F (,, l t! t +i F ( + i,, l ( + i! 0 l0 0 0 l0 0 l 0 i l t l j!s(l + j, j; λ B (j (l + j! (x; λt t i l! i l (y; λ (i l! l 0 0 i l t j 2 α λe t e xt 1 λe t e yt 1 1 (λe t + 1 t j 1 j t l+j j!s(l + j, j; λ (l + j! l j 2 α λe t e (x+yt A. (

7 The geeratig fuctio for the secod part of (2.9 is 1 0 l j 0 1 l j ( F (,, l t 1! l j 0 t l j!s(l + j, j; λ (l + j! t λe t 1 A B. j e xt 1 l j B (j 0 F (,, l t 1! (x; λt! t l j!s(l + j, j; λ (l + j! The geeratig fuctio for the third part of (2.9 is 1 l 0 l j +1 F (,, l t 1! 1 l j 1 ax{+l,0} 1 l j l i 0 t +i F ( + i,, l ( + i! l j 0 i 0 i l ( ( 2 λe t + 1 F (,, l t! F (,, l t 1! + i l t (y; λ (i l! α l 1 e yt 1 l j l +l i0 i (y; λ ti i! F (,, l t!. (2.11 Accordig to (1.5, whe j N ad λ 1, B (j (x; λ 0 for 0 j 1. The the first ter of (2.11 vaishes, ad this geeratig fuctio reduces to 1 l 1 l j l i0 1 l j 1 l j F ( + l + i,, l ( + l + i! j! S(l + j, j; λ (l + j! l l 1 i0 j! (l + j! t +l+i B (j (x; λt! S(l + j, j; λe(α i (y; λ ti+l i! Cobied with the coputatio above, we have as desired. l 0 l j 0 l 1 i0 i (y; λ ti+l i! B (j 0 (x; λt! B. F (,, l t 2 α! λe t e (x+yt, ( Whe j N ad λ 1, the ubers S(, j; 1 tur out to be S(, j, where S(, j are the Stirlig ubers of the secod id. Moreover, S(, j 0 for 0 j 1. The the secod ad third parts of forula (2.9 vaish, ad l F (,, l l j 0 l F (,, l. l0 0 I additio to these, A also equals zero. Therefore, equatio (2.12 holds. Whe j 0, (2.9 reduces to l l0 0 F (,, l. Sice A still vaishes here, equatio (2.10 idicates that (2.12 still holds. This copletes the proof. 7

8 Corollary 2.6. For α, λ C ad, j N 0, we have the relatioships: 2 (x + y; λ E (α (y; λ E(α +1 (y; λ B (x; λ 0 + λ E(α 0 (y; λb +1(x; λ, (2.13 ( l + j 1 E (α (x + y S(l + j, j l j l (y B (j (x. ( l0 Proof. (2.14 is a iediate cosequece of Theore 2.5 by settig λ 1. To get (2.13, set j 1 i (2.9. The those three parts equal λ ( B (x; λ (y + 1; λ E(α +1 (y; λ, respectively. Therefore, (x + y; λ 0 0 λ B (x; λ +1 (y; λ, λ E(α 0 (y; λb +1(x; λ, 1 λ (y + 1; λ E(α +1 (y; λ B (x; λ + λ E(α 0 (y; λb +1(x; λ. I view of the recurrece relatio (1.12, assertio (2.13 holds. Rear. Siilar to Corollary 2.2, the two relatioships give by Corollary 2.6 are ai results of [5] ad [11], respectively (see [5, Sectio 3, Theore 2] ad [11, Sectio 3, Theore 2]. However, the represetatios there both have probles. Oe is lacig a ter (i.e., the secod ter λ 1 0 (y; λb +1(x; λ o the right side of (2.13, ad the other is ot i the eatest +1 E(α for. The coo special case of (2.13 ad (2.14 is ( E (α 2 (x + y ( E (α 1 +1 (y +1 (y B (x, which has already bee obtaied i [10, Sectio 3, Theore 2]. Corollary 2.7. For λ C ad, j N 0, we have the relatioships: x!j! S(l + j, j; λb(j (l + j!( l! l (x; λ, (2.15 l j l + j 1 x S(l + j, jb (j l j l (x. (2.16 l0 Proof. The special case of Theore 2.5 whe α 0 is (x + y l l j 0!j!!(l + j!( l! S(l + j, j; λy l B (j (x; λ. With the substitutio y 0 i the last equatio, we get (2.15. Equatio (2.16 is a further special case of (2.15 whe λ 1. 8

9 May other idetities ca be obtaied fro Theores 2.1 ad 2.5. For exaple, whe β 0 ad j 0, Theores 2.1 ad 2.5 will reduce to (1.13 ad (1.14 respectively. Other special cases ca be foud i the refereces [5, 9, 10, 11]. 3. Explicit expressios for sus of products Oe of the ost rearable idetities for the Beroulli ubers is the covolutio idetity B j B j B 1 ( 1B, ( 1, j j0 which is equivalet to the for 1 2 B 2j B 2 2j (2 + 1B 2, ( 2. 2j j1 These two idetities have bee geeralized i ay wors (see [13] or [14] for a review o this subject. Particularly, i [13], explicit expressios are obtaied for sus of products of arbitrarily ay Beroulli ubers. Correspodig results are also derived for Beroulli polyoials, ad for Euler ubers ad polyoials. I this sectio, we first derive several idetities for the geeralized Apostol-Beroulli ad Apostol-Euler polyoials. Based o these idetities, we further ivestigate the sus of products of the Apostol-Beroulli polyoials ad of the Apostol-Euler polyoials. Before statig the results, we itroduce a lea ([15, p. 147, equatios (85 ad (86], see also [16, pp. 96 ad 99]. Lea 3.1. The geeralized Beroulli polyoials satisfy B (+1 (x (x 1(x 2 (x (x 1,! (! B(+1 (x D x[(x 1(x 2 (x ] Dx(x 1, where D x is the differetial operator defied by D x f(x : d dx f(x. The the followig theore holds. Theore 3.2. For + 1 we have B (+1 ( 1 B + (x; λ (x; λ ( ! + D x(x 1 (3.1 0 ( + 1 ( 1 B (x; λ B ( (x. (3.2 0 Proof. We prove (3.1 by iductio. It is clearly true for 1 i view of the recurrece relatio (1.15. Suppose that it is true for 1, the by aig use of (1.15 agai, we have B (+1 (x; λ (1 B( ( ( + 1 ( + ( (x; λ + (x B( 1 (x; λ ( 1 B + (x; λ + 1! + 1 ( 1 0! D 1 x (x 1 1 B + (x; λ + (x D x(x

10 Accordig to the Leibiz s rule, D x(x 1 j0 D j j x(x 1 1 Dx j (x (x D x(x D 1 x (x 1 1. Therefore, equatio (3.1 holds. Moreover, based o Lea 3.1, (3.2 also holds. This copletes the proof. By appealig to the recurrece relatio (1.17, we ca establish the correspodig results for the geeralized Apostol-Euler polyoials. Theore 3.3. For + 1 we have E (+1 (x; λ 2 ( 1 E + (x; λd!! x(x 1 ( ( 1 E + (x; λb (+1! (x. (3.4 0 Whe λ 1, Theores 3.2 ad 3.3 will reduce to the correspodig idetities for the geeralized Beroulli ad Euler polyoials (see [15, p. 148, equatios (87 ad (88]. Now, we cosider the sus of products of the Apostol-Beroulli polyoials ad of the Apostol-Euler polyoials. I aalogy to [13], we deote for 2, S (; x 1,, x B j1 (x 1 ; λ B j (x ; λ, j 1,, j where the su is tae over all oegative itegers j 1,..., j such that j 1 + +j. The the ext theore holds. Theore 3.4. Let y : x x. The for we have S (; x 1,, x ( ( 1 1 ( ( ( 1 ( j0 B ( (y B (y; λ 1 + j s(, + jy j j where s(, are the Stirlig ubers of the first id (see, e.g., [12, p. 212]. B (y; λ (3.5, (3.6 Proof. This is i fact the sae as the proofs due to Dilcher [13, Sectio 3, Lea 4 ad Theore 3]. Fro the geeratig fuctio of the Apostol-Beroulli polyoials, we have Thus by (3.2 ad the idetity S (; x 1,, x B ( (x x ; λ B ( (y; λ. ( 1 B ( (y 1 + j s(, + jy j j (see [13, p. 32, equatio (3.7], the desired results ca be obtaied. j0 10

11 For exaple, whe 2, (3.6 gives B (x; λb (y; λ (x + y 1B 1 (x + y; λ ( 1B (x + y; λ. 0 Corollary 3.5. If x x 0 the for we have 1 S (; x 1,, x ( 1 1 ( 1 s(, B (λ. (3.7 This follows fro (3.6 with the substitutio y 0 ad the fact that B (0; λ B (λ. I particular, by settig x 1 x 0, the right side of (3.7 leads us at oce to a expressio for B j1 (λ B j (λ. j 1,, j We ca deal with the sus of the products of the Apostol-Euler polyoials i a aalogous way. Let us deote T (; x 1,, x E j1 (x 1 ; λ E j (x ; λ, j 1,, j where the su is agai tae over all oegative itegers j 1,..., j such that j 1 + +j. I light of the assertio (3.4 of Theore 3.3, the ext theore ca be obtaied. Theore 3.6. Let y : x x. The for we have T (; x 1,, x ( 1 B ( ( 1! (ye + 1 (y; λ ( j ( 1 s(, + jy j ( 1! j E + 1 (y; λ. ( j0 For exaple, whe 2, (3.9 gives E (x; λe (y; λ 2(1 x ye (x + y; λ + 2E +1 (x + y; λ. 0 Moreover, Theore 3.6 has the followig special case. Corollary 3.7. If x x 0 the for we have T (; x 1,, x ( ( 1 s(, + 1E + (0; λ. ( 1! 0 I additio to these, sice E (λ 2 E ( 1 2 ; λ, we have a explicit expressio for the sus of products of the Apostol-Euler ubers. Corollary 3.8. For we have E j1 (λ E j (λ j 1,, j j ( j ( 2 s(, + j ( 1! j 2 E + 1 (λ. 0 j0 11

12 Acowledget The authors would lie to tha the aoyous referee for detailed suggestios which have iproved the presetatio of the paper. Refereces [1] Y. L. Lue, The Special Fuctios ad Their Approxiatios, Vol. I, Acadeic Press, New Yor-Lodo, (1969. [2] H. M. Srivastava, J. Choi, Series Associated with the Zeta ad Related Fuctios, Kluwer Acadeic Publishers, Dordrecht, (2001. [3] Q.-M. Luo, Apostol-Euler polyoials of higher order ad Gaussia hypergeoetric fuctios, Taiwaese J. Math. 10 (4 ( [4] Q.-M. Luo, H. M. Srivastava, Soe geeralizatios of the Apostol-Beroulli ad Apostol- Euler polyoials, J. Math. Aal. Appl. 308 (1 ( [5] Q.-M. Luo, H. M. Srivastava, Soe relatioships betwee the Apostol-Beroulli ad Apostol-Euler polyoials, Coput. Math. Appl. 51 (3 4 ( [6] T. M. Apostol, O the Lerch zeta fuctio, Pacific J. Math. 1 ( [7] Q.-M. Luo, O the Apostol-Beroulli polyoials, Cet. Eur. J. Math. 2 (4 ( [8] H. M. Srivastava, Soe forulas for the Beroulli ad Euler polyoials at ratioal arguets, Math. Proc. Cabridge Philos. Soc. 129 (1 ( [9] G.-S. Cheo, A ote o the Beroulli ad Euler polyoials, Appl. Math. Lett. 16 (3 ( [10] H. M. Srivastava, Á. Pitér, Rears o soe relatioships betwee the Beroulli ad Euler polyoials, Appl. Math. Lett. 17 (4 ( [11] W. Wag, T. Wag, A ote o the relatioships betwee the geeralized Beroulli ad Euler polyoials, Ars Cobi. to appear. [12] L. Cotet, Advaced Cobiatorics, D. Reidel Publishig Co., Dordrecht, (1974. [13] K. Dilcher, Sus of products of Beroulli ubers, J. Nuber Theory 60 (1 ( [14] T. Agoh, K. Dilcher, Covolutio idetities ad lacuary recurreces for Beroulli ubers, J. Nuber Theory 124 (1 ( [15] N. E. Nörlud, Vorlesuge über Differezerechug, Spriger, Berli, (1924. [16] S. Roa, The Ubral Calculus, Acadeic Press, Ic., New Yor, (