and Genocchi Polynomials

Transcription

1 Applied Mathematics & Iformatio Scieces , A Iteratioal Joural c 011 NSP Some Geeralizatios ad Basic or - Extesios of the Beroulli, Euler ad Geocchi Polyomials H. M. Srivastava Departmet of Mathematics ad Statistics, Uiversity of Victoria, British Columbia V8W 3R4, Caada Address: harimsri@math.uvic.ca Received Ja 4, 011; Accepted April 1, 011 I the vast literature i Aalytic Number Theory, oe ca fid systematic ad extesive ivestigatios ot oly of the classical Beroulli, Euler ad Geocchi polyomials ad their correspodig umbers, but also of their may geeralizatios ad basic or - extesios. Our mai object i this presetatio is to itroduce ad ivestigate some of the pricipal geeralizatios ad uificatios of each of these polyomials by meas of suitable geeratig fuctios. We preset several iterestig properties of these geeral polyomial systems icludig some explicit series represetatios i terms of the Hurwitz or geeralized zeta fuctio ad the familiar Gauss hypergeometric fuctio. By itroducig a aalogue of the Stirlig umbers of the secod id, that is, the so-called λ-stirlig umbers of the secod id, we derive several properties ad formulas ad cosider some of their iterestig applicatios to the family of the Apostol type polyomials. We also give a brief expository ad historial accout of the various basic or - extesios of the classical Beroulli polyomials ad umbers, the classical Euler polyomials ad umbers, the classical Geocchi polyomials ad umbers, ad also of their such geeralizatios as for example the above-metioed families of the Apostol type polyomials ad umbers. Relevat coectios of the defiitios ad results preseted here with those i earlier as well as forthcomig ivestigatios will be idicated. Keywords: Beroulli polyomials ad umbers; Euler polyomials ad umbers; Taylor-Maclauri series expasio; Basic or - extesios. 010 MSC: Primary 05A30, 11B68, 11B83, 11M35, 33C05, 33E0; Secodary 11B73, 6C05, 30B40.

2 Some Geeralizatios ad Basic or - Extesios of the Beroulli, Itroductio, Defiitios ad Motivatio Throughout this presetatio, we use the followig stadard otatios: N := {1,, 3, }, N 0 := {0, 1,, 3, } = N {0} ad Z := { 1,, 3, } = Z 0 \ {0}. Also, as usual, Z deotes the set of itegers, R deotes the set of real umbers ad C deotes the set of complex umbers. Furthermore, {λ} 0 = 1 ad {λ} = λλ 1 λ + 1 N 0 ; λ C deotes the fallig factorial ad λ 0 = 1 ad λ = λλ + 1 λ + 1 N 0 ; λ C deotes the risig factorial. The classical Beroulli polyomials B x, the classical Euler polyomials E x ad the classical Geocchi polyomials G x, together with their familiar geeralizatios B α x, E α x ad G α x of real or complex order α, are usually defied by meas of the followig geeratig fuctios see, for details, [6, p ] ad [68, p. 61 et se.]; see also [7, p. 397, Problem 7] ad [73] ad the refereces cited therei: α z e z e xz = 1 α e z e xz = + 1 ad α z e z e xz = + 1 B α x z E α x z G α x z z < π; 1 α := 1, 1.1 z < π; 1 α := 1 1. z < π; 1 α := 1, 1.3 so that, obviously, the classical Beroulli polyomials B x, the classical Euler polyomials E x ad the classical Geocchi polyomials G x are give, respectively, by B x := B 1 x, E x := E 1 x ad G x := G 1 x N For the classical Beroulli umbers B, the classical Euler umbers E ad the classical Geocchi umbers G of order, we have respectively. B := B 0 = B 1 0, E := E 0 = E 1 0, G := G 0 = G 1 0,

3 39 H. M. Srivastava Some iterestig aalogues of the classical Beroulli polyomials ad umbers were first ivestigated by Apostol [, p. 165, E. 3.1] ad more recetly by Srivastava [66, pp ]. We begi by recallig here Apostol s defiitios as follows. Defiitio 1 Apostol []; see also Srivastava [66]. The Apostol-Beroulli polyomials B x; λ λ C are defied by meas of the followig geeratig fuctio: ze xz λe z 1 = B x; λ z z < π upwhe λ = 1; z < log λ upwhe λ with, of course, B x = B x; 1 ad B λ := B 0; λ, 1.6 where B λ deotes the so-called Apostol-Beroulli umbers. Apostol [] ot oly gave elemetary properties of the polyomials B x; λ, but also obtaied the followig iterestig recursio formula for the umbers B λ see [, p. 166, E. 3.7]: 1 B λ =! λ λ 1 +1 S 1, N0 ; λ C \ {1}, 1.7 where S, deotes the Stirlig umbers of the secod id defied by meas of the followig expasio see [15, p. 07, Theorem B]: so that x = x! S,, 1.8 S, 0 = δ,0, S, 1 = S, = 1 ad S, 1 =, 1.9 δ, beig the Kroecer symbol. Recetly, Luo ad Srivastava [5] further exteded the Apostol-Beroulli polyomials as the so-called Apostol-Beroulli polyomials of order α. Luo [45], o the other had, gave a aalogous extesio of the geeralized Euler polyomials as the so-called Apostol- Euler polyomials of order α. Defiitio cf. Luo ad Srivastava [5]. The Apostol-Beroulli polyomials B α x; λ λ C

4 Some Geeralizatios ad Basic or - Extesios of the Beroulli, of real or complex order α are defied by meas of the followig geeratig fuctio: α z λe z e xz = B α x; λ z with, of course, z < π upwhe λ = 1; z < log λ upwhe λ 1 B α x = B α x; 1 ad B α λ := B α 0; λ, 1.11 where B α λ deotes the so-called Apostol-Beroulli umbers of order α. Defiitio 3 cf. Luo [45]. The Apostol-Euler polyomials E α x; λ λ C of real or complex order α are defied by meas of the followig geeratig fuctio: α λe z e xz = E α x; λ z z < log λ with, of course, E α x = E α x; 1 ad E α λ := E α 0; λ, 1.13 where E α λ deotes the so-called Apostol-Euler umbers of order α. Remar 1. The costraits o z, which we have used i Defiitios 1, ad 3 above, are meat to esure that the geeratig fuctios i 1.5, 1.10 ad 1.1 are aalytic throughout the prescribed ope diss i the complex z-plae cetred at the origi z = 0 i order to have the correspodig coverget Taylor-Maclauri series expasios about the origi z = 0 occurrig o their right-had sides each with a positive radius of covergece. Moreover, throughout this ivestigatio, log z is tacitly assumed to deote the pricipal brach of the may-valued fuctio log z with the imagiary part I log z costraied by π < I log z π. More importatly, throughout this presetatio, wherever log λ ad log λ appear as the radii of the ope diss i the complex z-plae cetred at the origi z = 0 i which the defiig geeratig fuctios are aalytic, it is tacitly assumed that the obviously exceptioal cases whe λ = 1 ad λ = 1, respectively, are to be treated separately. Naturally, therefore, the correspodig costraits o z i the earlier ivestigatios see, for example, [45], [5], [53] ad [66] should also be modified accordigly. Remar. The classical Euler umbers Ẽ are usually defied by meas of the followig geeratig fuctio see, for example, [68, p. 64, E ]: e z e z + 1 = sech z = z Ẽ z < π, 1.14

5 394 H. M. Srivastava which, whe compared with the geeratig fuctio i 1., yields the followig relatioships [cf. Euatio 1]: 1 1 Ẽ = E = E with the Euler umbers E ad the Euler polyomials E α x used i this paper. For the Apostol-Euler umbers Ẽ α λ λ C of order α, which correspod to the classical Euler umbers Ẽ, Luo [45] made use of the followig defiitio: e z λe z + 1 α = Ẽ α λ z z < 1 log λ However, for the sae of simplicity of the results preseted i this paper, we fid it to be coveiet to use the Apostol-Euler umbers E α λ λ C of order α, correspodig to the Euler umbers E, which are defied by meas of the followig geeratig fuctio [cf. Euatio 1.13]: α λe z = E α λ z + 1 z < log λ Of course, if ad whe it is eeded, the iterested reader will fid it to be fairly straightforward to apply the followig explicit relatioships betwee the Apostol-Euler umbers E α λ λ C ad Ẽ α λ λ C i order to covert ay of these results ito their desired forms. λ = E α 0; λ Ẽα E λ = E 0; λ Ẽ λ = E 1 ; λ Ẽ λ = E α λ = E α E α = E α 0 Ẽ α = E α α E = E 0 Ẽ = E 1 α ; λ Ẽ α λ = Ẽ α = α E α λ E λ α E α Ẽ = E Sice the publicatio of the wors by Luo ad Srivastava see [44], [45], [5], ad [53], may further ivestigatios of the above-metioed Apostol type polyomials have appeared i the literature. Boyadzhiev [4] gave some properties ad represetatios of the Apostol-Beroulli polyomials ad the Euleria polyomials. Garg et al. [17] studied the Apostol-Beroulli polyomials of order α ad obtaied some ew relatios ad formulas ivolvig the Apostol type polyomials ad the Hurwitz or geeralized zeta fuctio ζs, a defied by 1.0 below. Luo see [46] ad [47] obtaied the Fourier expasios ad itegral represetatios for the Apostol-Beroulli ad the Apostol-Euler polyomials, ad gave the multiplicatio formulas for the Apostol-Beroulli ad the Apostol-Euler polyomials of order α. Prévost [59] ivestigated the Apostol-Beroulli ad the Apostol-Euler polyomials by usig the Padé approximatio methods. Wag et al. see [78] ad [79]

6 Some Geeralizatios ad Basic or - Extesios of the Beroulli, further developed some results of Luo ad Srivastava [53] ad obtaied some formulas ivolvig power sums of the Apostol type polyomials. Zhag ad Yag [81] gave several idetities for the geeralized Apostol-Beroulli polyomials. O the other had, Ceci ad Ca [8] gave a -aalogue of the Apostol-Beroulli polyomials B x; λ. Choi et al. [11] gave the -extesios of the Apostol-Beroulli polyomials of order α ad the Apostol-Euler polyomials of order α see also [1]. Hwag et al. [4] ad Kim et al. [35] also gave -extesios of Apostol s type Euler polyomials. O the subject of the Geocchi polyomials G x ad their various extesios, a remarably large umber of ivestigatios have appeared i the literature see, for example, [9], [1], [1], [], [3], [5], [8], [30], [34], [38], [39], [40], [43], [41] [48], [49], [50], [58] ad [80]; see also the refereces cited i each of these wors. Moreover, Luo see [48] ad [50] itroduced ad ivestigated the Apostol-Geocchi polyomials of a real or complex order α, which are defied as follows. Defiitio 4. The Apostol-Geocchi polyomials G α x; λ λ C of real or complex order α are defied by meas of the followig geeratig fuctio: with, of course, α z λe z e xz = + 1 G α x; λ z z < log λ 1.18 G α x = G α x; 1, G α λ := G α 0; λ, G x; λ := G 1 x; λ ad G λ := G 1 λ, 1.19 where G λ, G α λ ad G x; λ deote the so-called Apostol-Geocchi umbers, the Apostol-Geocchi umbers of order α ad the Apostol-Geocchi polyomials, respectively. The mai object of this presetatio is to first preset some elemetary properties of the Apostol-Geocchi polyomials G α x; λ of order α i Sectio. We derive several explicit series represetatios of G α x; λ i terms of the Gaussia hypergeometric fuctio i Sectio 3. We fid some relatioships betwee the various Apostol type polyomials i Sectio 4. We obtai the series represetatios for the Apostol type polyomials ivolvig the Hurwitz or geeralized zeta fuctio ζs, a i Sectio 5. We itroduce the λ-stirlig umbers S, ; λ of the secod id, which aid us to prove some basic properties ad formulas i Sectio 6 i which we also pose two iterestig ope problems related to our preset ivestigatio. Fially, i Sectio 7, we give some iterestig applicatios of the λ-stirlig umbers S, ; λ of the secod id to the family of the Apostol type polyomials. For example, by closely followig the wor of Srivastava [66] dealig with the

7 396 H. M. Srivastava special case α = 1, we will derive various explicit series represetatios for p p p p G α ; eπiξ, G l ; eπiξ, E α ; eπiξ ad E l ; eπiξ, l N; p Z; ξ R; α C, ivolvig either the Stirlig umbers S, of the secod id defied by 1.8 or the λ-stirlig umbers S, ; λ of the secod id defied below by 6.1 ad the Hurwitz or geeralized zeta fuctio ζ s, a defied by cf. [3, p. 49] ad [68, p. 88] so that ζ s, a := for the Riema zeta fuctio ζ s. 1 + a s R s > 1; a C \ Z 0, 1.0 ζ s, 1 =: ζ s = 1 s 1 ζ s, Elemetary Properties of the Apostol-Geocchi Polyomials x; λ of Order α G α The followig elemetary properties of the Apostol-Geocchi polyomials G α x; λ of order α are readily derived from We, therefore, choose to omit the details ivolved. Property 1. Special values of the Apostol-Geocchi polyomials or the Apostol-Geocchi umbers of order α: G α λ =G α 0; λ, G 0 x; λ = x, G 0 λ =δ,0 ad G α 0 x; λ = G α 0 λ = δ α,0 N 0 ; α C, where δ, deotes the Kroecer symbol..1 Property. Summatio formulas for the Apostol-Geocchi polyomials of order α: ad G α x; λ = G α x; λ = Property 3. Differece euatio: λg α G α λ x. G α 1 λg x; λ..3 x + 1; λ + G α x; λ = G α 1 x; λ N..4 1

8 Some Geeralizatios ad Basic or - Extesios of the Beroulli, Property 4. Differetial relatios: { } G α x; λ = G α 1 x; λ N.5 x ad p { } x p G α x; λ = p! Gα px; λ,, p N 0 ; 0 p..6 Property 5. Itegral formulas: b a G α x; λup dx = Gα +1 b; λ Gα +1 a; λ ad b a G α x; λup dx = 1 G α + 1 λ b +1 a Property 6. Additio theorem of the argumet: G α+β x + y; λ = Property 7. Complemetary additio theorems: G α x; λgβ y; λ..9 G α α x; λ = 1+α λ α G α x; λ 1.10 ad G α α + x; λ = 1+α λ α G α x; λ Property 8. Recursio formulas: α G α x; λ = x G α αλ 1 x; λ Gα+1 x + 1; λ.1 ad α Gα+1 x; λ = α x G α 1 x; λ + α Gα x; λ..13 Whe we set α = 1, λ = 1 ad α = λ = 1 i the formulas.1 to.13, we get the correspodig formulas for the Apostol-Geocchi polyomials G x; λ, the geeralized Geocchi polyomials G α x ad the classical Geocchi polyomials G x, respectively.

9 398 H. M. Srivastava 3 Explicit Represetatios Ivolvig the Gaussia Hypergeometric Fuctio By usig Defiitio 4 i cojuctio with the geeratig fuctio 1.3, we have G l x; λ z log λ = e x = e x log λ l z + log λ e z+log λ + 1 = e x log λ = e x log λ z z + log λ l xz+log λ e G l x z + log λ l z l! G l l z x log λ l! z + l + 1 G l + log λ x,! which yields Lemma 1 below assertig a relatioship betwee the Apostol-Geocchi polyomials G l x; λ of order l N 0 ad the Geocchi polyomials G l x of order l N 0. Lemma 1. The followig relatioship holds true: G l x log λ x; λ = e + l +, l N 0 ; λ C. 1 G l log λ + x! 3.1 By 1.1 ad 1.18 with α = l N 0, we readily obtai Lemma below. Lemma. The followig relatioship holds true: G l x; λ = {} l E l l x; λ = l! E l l x; λ, l N 0; 0 l ; λ C or, euivaletly, E l x; λ = 1 { + l} l G l +l x; λ = + l! Gl +l x; λ, l N 0; λ C betwee the Apostol-Geocchi polyomial of order l ad the Apostol-Euler polyomial of order l. Moreover, sice the parameter λ C, by comparig Defiitio 4 with our Defiitio, we are led easily to Lemma 3 below. Lemma 3. The followig relatioship holds true: G α x; λ = α B α x; λ α, λ C; 1 α := 1 3.4

10 or, euivaletly, Some Geeralizatios ad Basic or - Extesios of the Beroulli, B α x; λ = 1 α G α x; λ α C; 1 α := betwee the Apostol-Geocchi polyomials G α x; λ ad the Apostol-Beroulli polyomials B α x; λ. Lemma 4 below follows easily from Lemma ad Lemma 3. Lemma 4. The followig relatioship holds true: B l x; λ = l! l E l l x; λ, l N 0; 0 l ; λ C 3.6 or, euivaletly, E l x; λ = l + l! Bl +l x; λ, l N 0; λ C 3.7 betwee the Apostol-Beroulli polyomial of order l ad Apostol-Euler polyomial of order l. I order to prove the mai assertios i this sectio, we recall each of the followig ow results see also the earlier ivestigatios o the subject of explicit hypergeometric represetatios by Todorov [77] ad Srivastava ad Todorov [76]. Lemma 5 Luo ad Srivastava [5, p. 93, Lemma 1 13]. The Apostol-Euler polyomials E α x; λ of order α are represeted by E α x log λ x; λ = e E α log λ + x N 0 ; λ, α C 3.8! i terms of the Euler polyomials of order α. Theorem A Luo [45, p. 90, Theorem 1]. Each of the followig explicit series represetatios holds true: α + 1 E α x; λ = α λ λ + 1 α+ 1 j j j=0 j x + j j F 1, ; + 1; 3.9 x + j N0 ; α C; λ C \ { 1} ad E α x log λ x; λ = e r r 1 j j j=0 log λ α + r 1! r r r r=0 j r x + j + r F 1 r, r; r + 1; j x + j 3.10

11 400 H. M. Srivastava N 0 ; α, λ C, where F 1 a, b; c; z deotes the Gaussia hypergeometric fuctio defied by cf., e.g., [1, p. 556 et se.] F 1 a, b; c; z = F 1 b, a; c; z := a b c z a, b C; c C \ Z 0 ; z < 1; z = 1, ad R c a b > 0; z = 1, ad R c a b > 1. We ow state the mai result i this sectio as Theorem 1 below. Theorem 1. The followig explicit series represetatios hold true: ad G l x; λ = l l! l l l l + 1 λ λ + 1 l+ j x + j l F 1 l +, ; + 1;, l N 0 ; λ C \ { 1} 1 j j j=0 j x + j G l x log λ + l + + l! log λ x; λ = e l! + l r 1 + l l + r 1 r 1 j r r r j r=0 j=0 j r x + j + r l j F 1 r + l, r; r + 1; x + j 3.1, l N 0 ; λ C, where F 1 a, b; c; z deotes the Gaussia hypergeometric fuctio defied by??. Proof. We mae use of the relatioship 3. i cojuctio with 3.9 ad 3.10 with, of course, α = l ad l, l N 0 ; 0 l. We thus readily obtai the assertios 3.11 ad 3.1 of Theorem 1.

12 Some Geeralizatios ad Basic or - Extesios of the Beroulli, Corollary 1. The followig explicit formula for the Apostol-Geocchi polyomials G α x; λ ivolvig the Stirlig umbers S, of the secod id holds true: G l x; λ = l l! l j l + j 1 j! λ j λ + 1 j+l S l, j x 3.13 l j=0, l N 0 ; λ C \ { 1}. Further, by settig λ = 1 i 3.13, we obtai the followig explicit formula for the geeralized Geocchi polyomials G l x l N 0 ivolvig the Stirlig umbers S, of the secod id: G l x = l l l + j 1 j j=0 l! j! 1 j S l, jx, l N By settig λ = 1 i 3.11, we obtai a explicit formula for the Geocchi polyomials G l x of order l N 0 i terms of the Gaussia hypergeometric fuctio. Corollary. The followig series represetatio holds true: G l x = l! l l 1 l l j j j=0 j x + j l j F 1 + l, ; + 1;, l N x + j By settig x = 0 i 3.11, we obtai the explicit series represetatio give by Corollary 3 below. Corollary 3. The followig explicit series represetatio holds true: G l λ = l l! l l + 1! λ, l N0 ; λ C \ { 1}. S l,, l λ + 1 If we set λ = 1 i 3.16, the we obtai the followig formula for the Geocchi umbers G l of order l N 0 ivolvig the Stirlig umbers of the secod id: G l l l + 1 = l!! 1 S l,, l N

13 40 H. M. Srivastava Corollary 4. The followig explicit series represetatio holds true for the Apostol- Geocchi polyomials G x; λ: 1 1 λ G x; λ = λ j j j=0 F 1 + 1, ; + 1; j x + j 1 j, N 0 ; λ C \ { 1}. x + j 3.18 Fially, we calculate a few values of the Apostol-Geocchi umbers G λ by applyig the formula 3.16 with l = 1 as follows: G 0 λ = 0, G 1 λ = λ + 1, G λ = 4λ λ + 1, G 6λλ 1 3λ = λ + 1 3, G 4 λ = 8λλ 4λ + 1 λ + 1 4, G 5 λ = 10λλ3 11λ + 11λ 1 λ + 1 5, G 6 λ = 1λλ4 6λ λ 6λ + 1 λ + 1 6, 3.19 ad so o. By applyig 3.3 with l = 1 ad x = 0 i cojuctio with 3.19, we have the correspodig values of the Apostol-Euler umbers E λ give by E 0 λ = λ + 1, E 1λ = λ λ + 1, E λλ 1 λ = λ + 1 3, E 3 λ = λλ 4λ + 1 λ + 1 4, E 4 λ = λλ3 11λ + 11λ 1 λ + 1 5, E 5 λ = λλ4 6λ λ 6λ + 1 λ + 1 6, 3.0 ad so o. 4 Relatioships Ivolvig the Apostol-Geocchi Polyomials G α x; λ of Order α I this sectio, we prove a iterestig relatioship betwee the geeralized Apostol- Geocchi polyomials G α x; λ ad the Apostol-Beroulli polyomials B x; λ. Theorem. The followig relatioship holds true: G α x + y; λ = [ ] + 1G α 1 y; λ G α y; λ B x; λ 4.1

14 Some Geeralizatios ad Basic or - Extesios of the Beroulli, α, λ C; N0 betwee the geeralized Apostol-Geocchi polyomials ad the Apostol-Beroulli polyomials. Proof. By applyig a aalogous method see the proof give by Luo ad Srivastava [53, p. 636, Theorem 1], we ca obtai the explicit formula 4.1 asserted by Theorem. The details ivolved are beig omitted here. { } I terms of the geeralized Apostol-Geocchi umbers G α λ, by settig y = 0 i Theorem, we obtai the followig explicit relatioship betwee the geeralized Apostol-Geocchi polyomials G α x; λ of order α ad the Apostol-Beroulli polyomials B x; λ. Corollary 5. The followig relatioship holds true: G α x; λ = [ ] + 1G α 1 λ G α λ B x; λ 4. α, λ C; N0 betwee the Apostol-Geocchi polyomials of order α ad the Apostol-Beroulli polyomials. By otig that G 0 y; λ = y N0 ; λ C ad usig the assertio 4.1 with α = 1, we deduce Corollary 6 below. Corollary 6. The followig relatioship holds true: [ G x + y; λ = + 1y G +1y; λ ] B x; λ N0 ; λ C betwee the Apostol-Geocchi polyomials ad the Apostol-Beroulli polyomials. By taig y = 0 i 4.3, ad i view of the fact that we get the followig relatioship: G x; λ = =1 + 1 G 1 y; λ = G 1 λ = λ + 1, G +1 λb x; λ + λ C \ { 1}; N. λ 1 B x; λ 4.4 λ + 1

15 404 H. M. Srivastava By settig λ = 1 i the formula 4.4, we obtai the followig relatioship betwee the classical Geocchi umbers ad the classical Beroulli polyomials: G x = G +1 B x N, =1 which, i its further special case whe x = 0, yields the followig relatioship betwee the classical Geocchi umbers ad the classical Beroulli umbers: G = G +1 B N =1 By settig λ = 1 i 4.1, we obtai a additio theorem for the Geocchi polyomials of order α give by Corollary 7 below. Corollary 7. The followig relatioship holds true: G α [ ] x + y = + 1G α 1 y G α y B x 4.7 α C; N 0. Lettig y = 0 i 4.7, we get the followig relatioship betwee the Geocchi polyomials of order α ad the classical Beroulli polyomials: G α x = [ ] + 1G α 1 G α +1 B x α C; N We ext recall a potetially useful result due to Luo ad Srivastava [53, p. 638, Theorem ]. Theorem B Luo ad Srivastava [53, p. 638, Theorem ]. The followig relatioship holds true: E α [ ] x + y; λ = E α 1 +1 y; λ E α y; λ B x; λ 4.9 α λ 1 + B +1x; λ α C; λ C \ { 1}; N λ + 1 betwee the geeralized Apostol-Euler polyomials ad the Apostol-Beroulli polyomials. Remar 3. The followig additioal term i 4.9: α λ 1 B +1x; λ, + 1 λ + 1 was first foud by Wag et al. see [78, Corollary.6.13].

16 Some Geeralizatios ad Basic or - Extesios of the Beroulli, { } I terms of the geeralized Apostol-Euler umbers E α λ, by settig y = 0 i Theorem B, we obtai the followig explicit relatioship betwee the geeralized Apostol- Euler polyomials ad the Apostol-Beroulli polyomials. Corollary 8. The followig relatioship holds true: E α [ ] x; λ = E α 1 +1 λ E α λ B x; λ α λ 1 + B +1x; λ α C; λ C \ { 1}; N λ + 1 betwee the geeralized Apostol-Euler polyomials ad the Apostol-Beroulli polyomials. Corollary 9 below provides the corrected versio of each of the five ow formulas due to Luo ad Srivastava [53, pp , Es. 56, 57, 60, 63 ad 64]. Corollary 9. Each of the followig relatioships holds true: [y E x + y; λ = +1 E +1y; λ ] B x; λ + 1 λ λ + 1 E x; λ = E +1 0; λb x; λ E x; λ = 1 B +1 x; λ, 4.1 λ 1 B +1 x; λ, + 1 λ [ B λ B ] λ B x; λ λ 1 + B +1 x; λ, λ + 1 ad E α x; λ = E α λ = [ E α 1 +1 x; λ E α +1 x; λ ] B λ λ α B +1λ 4.15 λ + 1 [ +1 E α 1 α ] ; λ E α +1 λ B λ α λ 1 + B +1λ λ + 1 α C; λ C \ { 1}; N0.

17 406 H. M. Srivastava 5 Explicit Represetatios Ivolvig the Hurwitz or Geeralized Zeta Fuctio ζs, a The Hurwitz-Lerch zeta fuctio Φz, s, a defied by see, for example, [68, p. 11 et se.] z Φz, s, a := + a s 5.1 a C \ Z 0 ; s C whe z < 1 ; Rs > 1 whe z = 1, which ca ideed be cotiued meromorphically to the whole complex s-plae, except for a simple pole at s = 1 with its residue 1, cotais as its special cases ot oly the Hurwitz or geeralized zeta fuctio ζs, a defied by 1.0 ad the Riema zeta fuctio ζs defied by 1.1, but also such other importat fuctios of Aalytic Number Theory as for example the Lipschitz-Lerch zeta fuctio ϕξ, a, s or L ξ, s, a defied by cf. [68, p. 1, E..5 11]: ϕξ, a, s := e πiξ + a s = Φ e πiξ, s, a =: L ξ, s, a 5. a C \ Z 0 ; Rs > 0 whe ξ R \ Z; Rs > 1 whe ξ Z which was first studied by Rudolf Lipschitz ad Matyáš Lerch i coectio with Dirichlet s famous theorem o primes i arithmetic progressios. For various extesios ad geeralizatios of the Hurwitz-Lerch zeta fuctio Φz, s, a defied by 5.1, the iterested reader may be referred to several recet wors icludig for example [19], [41] ad [75] ad the refereces cited i each of these wors see also [13] ad [67]. Precisely oe decade ago, Srivastava [66] made use of Lerch s fuctioal euatio: { ϕξ, a, 1 s = Γs [ ] 1 π s exp s aξ πi ϕ a, ξ, s + exp [ 1 ] } s + a1 ξ πi ϕa, 1 ξ, s s C; 0 < ξ < 1 i cojuctio with Apostol s formula [, p. 164]: ϕξ, a, 1 = Φe πiξ, 1, a = B a; e πiξ 5.3 N, 5.4 i order to obtai a elegat formula for the Apostol-Beroulli polyomials B x; λ, which we recall here as Theorem C below.

18 Some Geeralizatios ad Basic or - Extesios of the Beroulli, Theorem C Srivastava s formula [66, p. 84, E. 4.6]. The Apostol-Beroulli polyomials B x; λ at ratioal argumets are give by { p B ; eπiξ = π ζ, ξ + j 1 [ ] ξ + j 1p exp πi + [ exp j ξp + πi] }, N \ {1} ; p Z; N; ξ R, ζ, j ξ 5.5 i terms of the Hurwitz or geeralized zeta fuctio ζs, a. Two aalogous formulas for the Apostol-Euler polyomials E x; λ ad the Apostol- Geocchi polyomials G x; λ at ratioal argumets are asserted by Theorem 3 ad Theorem 4, respectively. Theorem 3. The followig represetatio of the Apostol-Euler polyomials at ratioal argumets holds true: { p E ; eπiξ = ξ + j 1 π +1 ζ + 1, [ + 1 exp ξ + j 1p [ exp ] πi + j ξ 1p πi ζ + 1, ] }, N; p Z; ξ R i terms of the Hurwitz or geeralized zeta fuctio ζs, a. j ξ Proof. First of all, we recall a useful relatioship betwee the Apostol-Euler polyomials ad the Apostol-Beroulli polyomials give by see [53, p. 636, E. 38] E 1 x; λ = [ x B x; λ B ; λ] N 5.7 or, euivaletly, by Taig E x; λ = + 1 x = p [ B +1 x; λ +1 B +1 x ; λ] N ad λ = e πiξ p Z; N; ξ R

19 408 H. M. Srivastava i the last formula 5.8, we fid from Srivastava s formula?? with that E p ; eπiξ { = + 1! + 1 π , ad ξ ξ ζ + 1, j ξ + 1! 4π +1 + = π +1 + ζ + 1, j ξ ζ + 1, ξ + j 1 [ + 1 exp [ exp πi] j ξp ζ + 1, ξ + j 1 [ + 1 exp [ exp ζ + 1, ξ + j 1 [ + 1 exp ζ + 1, ξ + j 1 [ + 1 exp ζ + 1, j ξ [ exp ζ + 1, j ξ [ exp πi] } j ξp ] ξ + j 1p πi ] ξ + j 1p πi ] ξ + j 1p πi ] ξ + j 1p πi ] j ξp πi The first sum i 5.9 ca obviously be rewritte the followig form: ζ + 1, ξ + j 1 [ + 1 exp = + ζ + 1, ξ + j 1 [ + 1 exp ζ + 1, ] j ξp πi. 5.9 ] ξ + j 1p πi [ ξ + j exp ] ξ + j 1p πi ] ξ + j 1p πi. 5.10

20 Some Geeralizatios ad Basic or - Extesios of the Beroulli, The third sum i 5.9 ca also be rewritte the followig form: ζ + 1, j ξ [ exp = + ζ + 1, ] j ξp πi [ j ξ 1 exp ζ + 1, j ξ [ exp ] j ξ 1p πi ] j ξp πi Upo first separatig the eve ad odd terms i 5.10 ad 5.11, ad the substitutig from 5.10 ad 5.11 ito 5.9, we are led evetually to the formula?? asserted by Theorem 3. Theorem 4. The followig represetatio of the Apostol-Geocchi polyomials at ratioal argumets holds true: { p G ; eπiξ = π ζ, + ζ, j ξ 1 [ ] ξ + j 1 ξ + j 1p exp πi [ exp j ξ 1p + πi] } 5.1 N \ {1}; p Z; N; ξ R i terms of the Hurwitz or geeralized zeta fuctio ζs, a. Proof. We apply the relatioship: G x; λ = E 1 x; λ 5.13 with, of course, x = p ad λ = e πiξ p Z; N; ξ R i cojuctio with the formula??. We thus obtai the assertio 5.1 of Theorem 4. For ξ Z, the formula?? ca easily be show to reduce to the followig ow result give earlier by Cvijović ad Kliowsi [16, p. 159, Theorem B] see also [66, p. 78, Theorem B]. Corollary 10. The followig represetatio of the classical Euler polyomials holds true: p E = 4 π +1 ζ + 1, j 1 j 1pπ si π,, N; p Z i terms of the Hurwitz or geeralized zeta fuctio ζs, a. 5.14

21 410 H. M. Srivastava A special case for the formula 5.1 whe ξ Z is stated here as Corollary 10 below. Corollary 11. The followig represetatio of the classical Geocchi polyomials holds true: G p = 4 π ζ, j 1 cos j 1pπ π N \ {1}; p Z; N i terms of the Hurwitz or geeralized zeta fuctio ζs, a. The followig formula for the Apostol-Beroulli polyomials B α x; λ of order α was prove by Luo ad Srivastava [5]. Theorem D Luo ad Srivastava [5, p. 300, Theorem ]. The Apostol-Beroulli polyomials B α x; λ of order α at ratioal argumets are give by p B α ; eπiξ = e πiξ 1 1 α 1 B 1 e πiξ ζ, ξ + j 1 exp + =! π [ ξ + j 1 p B α 1 ] πi e πiξ ζ, j ξ [ exp ] j ξ p + πi N \ {1} ; p Z; N; ξ R \ Z; α C 5.15 holds true i terms of the Hurwitz or geeralized zeta fuctio ζs, a. For α = 1, the formula 5.15 reduces to Srivastava s formula??. Whe ξ Z i??, Srivastava s formula 5.15 ca easily be show to reduce to a ow result give earlier by Cvijović ad Kliowsi [16, p. 159, Theorem A] see also [66, p. 78, Theorem A]: B p = π = ζ, j jpπ cos π, N \ {1} ; p Z; N. The followig formula is a complemet of 5.15 whe ξ Z: p p B α = B α B α 1 1! π B α 1 ζ, j jpπ cos π

22 Some Geeralizatios ad Basic or - Extesios of the Beroulli, N \ {1} ; p Z; N; α C. By applyig?? ad 5.1, we ow derive the followig represeatio formulas for the Apostol-Euler polyomials of order α ad the Apostol-Geocchi polyomials of order α, respectively. Theorem 5. The followig represetatio of the Apostol-Euler polyomials of order α holds true: p E α ; eπiξ = e πiξ + 1 E α 1 e πiξ! + π +1 E α 1 eπiξ =1 { [ ] ξ + j ξ + j 1p ζ + 1, exp πi [ j ξ 1 + ζ + 1, exp + 1 ] } j ξ 1p + πi, N; p Z; ξ R \ Λ Λ := { + 1 } : Z ; α C i terms of the Hurwitz or geeralized zeta fuctio ζs, a. Proof. We apply the ow result [45, p. 919, E. 9 with α α 1 ad β = 1]: E α x; λ = E α 1 λe x; λ ad the special values of E x; λ give by E 0 x; λ = E 0 λ = λ Upo separatig the = 0 term i cojuctio with the formula??, the represetatio formula 5.18 follows readily. Theorem 6. The followig represetatio of the Apostol-Geocchi polyomials of order α at ratioal argumets holds true: p G α ; eπiξ = e πiξ + 1 Gα 1 1 e πiξ! + π G α 1 eπiξ = { [ ] ξ + j 1 ξ + j 1p ζ, exp πi + ζ, [ j ξ 1 exp j ξ 1p + πi] } 5.19

23 41 H. M. Srivastava N \ {1}; p Z; N; ξ R \ Λ Λ := { + 1 } : Z ; α C i terms of the Hurwitz or geeralized zeta fuctio ζs, a. Proof. We apply the formula.3 ad ote that G 0 x; λ = G 0 λ = 0 ad G 1 x; λ = G 1 λ = λ Upo first separatig the = 0 ad = terms, ad the usig the formula 5.1, we arrive at the represetatio 5.19 asserted by Theorem 6. I their special cases whe ξ Z, Theorems 5 ad 6 readily yield Corollaries 1 ad 13, respectively, which provide the correspodig represetatios of the Euler polyomials of order α ad the Geocchi polyomials of order α at ratioal argumets. Corollary 1. The followig represetatio of the geeralized Euler polyomials at ratioal argumets holds true: E α p = E α 1 + 4! π +1 =1 j 1pπ si π E α 1 i terms of the Hurwitz or geeralized zeta fuctio ζs, a. ζ + 1, j 1,, N; p Z; α C 5.1 Corollary 13. The followig represetatio of the geeralized Geocchi polyomials at ratioal argumets holds true: G α p = G α = 4! π G α 1 N \ {1}; p Z; N; α C ζ, j 1 j 1pπ cos π 5. i terms of the Hurwitz or geeralized zeta fuctio ζs, a. Clearly, by settig α = 1 i 5.1 ad 5., we agai obtai the formulas 5.14 ad 11, respectively. O the other had, if we apply the formulas 3.4 of Lemma 3 ad 3.7 of Lemma 4 i cojuctio with the assertio 5.15 of Theorem D of Luo ad Srivastava [5], we obtai the series represetatios of G α are give by Theorems 7 ad 8 below. x; λ ad E α x; λ, respectively, which

24 Some Geeralizatios ad Basic or - Extesios of the Beroulli, Theorem 7. The followig series represetatio holds true for the Apostol-Geocchi polyomials of reder α: p G α ; eπi = α + e πiξ + 1 Bα 1 1 ζ, ζ, e πiξ = ξ + j 1 exp! π B α 1 [ ξ + j 1 p e πiξ ] πi [ j ξ 1 exp ] j ξ 1 p + πi 5.3 N \ {1}; p Z; N; ξ R \ Λ Λ := { + 1 } : Z ; α C i terms of the Hurwitz or geeralized zeta fuctio. Theorem 8. The followig series represetatio holds true for the Apostol-Euler polyomials of order l: p E l ; eπiξ l = + l 1!e πiξ + 1 Bl 1 B l 1 +l e πiξ + ζ, ζ, +l 1 e πiξ +l! + l π = [ ξ + j 1 ξ + j 1 p exp ] πi [ j ξ 1 exp ] j ξ 1 p + πi 5.4 N \ {1}; p Z; l, N; ξ R \ Λ Λ := { + 1 } : Z i terms of the Hurwitz or geeralized zeta fuctio. Remar 4. It is ot difficult to apply the relatioships 3.5 of Lemma 3 ad 3.6 of Lemma 4 i cojuctio with the above formulas 5.19 ad 5.18, respectively, i order to obtai the correspodig series represetatios for the Apostol-Beroulli polyomials B α x; λ of order α C..

25 414 H. M. Srivastava 6 The λ-stirlig Numbers of the Secod Kid ad Their Elemetary Properties I this sectio, we first itroduce a aalogue of the familiar Stirlig umbers S, of the secod id, which we choose to call the λ-stirlig umbers of the secod id. We the derive several elemetary properties icludig recurrece relatios for them. We also pose two ope problems relevat to our preset ivestigatio. Defiitio 5. The λ-stirlig umbers S, ; λ of the secod id is defied by meas of the followig geeratig fuctio: λe z 1! = S, ; λ z N 0 ; λ C, 6.1 so that, obviously, S, := S, ; 1 for the Stirlig umbers S, of the secod id defied by 1.8 see [15, p. 06, Theorem A]. Theorem 9. The λ-stirlig umbers S, ; λ of the secod id ca also be defied as follows: x λ x x =! S, ; λ N 0 ; λ C. 6. Proof. By usig 6.1 ad the biomial theorem, we easily obtai the assertio 6. of Theorem 7. Theorem 10. The followig explicit represetatio formulas hold true: S, ; λ = 1! 1 j λ j j, N 0 ; λ C 6.3 j j=0 ad S, ; λ = 1! 1 j λ j j, N 0 ; λ C. 6.4 j j=0 Proof. Just as i our demostratio of Theorem 7, we ca easily derive 6.3 ad 6.4 by usig 6.1 ad the biomial theorem. Theorem 11. The λ-stirlig umbers S, ; λ of the secod id satisfy the followig triagular ad vertical recurrece relatios: S, ; λ = S 1, 1; λ + S 1, ; λ, N 6.5

26 ad respectively. Some Geeralizatios ad Basic or - Extesios of the Beroulli, S, ; λ = λ j 1 Sj, 1; λ, N, 6.6 j j=0 Proof. By differetiatig both sides of 6.1 with respect to the variable z, we readily arrive at the recursio formulas 6.5 ad 6.6 asserted by Theorem 9. Theorem 1. The followig explicit relatioships hold true: ad S, ; λ = S, ; λ = j= j=0 j log λ j Sj,, N 0 ; λ C 6.7 j! λ j λ 1 j j! S, j, N 0 ; λ C 6.8 betwee the λ-stirlig umbers S, ; λ of the secod id ad the Stirlig umbers S, of the secod id. Proof. By applyig 6.1, it is failrly straightforward to derive the formulas 6.7 ad 6.8. By meas of the formula 6.3 or 6.8 i cojuctio with 6.1, we ca compute several values of S, ; λ give by S0, 0; λ = 1, S1, 0; λ = 0, S1, 1; λ = λ, S, 0; λ = 0, S, 1; λ = λ, S, ; λ = λλ 1, S3, 0; λ = 0, S3, 1; λ = λ, S3, ; λ = λ4λ 1, S3, 3; λ = 1 λ9λ 8λ + 1, S4, 0; λ = 0, S4, 1; λ = λ, S4, ; λ = λ8λ 1, S4, 3; λ = 1 λ7λ 16λ + 1, S4, 4; λ = 1 6 λ64λ3 81λ + 4λ 1, S5, 0; λ = 0, S5, 1; λ = λ, S5, ; λ = λ16λ 1, S5, 3; λ = 1 λ81λ 3λ + 1, S5, 4; λ = 1 6 λ56λ3 43λ + 48λ 1, S5, 5; λ = 1 4 λ65λ5 104λ λ 64λ + 1, 6.9 ad S0, ; λ = λ 1!, S, 0; λ = δ,0 ad S, 1; λ = λ, N 0, 6.10

27 416 H. M. Srivastava ad so o, δ m, beig the Kroecer symbol. Whe λ = 1, 6.1 ad 6. become the correspodig rather familiar defiitios for the Stirlig umbers S, of the secod id see, for details, [15, p. 06, Theorem A; p. 07 Theorem B]. Similarly, i their special case whe λ = 1, the formulas 6.3, 6.4, 6.5 ad 6.6 would yield the correspodig well-ow results for the Stirlig umbers S, of the secod id see, for details, [15, p. 04, Theorem A; p. 08, Theorem A; p. 09, Theorem B]. Each of the followig special values of S, is ow see [15, pp. 6 7, Ex. 16] ad [60, p. 31]: S, = 1, S, 1 =, S, = ad S, 3 = , 6.11 so that, if we mae use of the formula 6.3 with λ = 1 i cojuctio with these special values of S,, we obtai the followig iterestig summatio formulas: ad j=0 j=0 1 j j = 1, 6.1 j j=0 1 j j +1 = 1 + 1! j, j j + = 1 +! j 1 1 j j j=0 j +3 = 1 + 3! More geerally, we have the followig formula recorded by Gould [18, p. 3, Etry 1.17]: 1 j j + = 1 +! j j=0 j=0 j j Ope Problem 1. Does there exist a aalogue of the sum give below? 1 j λ j j + j j=0 N; N 0 ; λ C. 1 S + j, j j! Ope Problem. Ca we fid a ratioal geeratig fuctio for the λ-stilig umbers S, ; λ of the secod id aalogous to a ow result [15, p. 07, Theorem C]?

28 Some Geeralizatios ad Basic or - Extesios of the Beroulli, Applicatios of the λ-stirlig Numbers S, ; λ of the Secod Kid to the Family of the Apostol Type Polyomials I the sectio, we give some applicatios of the λ-stirlig umbers S, ; λ of the secod id to the Apostol type polyomials ad Apostol type umbers. We obtai some iterestig series represetatios for the Apostol-Geocchi polyomials ivolvig the λ- Stirlig umbers S, ; λ of the secod id ad the Hurwitz or geeralized zeta fuctio ζs, a. We begi by recallig that Wag et al. [78] gave the followig results for the Apostol-Euler polyomials of order α usig the λ-stirlig umbers S, ; λ of the secod id defied by 6.1. ad E α x + y; λ = x = l= j l l= j j!!l + j! l! Sl + j, j; λe α l y; λbj x; λ, j N 0; α, λ C 7.1 j! Sl + j, j; λbj l l + j! l! x; λ, j N 0; λ C. 7. Applicatio 1. First of all, we give some recurrece relatioships for the Apostol-Beroulli umbers of order l l N by usig the λ-stirlig umbers of the secod id. Theorem 13. Let S, ; λ deote the λ-stirlig umbers of the secod id defied by 6.1. The +l + l S + l, l; λb l λ = 0, l N; λ C. 7.3 Proof. By applyig 1.10 with α = l N ad x = 0 ad 6.1, we fid that 1 = z l λe z 1 l = z l l! = [ + l l B l λ z S, l; λ z 1 +l B l λ z ] S + l, l; λb l λ + l z. 7.4 Now, by comparig the coefficiets of z N o both sides of 7.4, we easily obtai the assertio 7.3 of Theorem 11.

29 418 H. M. Srivastava Remar 5. By settig λ = 1 i 7.3 ad observig that +l = + +l =+1 ad S + l, l = l, we have the followig recurrece relatio for the Beroulli umbers of order l or, euivaletly, the Nörlud umbers [56]: B l + l = l S + l, lb l. 7.5 Remar 6. Whe λ 1 i 7.3, if we apply the followig values for the λ-stirlig umbers S, ; λ: S0, ; λ = λ 1! ad S, 1; λ = λ, N i cojuctio with 7.3, we have the followig recurrece relatio for the Apostol- Beroulli umbers of order l or, euivaletly, the geeralized Nörlud umbers [56]: B l l! +l λ = λ 1 l +l 1 + l S + l, l; λb l λ. 7.7 Remar 7. By settig l = 1 i 7.5 ad otig that S, 1 = 1, we deduce the followig familiar recurrece relatios for the classical Beroulli umbers B : B 0 = 1 ad B = B N. 7.8 Remar 8. By settig l = 1 i 7.7 ad otig that S, 1; λ = λ, we deduce the followig ow recurrece relatios for the Apostol-Beroulli umbers B λ: B 0 λ = 0, B 1 λ = 1 λ 1 ad B λ = λ 1 λ 1 B λ N \ {1}. 7.9 Applicatio. If we tae α = l l N i 1.10, the Defiitio assumes the followig form: λe z l 1 e xz = B l x; λ z z. 7.10

30 Some Geeralizatios ad Basic or - Extesios of the Beroulli, By 7.10 ad 6.1, we thus have B l x; λ z = z l λe z 1 l e xz = z l l! = [ + l which leads us to Theorem 1 below. l S, l; λ z 1 +l zx + l ]z S, l; λx +l, 7.11 Theorem 14. The followig relatioship holds true: B l x; λ = + l l 1 +l + l S, l; λx +l, l N 7.1 betwee the geeralized Apostol-Beroulli polyomials of order l l N ad the λ- Stirlig umbers of the secod id. Remar 9. Taig λ = 1 i 7.1, we have B l x = which upo settig l =, yields or, euivaletly, + l B x = B x = l 1 +l + l S, lx +l, 7.13 S, x 7.14 S +, x Remar 10. Puttig x = 0 i 7.1, we have 1 + l B l λ = S + l, l; λ l Further, upo lettig λ = 1 i 7.16 or settig x = 0 i 7.13, we obtai which, for l =, yields B l = B = 1 + l S + l, l, 7.17 l 1 S,, 7.18

31 40 H. M. Srivastava or, euivaletly, B = S, Applyig the recursio formula 7.19 ad the ow formulas i [56, p. 146], we ca calculate the first five values of B ad B N as give below: B 1 1 = 1, B = 7 6, B 3 3 = 9, B 4 4 = 43 10, B 5 B 1 1 = 1, B = 5 6, B3 3 = 9 4, B4 4 = B 1 1 = 1, B = 5 6, B3 3 = 9 4, B4 4 = = , ad B 5 5 = ad B 5 5 = Next, by applyig 1.10 with α = l N ad 6.1, we have x z+l = λe z 1 l = l! [ = l! S, l; λ z B l x; λ z B l x; λ z S, l; λb l x; λ ] z, 7.1 which leads us to a euivalet versio of 7. give by Theorem 13 below. Theorem 15. The followig expasio formula holds true: x l = l 1 Remar 11. Whe λ = 1 i 7., we have x l = l 1 S, l; λb l x; λ, l N 0; l. 7. S, lb l x, l N 0; l. 7.3 Remar 1. Upo settig l = 1 i 7., if we apply 6.10, we deduce the followig ow differece euatio: x 1 = λb x + 1; λ B x; λ, 7.4 which, i the further special case whe λ = 1, is a well-ow rather classical result. Applicatio 3. We here obtai some series represetatios of the Apostol-Geocchi polyomials of higher order by applyig the λ-stirlig umbers of the secod id. Ideed, by

32 Some Geeralizatios ad Basic or - Extesios of the Beroulli, usig 1.10, 1.18 ad 6.1, we obtai l G l x; λ z = z λ e z e xz λe z 1 l 1 x = B l ; λ z l! [ = l! r r=0 which leads us to the followig lemma. Lemma 6. The followig relatioship holds true: G l x; λ = l! r=0 r S r, l; λb r l r r S r, l; λb l r S, l; λ z x ; λ] z, 7.5 x ; λ, l N 0 ; λ C 7.6 betwee the geeralized Apostol-Geocchi polyomials ad the λ-stirlig umbers of the secod id. By applyig 5.15 ad 7.6, we easily obtai the followig series represetatio for the geeralized Geocchi polyomials G l x. Theorem 16. The Apostol-Geocchi polyomials G l x; λ of order l at ratioal argumets are give by p G l ; eπiξ = r l! r e 4πiξ 1 r r= r! l! r r= = 4π ζ B l 1 r 1 r e 4πiξ S r, l; e πiξ r B l 1 r e 4πiξ S r, l; e πiξ, ξ + j 1 [ ] ξ + j 1 p exp πi + ζ, j ξ [ exp ] j ξ p + πi, l N \ {1}; p Z; N; ξ R \ Λ { } Λ := : Z 7.7 i terms of the λ-stirlig umbers S, ; λ of the secod id ad the Hurwitz or geeralized zeta fuctio ζs, a.

33 4 H. M. Srivastava By applyig 7.6 with λ = 1 ad 5.17, we ca obtai the followig series represetatio for the geeralized Geocchi polyomials G l x, which is actually a complemet of 7.7 for ξ Z. Corollary 14. The geeralized Geocchi polyomials G l x of ratioal argumets are give by G l p + r= p = l! S, l + l! S 1, l r= = [ r S r, l B r l 1 + r r r r+1! 4π r r B l 1 r l p 1 S r, l ] B l 1 r 1 N \ {1};, l N; p Z ζ, j jpπ cos π 7.8 i terms of the Stirlig umbers S, of the secod id ad the Hurwitz or geeralized zeta fuctio ζs, a. By lettig l = 1 i 7.8, we obtai the followig explicit series represetatio for the classical Geocchi polyomials. Corollary 15. The classical Geocchi polyomials G x at ratioal argumets are give by G p p = 1 + 1! π ζ, j jpπ cos π = N \ {1};, l N; p Z i terms of the Hurwitz or geeralized zeta fuctio ζs, a. 7.9 Remar 13. It is ot difficult to derive the correspodig formulas for the Apostol-Euler polyomials ad the Apostol-Beroulli polyomials at ratioal argumets by applyig the relatioships 3.3 ad 3.5 i cojuctio with 7.7, 7.8 ad 7.9. The details are beig omitted here. 8 Further Results ad Observatios I this sectio, we apply Srivastava s formula Theorem C above ad some relatioships i order to obtai several differet series represetatios for the Geocchi polyomials of order α ad the Euler polyomials of order α.

34 Some Geeralizatios ad Basic or - Extesios of the Beroulli, We first rewrite the formulas 4. ad 4.11 i coveiet forms give by Lemmas 7 ad 8, respectively. Lemma 7. The followig series represetatio holds true: G α x; λ = + 1 [ + 1G α 1 α, λ C; N0. ] λ Gα +1 λ B x; λ 8.1 Lemma 8. The followig series represetatio holds true: E α x; λ = + 1 λ [ ] E α 1 α +1 λ E +1 λ B x; λ λ + 1 α B +1x; λ α, λ C; N0. 8. Theorem 17. The Apostol-Geocchi polyomials G α x; λ of order α at ratioal argumets are give by p G α ; [ eπiξ = e πiξ 1! + 1 π = G α 1 1 eπiξ G α [ ] e πiξ + 1G α 1 eπiξ G α ζ, ξ + j 1 [ ] ξ + j 1p exp πi + ζ, j ξ [ exp ] j ξp + πi N \ {1}; N; p Z; ξ R \ Z; α C +1 eπiξ ] 8.3 i terms of the Hurwitz or geeralized zeta fuctio ζs, a. Proof. Upo separatig the = 0 ad = 1 terms i 8.1 ad applyig Srivastava s formula?? with N \ {1}, if we ote that B 0 x; λ = B 0 λ = 0 ad B 1 x; λ = B 1 λ = 1 λ 1, 8.4 we arrive at the formula 8.3 asserted by Theorem 15.

35 44 H. M. Srivastava Theorem 18. The Apostol-Euler polyomials E α x; λ of order α at ratioal argumets are give by E α p ; eπiξ = + [ e πiξ 1! + 1 π = E α 1 ] e πiξ E α e πiξ [ ] E α 1 +1 eπiξ E α +1 eπiξ ζ, ξ + j 1 [ ] ξ + j 1p exp πi ζ, j ξ [ exp ] j ξp + πi α eπiξ 1 + 1! + 1 π +1 e πiξ + 1 [ ] + 1 ξ + j 1p exp πi + ζ + 1, j ξ [ exp N \ {1}; N; p Z; ξ R \ {Z Λ} i terms of the Hurwitz or geeralized zeta fuctio ζs, a. ζ + 1, ξ + j 1 ] j ξp πi 8.5 Λ := { + 1 } : Z ; α C Proof. Just as i our demostratio of Theorem 16, the represetatio formula 8.5 ca be prove by applyig 8. ad??. By meas of 8.1 with λ = 1 ad 8. with λ = 1 i cojuctio with the formula 5.16, we ca deduce Corollaries 16 ad 17 below assertig series represetatios for the Geocchi polyomials of order α ad the Euler polyomials of order α, respectively. Corollary 16. The geeralized Geocchi polyomials G α x of order α at ratioal argumets are give by [ G α 1 G α p = [ ] + 1G α 1 G α p ] G α 4! [ ] + 1 π + 1G α 1 G α +1 = ζ, j jpπ cos π N \ {1}; N; p Z; α C 8.6

36 Some Geeralizatios ad Basic or - Extesios of the Beroulli, i terms of the Hurwitz or geeralized zeta fuctio ζs, a. Corollary 17. The geeralized Euler polyomials E α x of order α at ratioal argumets are give by E α p = [ ] E α 1 +1 E α p [ ] E α 1 E α 4! [ ] + 1 π E α 1 +1 Eα +1 = ζ, j jpπ cos π N \ {1}; N; p Z; α C i terms of the Hurwitz or geeralized zeta fuctio ζs, a. 8.7 Remar 14. The series represetatio formulas 8.6 ad 8.7 are, respectively, the complemet of 8.3 ad 8.5 for ξ Z. Furthermore, by lettig α = 1 i 8.6 ad 8.7, we obtai the followig explicit series represetatios for the classical Geocchi polyomials ad the classical Euler polyomials, respectively. Corollary 18. The classical Geocchi polyomials G x at ratioal argumets are give by G p = 1 p + 1 G +1 1 G + ζ, j cos jpπ π = i terms of the Hurwitz or geeralized zeta fuctio ζs, a. 4G +1! + 1 π N \ {1}; N; p Z 8.8 Corollary 19. The classical Euler polyomials E x at ratioal argumets are give by p E = p + 1 E E +1! E π = ζ, j jpπ cos π N \ {1}; N; p Z 8.9 i terms of the Hurwitz or geeralized zeta fuctio ζs, a. Remar 15. It is fairly easy to apply the relatioships 3.5 of Lemma 3 ad 3.6 of Lemma 4 i cojuctio with the above formulas 8.3 ad 8.5, respectively, i order to obtai the correspodig series represetatios for the Apostol-Beroulli polyomials B α x; λ of order α at ratioal argumets. The details ivolved are beig left as a exercise for the iterested reader.

37 46 H. M. Srivastava We ow separate the eve ad odd terms of the formula 8.9. By otig that we thus obtai p E 1 ad E p E = 0 ad E 1 = 1 G p = 1 E 1 + ζ 1, j = ζ + 1, j si E +1 + cos N, 4 E ! π 1 1 = jpπ N \ {1}; N; p Z 8.10 =1 jpπ 4 E ! π N \ {1}; N; p Z O the other had, by separatig the eve ad odd terms of the formula 5.14, we get see [16, p. 159, Theorem B] ad [66, p. 78, Theorem B]; see also Corollary 10 above E 1 p 4 1! = 1 π ζ, j 1 j 1pπ cos N \ {1}; N; p Z 8.1 ad E p = 1 4 π +1 ζ + 1, j 1 j 1pπ si 8.13 N \ {1}; N; p Z. Fially, by comparig the formulas 8.1 ad 8.10 ad the formulas 8.13 ad 8.11, respectively, we obtai the followig iterestig relatioships ivolvig the eve ad odd Hurwitz or geeralized zeta fuctios: 1 +1 π !, j 1 = cos ζ 1, j si j 1pπ jpπ + π 1! N \ {1}; N; p Z E +1 = 1 ζ p 1 E

38 ad Some Geeralizatios ad Basic or - Extesios of the Beroulli, π ! =1 + 1, j 1 ζ, j cos si jpπ j 1pπ N \ {1}; N; p Z. E +1 = 1 ζ + π+1 + 1! E We ow recall the followig iterestig itegral represetatios for the Apostol- Beroulli polyomials ad the Apostol-Euler polyomials, which were give recetly by Luo [46]. Lemma 9 Luo [46, p. 198, Theorem ; p. 199, Theorem ]. The followig itegral represetatio holds true for the Apostol-Beroulli polyomials: B z; e πiξ = z; ξ e πizξ U; z, t coshπξt + i V ; z, t sihπξt t 1 up dt, 0 cosh πt cos πx 8.16 N; 0 Rz 1; ξ < 1 ξ R, where z; ξ is give by 0 ξ = 0 z; ξ = 1 πiξ e πizξ ξ 0, U; z, t = [ cos πz π π cos e πt] ad V ; z, t = [ si πz π π + si e πt]. Furthermore, the followig itegral represetatio holds true for the Apostol-Euler polyomials: E z; e πiξ = e πizξ 8.17 X; x, t coshπξt + i Y ; z, t sihπξt t up updt cosh πt cos πz 0 N; 0 Rz 1; ξ < 1 ξ R,

39 48 H. M. Srivastava where ad X; z, t = Y ; z, t = [ e πt si πz + π [ e πt cos πz + π + e πt si πz π e πt cos πz π ] ]. We apply the relatioships 3. of Lemma ad 3.4 of Lemma 3 i cojuctio with the above formulas 8.16 ad 8.17, respectively. We thus obtai the correspodig itegral represetatios for the Apostol-Geocchi polyomials G α z; λ. Theorem 19. The followig itegral represetatio holds true for the Apostol-Geocchi polyomials: G z; e πiξ = e πizξ 8.18 M; z, t coshπξt + i N; x, t sihπξt t 1 up dt coshπt cosπz 0 N; 0 Rz 1; ξ < 1 ξ R, where ad M; z, t = N; z, t = [ e πt cos πz π [ e πt si πz π e πt cos πz + π + e πt si πz + π ] ]. Remar 16. Upo lettig ξ Z i 8.16 ad 8.17, we easily deduce that B z = 0 cos πz π e πt cos π coshπt cosπz N; 0 Rz 1 t 1 up dt 8.19 ad E z = 0 e πt si πz π + e πt si πx + π coshπt cosπz t up dt 8.0 N; 0 Rz 1 for the classical Beroulli polyomials ad the classical Euler polyomials, respectively. Moreover, by settig z = p i 8.19 ad 8.0, ad otig the formulas 5.16 ad

40 Some Geeralizatios ad Basic or - Extesios of the Beroulli, , we ca get the followig itegral represetatios for the Hurwitz or geeralized zeta fuctio ζs, a: ad ζ, j jpπ cos π = π 1! 0 cos pπ ζ + 1, j 1 j 1pπ si = π+1 π e πt cos π t 1 up dt coshπt cos pπ N \ {1} ; p N 0 ; N, p 0 eπt si pπ π π coshπt cos + e πt si pπ N; p N 0 ; N, p. pπ + π 8.1 t up dt, 8. Remar 17. By lettig i 8.1 ad 8., we obtai the followig iterestig itegral represetatios ivolvig the eve Hurwitz or geeralized zeta fuctio ζ, a ad the odd Hurwitz or geeralized zeta fuctio ζ + 1, a, respectively: ad ζ, j jpπ cos = π 1! ζ + 1, j 1 si π +1 si = 0 cos pπ e πt t 1 up dt 8.3 coshπt cos pπ N \ {1} ; p N 0 ; N; p j 1pπ pπ 0 coshπt coshπt cos N; p N 0 ; N; p. pπ t up dt 8.4

41 430 H. M. Srivastava Remar 18. The formulas??,?? ad 5.1 lead us easily to the followig represetatios for the Apostol-Beroulli polyomials, the Apostol-Euler polyomials ad the Apostol-Geocchi polyomials at ratioal argumets: B p ; eπiξ [ exp exp = π ξ + j 1p [ j ξ 1p + πi ζ, ] + πi ]} ξ + j 1 ζ, N \ {1} ; p Z; N; ξ R, j ξ E p ; eπiξ + { = π +1 ζ + 1, ξ + j [ + 1 exp ζ + 1, j ξ 1 exp [ + 1 +, N; p Z; ξ R ] ξ + jp πi j ξ 1p πi] } 8.6 ad G p ; eπiξ + { = π ζ, ξ + j [ ] ξ + jp exp πi ζ, j ξ 1 exp [ j ξ 1p + πi] } 8.7 respectively. N \ {1}; p Z; N; ξ R, By applyig Lemma, Lemma 3 ad Lemma 4 ad the above formulas??, 8.6 ad 8.7 i cojuctio with the results of this paper ad of the earlier wors see, for example, [44], [5], [45], [53], [50], [46], [47], [48] ad [49], we ca also derive a large umber of iterestig formulas ad relatioships. For example, if we apply the relatioships 3. ad 3.4 i cojuctio with the ow Fourier expasios of the Apostol-Beroulli polyomials ad the Apostol-Euler polyomials see, for details, [46, p. 195, Theorem.1. ad.3; p. 196, Theorem..8 ad Theorem..9], we obtai the correspodig Fourier expoetial series expasios for the Apostol-Geocchi polyomials G x; λ as follows.