Some properties of the generalized Apostol-type polynomials

Transcription

1 Lu ad Luo Boudary Value Probles 2013, 2013:64 R E S E A R C H Ope Access Soe properties of the geeralized Apostol-type polyoials Da-Qia Lu 1 ad Qiu-Mig Luo 2* * Correspodece: luoath2007@163.co 2 Departet of Matheatics, Chogqig Noral Uiversity, Chogqig Higher Educatio Mega Ceter, Huxi Capus, Chogqig, , People s Republic of Chia Full list of author iforatio is available at the ed of the article Abstract I this paper, we study soe properties of the geeralized Apostol-type polyoials see Luo ad Srivastava i Appl. Math. Coput. 217: , 2011, icludig the recurrece relatios, the differetial equatios ad soe other coected probles, which exted soe kow results. We also deduce soe properties of the geeralized Apostol-Euler polyoials, the geeralized Apostol-Beroulli polyoials, ad Apostol-Geocchi polyoials of high order. MSC: Priary 11B68; secodary 33C65 Keywords: geeralized Apostol type polyoials; recurrece relatios; differetial equatios; coected probles; quasi-ooial 1 Itroductio, defiitios ad otivatio The classical Beroulli polyoials B x, the classical Euler polyoials E x adthe classical Geocchi polyoials G x, together with their failiar geeralizatios B x, E x adg x of real or coplex order α, are usually defied by eas of the followig geeratig fuctios see, for details, [1], pp ad [2], p.61 et seq.; see also [3] ad the refereces cited therei: z α e xz = e z 1 2 α e xz = e z +1 =0 =0 B xz! E xz! z <2π, 1.1 z < π 1.2 ad 2z α e xz = e z +1 =0 G xz! z < π. 1.3 So that, obviously, the classical Beroulli polyoials B x, the classical Euler polyoials E x ad the classical Geocchi polyoials G x are give, respectively, by B x:=b 1 x, E x:=e 1 x ad G x:=g 1 x N Lu ad Luo; licesee Spriger. This is a Ope Access article distributed uder the ters of the Creative Coos Attributio Licese which perits urestricted use, distributio, ad reproductio i ay ediu, provided the origial work is properly cited.

2 Lu ad Luo Boudary Value Probles 2013, 2013:64 Page 2 of 13 For the classical Beroulli ubers B, the classical Euler ubers E ad the classical Geocchi ubers G of order,wehave B := B 0 = B 1 0, E := E 0 = E 1 0 ad G := G 0 = G 1 0, 1.5 respectively. Soe iterestig aalogues of the classical Beroulli polyoials ad ubers were first ivestigated by Apostol see [4], p.165, Eq. 3.1 ad ore recetly by Srivastava see [5], pp We begi by recallig here Apostol s defiitios as follows. Defiitio 1.1 Apostol [4]; see also Srivastava [5] The Apostol-Beroulli polyoials B x; λλ C are defied by eas of the followig geeratig fuctio: ze xz λe z 1 = B x; λ z! =0 z <2π whe λ =1; z < log λ whe λ with, of course, B x=b x;1 ad B λ:=b 0; λ, 1.7 where B λ deotes the so-called Apostol-Beroulli ubers. Recetly,Luo ad Srivastava [6] further exteded the Apostol-Beroulli polyoials as the so-called Apostol-Beroulli polyoials of order α. Defiitio 1.2 Luo ad Srivastava [6] The Apostol-Beroulli polyoials B x; λλ Coforderα N 0 are defied by eas of the followig geeratig fuctio: z α e xz = B λe z x; λz 1! =0 z <2π whe λ =1; z < log λ whe λ with, of course, B x=b x;1 ad B λ:=b 0; λ, 1.9 where B λ deotes the so-called Apostol-Beroulli ubers of order α. O the other had, Luo [7], gave a aalogous extesio of the geeralized Euler polyoials as the so-called Apostol-Euler polyoials of order α. Defiitio 1.3 Luo [7] The Apostol-Euler polyoials E x; λλ Coforderα N 0 are defied by eas of the followig geeratig fuctio: 2 α e xz = λe z +1 E =0 x; λz! z < log λ 1.10

3 Lu ad Luo Boudary Value Probles 2013, 2013:64 Page 3 of 13 with, of course, E x=e x;1 ad E λ:=e 0; λ, 1.11 where E λ deotes the so-called Apostol-Euler ubers of order α. O the subject of the Geocchi polyoials G x ad their various extesios, a rearkably large uber of ivestigatios have appeared i the literature see, for exaple, [8 14]. Moreover, Luo see [12 14] itroduced ad ivestigated the Apostol-Geocchi polyoials of real or coplex order α, which are defied as follows: Defiitio 1.4 The Apostol-Geocchi polyoials G x; λλ Coforderα N 0 are defied by eas of the followig geeratig fuctio: 2z α e xz = λe z +1 with, of course, G =0 x; λz! z < log λ 1.12 G x=g x;1, G λ:=g 0; λ, G x; λ:=g 1 x; λ ad G λ:=g 1 λ, 1.13 where G λ, G λ adg x; λ deote the so-called Apostol-Geocchi ubers, the Apostol-Geocchi ubers of order α ad the Apostol-Geocchi polyoials, respectively. Recetly, Luo ad Srivastava [15] itroduced a uificatio ad geeralizatio of the above-etioed three failies of the geeralized Apostol type polyoials. Defiitio 1.5 Luo ad Srivastava [15] The geeralized Apostol type polyoials F x; λ; u, vα N 0, λ, u, v Coforderα are defied by eas of the followig geeratig fuctio: 2 u z v α e xz = λe z +1 F =0 x; λ; u, vz! z < log λ, 1.14 where F λ; u, v:=f 0; λ; u, v 1.15 deote the so-called Apostol type ubers of order α. So that, by coparig Defiitio 1.5 with Defiitios 1.2, 1.3 ad 1.4,we have B E G x; λ= 1α F x; λ; 0, 1, 1.16 x; λ=f x; λ; 1, 0, 1.17 x; λ=f x; λ; 1,

4 Lu ad Luo Boudary Value Probles 2013, 2013:64 Page 4 of 13 A polyoial p x N, x C is said to be a quasi-ooial [16], wheever two operators ˆM, ˆP, called ultiplicative ad derivative or lowerig operators respectively, ca be defied i such a way that ˆPp x=p 1 x, 1.19 ˆMp x=p +1 x, 1.20 which ca be cobied to get the idetity ˆM ˆPp x=p x The Appell polyoials [17] ca be defied by cosiderig the followig geeratig fuctio: where Ate xt = =0 R x t, 1.22! At= R k k! tk A is aalytic fuctio at t =0. Fro [18], we kow that the ultiplicative ad derivative operators of R xare 1 α k ˆM =x + α 0 + k! D k x, 1.24 ˆP = D x, 1.25 where A t At = t α! =0 By usig 1.21, we have the followig lea. Lea 1.6 [18] The Appell polyoials R x defied by 1.22 satisfy the differetial equatio: α 1 1! y + α 2 2! y α 1 1! y +x + α 0 y y = 0, 1.27 where the uerical coefficiets α k, k =1,2,..., 1are defied i 1.26, ad are liked to the values R k by the followig relatios: R k+1 = k h=0 k R h α k h. h

5 Lu ad Luo Boudary Value Probles 2013, 2013:64 Page 5 of 13 Let P be the vector space of polyoials with coefficiets i C. A polyoial sequece {P } 0 be a polyoial set. {P } 0 is called a σ -Appell polyoial set of trasfer power series A is geerated by Gx, t=atg 0 x, t= =0 where G 0 x, t is a solutio of the syste: σ G 0 x, t=tg 0 x, t, G 0 x,0=1. P x t, 1.28! I [19], the authors ivestigated the coectio coefficiets betwee two polyoials. Ad there is a result about coectio coefficiets betwee two σ -Appell polyoial sets. Lea 1.7 [19] Let σ 1. Let {P } 0 ad {Q } 0 be two σ -Appell polyoial sets of trasfer power series, respectively, A 1 ad A 2. The where Q x= =0 A 2 t A 1 t = α k t k.!! α P x, 1.29 I recet years, several authors obtaied ay iterestig results ivolvig the related Beroulli polyoials ad Euler polyoials [5, 20 40]. Ad i [29], the authors studied soe series idetities ivolvig the geeralized Apostol type ad related polyoials. I this paper, we study soe other properties of the geeralized Apostol type polyoials F x; λ; u, v, icludig the recurrece relatios, the differetial equatios ad soe coectio probles, which exted soe kow results. As special, we obtai soe properties of the geeralized Apostol-Euler polyoials, the geeralized Apostol-Beroulli polyoials ad Apostol-Geocchi polyoials of high order. 2 Recursio forulas ad differetial equatios Fro the geeratig fuctio 1.14, we have x F x; λ; u, v=f 1 x; λ; u, v. 2.1 A recurrece relatio for the geeralized Apostol type polyoials is give by the followig theore. Theore 2.1 For ay itegral 1, λ C ad α N, the followig recurrece relatio for the geeralized Apostol type polyoials F x; λ; u, v holds true: αv +1 1 F +1 x; λ; u, v=αλ 2! u + v! F +v α+1 x +1;λ; u, v xf x; λ; u, v. 2.2

6 Lu ad Luo Boudary Value Probles 2013, 2013:64 Page 6 of 13 Proof Differetiatig both sides of 1.14 withrespecttot, ad usig soe eleetary algebra ad the idetity priciple of power series, recursio 2.2easily follows. By settig λ := λ, u =0adv =1iTheore2.1, ad the ultiplyig 1 α o both sides of the result, we have: Corollary 2.2 For ay itegral 1, λ C ad α N, the followig recurrece relatio for the geeralized Apostol-Beroulli polyoials B x; λ holds true: [ ] α +1 B +1 x; λ=αλbα+1 +1 x +1;λ xb x; λ. 2.3 By settig u =1adv =0iTheore2.1, we have the followig corollary. Corollary 2.3 For ay itegral 1, λ C ad α N, the followig recurrece relatio for the geeralized Apostol-Euler polyoials E x; λ holds true: E +1 x; λ=xe x; λ αλ 2 E α+1 x +1;λ. 2.4 By settig u =1adv =1iTheore2.1, we have the followig corollary. Corollary 2.4 For ay itegral 1, λ C ad α N, the followig recurrece relatio for the geeralized Apostol-Geocchi polyoials G x; λ holds true: 2 [ α +1 ] G +1 x; λ=αλgα+1 +1 x +1;λ 2 +1xG x; λ. 2.5 Fro 1.14 ad1.22, we kow that the geeralized Appostol type polyoials F x; λ; u, v is Appell polyoials with 2 u t v α At=. 2.6 λe t +1 Fro the Eq. 23 of [15], we kow that G 0 1; λ=0.sofro2.6ad1.12, we ca obtai that if v =0,wehave A t At = λα 2 =0 G +1 1; λ +1 t!. 2.7 By usig 1.24ad1.26, we ca obtai the ultiplicative ad derivative operators of the geeralized Appostol type polyoials F x; λ; u, v ˆM = x + λα 2 G 11; λ + λα 2 1 G k+1 1; λ k +1! D k x, 2.8 ˆP = D x. 2.9 Fro 2.1, we ca obtai p x F p x; λ; u, v=! p! F p x; λ; u, v The by usig 1.20, 2.8ad2.10, we obtai the followig result.

7 Lu ad Luo Boudary Value Probles 2013, 2013:64 Page 7 of 13 Theore 2.5 For ay itegral 1, λ C ad α N, the followig recurrece relatio for the geeralized Apostol type polyoials F x; λ; u,0holds true: F +1 x x; λ; u,0= + λα 2 G 11; λ F x; λ; u,0 + λα 2 1 G k+1 1; λ k k +1 F k x; λ; u, By settig u =1iTheore2.5, we have the followig corollary. Corollary 2.6 For ay itegral 1, λ C ad α N, the followig recurrece relatio for the geeralized Apostol-Euler polyoials E x; λ holds true: E +1 x x; λ= + λα 2 G 11; λ E x; λ+λα 2 1 G k+1 1; λ k k +1 E k x; λ Furtherore, applyig Lea 1.7 to F x; λ; u, 0, we have the followig theore. Theore 2.7 The geeralized Apostol type polyoials F x; λ; u,0 satisfy the differetial equatio: λα 2 G 1; λ y + λα! 2 + λα 2 G 2 1; λ y + 2 G 1 1; λ y 1 + 1! x + λα 2 G 11; λ y y = Specially, by settig u =1 i Theore 2.7, thewe have thefollowigcorollary. Corollary 2.8 The geeralized Apostol-Euler polyoials E x; λ satisfy the differetial equatio: λα 2 G 1; λ y + λα! 2 + λα 2 G 2 1; λ y + 2 G 1 1; λ y 1 + 1! x + λα 2 G 11; λ y y = Coectio probles Fro 1.14 ad1.28, we kow that the geeralized Apostol type polyoials F x; λ; u, varead x -Appell polyoial set, where D x deotes the derivative operator. Fro Table 1 i [19], we kow that the derivative operators of ooials x ad the Gould-Hopper polyoials g x, h [30] arealld x. Ad their trasfer power series At are 1 ad e ht,respectively. Applyig Lea 1.7 to P x=x ad Q x=f x; λ; u, v, we have the followig theore.

8 Lu ad Luo Boudary Value Probles 2013, 2013:64 Page 8 of 13 Theore 3.1 F x; λ; u, v= F λ; u, vx, 3.1 =0 where F λ; u, v is the so-called Apostol type ubers of order α defied by By settig λ := λ, u =0adv =1iTheore3.1, ad the ultiplyig 1 α o both sides of the result, we have the followig corollary. Corollary 3.2 B x; λ= B λx, 3.2 =0 which is just Eq.3.1of [23]. By settig u =0adv =0iTheore3.1, we have the followig corollary. Corollary 3.3 E x; λ= E λx. 3.3 =0 By settig u =1adv =1iTheore3.1, we have the followig corollary. Corollary 3.4 G x; λ= G λx, 3.4 =0 which is just Eq.24of [15]. Applyig Lea 1.7 to P x=f x; λ; u, vadq x=f x; λ; u, v, we have the followig theore. Theore 3.5 F x; λ; u, v= F α 1 λ; u, vf x; λ; u, v, 3.5 =0 where F λ; u, v is the so-called Apostol type ubers of order α defied by By settig λ := λ, u =0adv =1iTheore3.5, ad the ultiplyig 1 α o both sides of the result, we have the followig corollary. Corollary 3.6 B x; λ= B α 1 λb x; λ, 3.6 =0 which is just Eq.3.2of [23].

9 Lu ad Luo Boudary Value Probles 2013, 2013:64 Page 9 of 13 By settig u =1adv =0iTheore3.5, we have the followig corollary. Corollary 3.7 E x; λ= E α 1 λe x; λ. 3.7 =0 By settig u =1adv =1iTheore3.5, we have the followig corollary. Corollary 3.8 G x; λ= G α 1 λg x; λ. 3.8 =0 Applyig Lea 1.7 to P x=g x, hadq x=f x; λ; u, v, we have the followig theore. Theore 3.9 F x; λ; u, v= r=0 [ [ r/]! r! 1 k h k k! r k! F r k λ; u, v ] g r x, h. 3.9 By settig λ := λ, u =0adv =1iTheore3.9, ad the ultiplyig 1 α o both sides of the result, we have the followig corollary. Corollary 3.10 B x; λ= r=0 [ [ r/]! r! 1 k h k k! r k! B r k λ ] g r x, h, 3.10 which is just Eq. 3.3 of [23]. By settig u =1adv =0iTheore3.9, we have the followig corollary. Corollary 3.11 E x; λ= r=0 [ [ r/]! r! 1 k h k k! r k! E r k λ ] g r x, h By settig u =1adv =1iTheore3.9, we have the followig corollary. Corollary 3.12 G x; λ= r=0 [ [ r/]! r! 1 k h k k! r k! G r k λ ] g r x, h Whe vα = 1, applyig Lea 1.7 to P x=e α 1 x; λ adq x=f x; λ; u, v, we have the followig theore.

10 Lu ad Luo Boudary Value Probles 2013, 2013:64 Page 10 of13 Theore 3.13 If vα =1,the we have F x; λ; u, v= 2 u 1α G λe α 1 x; λ =0 By settig λ := λ, u =0adv =1iTheore3.13, ad the ultiplyig 1 α o both sides of the result, we have the followig corollary. Corollary 3.14 B x; λ= 1 2 =0 G λx By settig u =1adv =1iTheore3.13, we have the followig corollary. Corollary 3.15 G x; λ= 1 2 =0 which is just the case of α =1i 3.4. G λx, 3.15 Whe v =1orα = 0, applyig Lea 1.7 to P x =G α 1 x; λ adq x =F x; λ; u, v,wecaobtaithefollowigtheore. Theore 3.16 If v =1or α =0,we have F x; λ; u, v= =0 2 u 1α G λg α 1 x; λ By settig λ := λ, u =0adv =1iTheore3.13, ad the ultiplyig 1 α o both sides of the result, we have the followig corollary. Corollary 3.17 B x; λ= 1 α G λg α 1 x; λ =0 Whe α =1i3.17, it is just By settig u =1adv =1iTheore3.16, we have the followig corollary. Corollary 3.18 G x; λ= =0 which is equal to 3.8. G λg α 1 x; λ, 3.18

11 Lu ad Luo Boudary Value Probles 2013, 2013:64 Page 11 of13 If α =0iTheore3.16,wehave: Corollary 3.19 x = =0 G λg 1 x; λ Herite-based geeralized Apostol type polyoials Fially, we give a geeratio of the geeralized Apostol type polyoials. The two-variable Herite-Kapé de Fériet polyoials 2VHKdFP H x, y aredefied by the series [31] [/2] x 2r y r H x, y=! r! 2r! r=0 4.1 with the followig geeratig fuctio: exp xt + yt 2 = =0 t! H x, y. 4.2 Ad the 2VHKdFP H x, y are also defied through the operatioal idetity exp y 2 {x } = H x 2 x, y. 4.3 Actig the operator exp y 2 x 2 o1.14, ad by the idetity [32] exp y 2 {exp ax 2 + bx } 1 = exp ax2 bx b 2 y, 4.4 x 2 1+4ay 1+4ay we defie the Herite-based geeralized Apostol type polyoials H F x, y; λ; u, v by the geeratig fuctio 2 u z v α e xt+yt2 = λe t +1 HF =0 x, y; λ; u, vt! t < log λ. 4.5 Clearly, we have ad HF x, y; λ; u, v= H F 1 x, y; λ; u, v. Fro the geeratig fuctio 4.5, we easily obtai x H F x, y; λ; u, v= HF 1 x, y; λ; u, v 4.6 y H F x, y; λ; u, v= 1 HF 2 x, y; λ; u, v, 4.7

12 Lu ad Luo Boudary Value Probles 2013, 2013:64 Page 12 of13 which ca be cobied to get the idetity 2 x 2 H F x, y; λ; u, v= y H F x, y; λ; u, v. 4.8 Actig with the operator exp y 2 x 2 o both sides of 3.1, 3.5, 3.13, 3.18, ad by usig 4.3, we obtai HF x; λ; u, v= =0 HF x; λ; u, v= =0 HF x; λ; u, v= =0 HG x; λ= =0 F λ; u, vh x, y, 4.9 F α 1 λ; u, v HF x; λ; u, v, u 1α G λ H E α 1 x; λ, where vα =1, 4.11 G λ H G α 1 x; λ, where v =1orα =0, 4.12 where H E x; λ ad H G x; λ are the Herite-based geeralized Apostol-Euler polyoials ad the Herite-based geeralized Apostol-Geocchi polyoials respectively, defied by the followig geeratig fuctios: 2 α e xt+yt2 = λe t +1 2t α e xt+yt2 = λe t +1 HE =0 =0 HG x; λt! x; λt! t < log λ, t < log λ. Copetig iterests The authors declare that they have o copetig iterests. Authors cotributios All authors cotributed equally i writig this paper, ad read ad approved the fial auscript. Author details 1 Departet of Matheatics, Yagzhou Uiversity, Yagzhou, Jiagsu , People s Republic of Chia. 2 Departet of Matheatics, Chogqig Noral Uiversity, Chogqig Higher Educatio Mega Ceter, Huxi Capus, Chogqig, , People s Republic of Chia. Ackowledgeets Dedicated to Professor Hari M Srivastava. The preset ivestigatio was supported, i part, by the Natioal Natural Sciece Foudatio of Chia uder Grat , Fud of Sciece ad Iovatio of Yagzhou Uiversity, Chia uder Grat 2012CXJ005, Research Project of Sciece ad Techology of Chogqig Educatio Coissio, Chia uder Grat KJ ad Fud of Chogqig Noral Uiversity, Chia uder Grat 10XLR017 ad 2011XLZ07. Received: 10 Deceber 2012 Accepted: 19 March 2013 Published: 28 March 2013 Refereces 1. Sádor, J, Crstici, B: Hadbook of Nuber Theory, vol. II. Kluwer Acadeic, Dordrecht Srivastava, HM, Choi, J: Series Associated with the Zeta ad Related Fuctios. Kluwer Acadeic, Dordrecht Srivastava, HM, Pitér, Á: Rearks o soe relatioships betwee the Beroulli ad Euler polyoials. Appl. Math. Lett. 17, Apostol, TM: O the Lerch zeta fuctio. Pac. J. Math. 1,

13 Lu ad Luo Boudary Value Probles 2013, 2013:64 Page 13 of13 5. Srivastava, HM: Soe forulas for the Beroulli ad Euler polyoials at ratioal arguets. Math. Proc. Cab. Philos. Soc. 129, Luo, Q-M, Srivastava, HM: Soe geeralizatios of the Apostol-Beroulli ad Apostol-Euler polyoials. J. Math. Aal. Appl. 308, Luo, Q-M: Apostol-Euler polyoials of higher order ad Gaussia hypergeoetric fuctios. Taiwa. J. Math. 10, Horada, AF: Geocchi polyoials. I: Proceedigs of the Fourth Iteratioal Coferece o Fiboacci Nubers ad Their Applicatios, pp Kluwer Acadeic, Dordrecht Horada, AF: Negative order Geocchi polyoials. Fiboacci Q. 30, Horada, AF: Geeratio of Geocchi polyoials of first order by recurrece relatios. Fiboacci Q. 30, Jag, L-C, Ki, T: O the distributio of the q-euler polyoials ad the q-geocchi polyoials of higher order. J. Iequal. Appl. 2008, Luo, Q-M: Fourier expasios ad itegral represetatios for the Geocchi polyoials. J. Iteger Seq. 12, Luo,Q-M: q-extesios for the Apostol-Geocchi polyoials. Ge. Math. 17, Luo, Q-M: Extesio for the Geocchi polyoials ad its Fourier expasios ad itegral represetatios. Osaka J. Math. 48, Luo, Q-M, Srivastava, HM: Soe geeralizatios of the Apostol-Geocchi polyoials ad the Stirlig ubers of the secod kid. Appl. Math. Coput. 217, Dattoli, G, Srivastava, HM, Zhukovsky, K: Orthogoality properties of the Herite ad related polyoials. J. Coput. Appl. Math. 182, Appell, P: Sur ue classe de polyoials. A. Sci. Éc. Nor. Super. 9, He, MX, Ricci, PE: Differetial equatio of Appell polyoials via the factorizatio ethod. J. Coput. Appl. Math. 139, Be Cheikh, Y, Chaggara, H: Coectio probles via lowerig operators. J. Coput. Appl. Math. 178, Garg, M, Jai, K, Srivastava, HM: Soe relatioships betwee the geeralized Apostol-Beroulli polyoials ad Hurwitz-Lerch zeta fuctios. Itegral Trasfors Spec. Fuct. 17, Li, S-D, Srivastava, HM, Wag, P-Y: Soe expasio forulas for a class of geeralized Hurwitz-Lerch zeta fuctios. Itegral Trasfors Spec. Fuct. 17, Liu, HM, Wag, WP: Soe idetities o the Beroulli, Euler ad Geocchi polyoials via power sus ad alterate power sus. Discrete Math. 309, Lu, D-Q: Soe properties of Beroulli polyoials ad their geeralizatios. Appl. Math. Lett. 24, Luo, Q-M: The ultiplicatio forulas for the Apostol-Beroulli ad Apostol-Euler polyoials of higher order. Itegral Trasfors Spec. Fuct. 20, Luo, Q-M, Srivastava, HM: Soe relatioships betwee the Apostol-Beroulli ad Apostol-Euler polyoials. Coput. Math. Appl. 51, Ozde, H, Sisek, Y, Srivastava, HM: A uified presetatio of the geeratig fuctios of the geeralized Beroulli, Euler ad Geocchi polyoials. Coput. Math. Appl. 60, Yag, S-L: A idetity of syetry for the Beroulli polyoials. Discrete Math. 308, Yag, SL, Qiao, ZK: Soe syetry idetities for the Euler polyoials. J. Math. Res. Expo. 303, Lu, D-Q, Srivastava, HM: Soe series idetities ivolvig the geeralized Apostol type ad related polyoials. Coput. Math. Appl. 62, Gould, HW, Hopper, AT: Operatioal forulas coected with two geeralizatios of Herite polyoials. Duke Math. J. 29, Appell, P, Kapé de Fériet, J: Fuctios hypergéoétriques et hypersphériques: Polyóes d Herite. Gauthier-Villars, Paris Dattoli, G, Kha, S, Ricci, PE: O Crofto-Glaisher type relatios ad derivatio of geeratig fuctios for Herite polyoials icludig the ulti-idex case. Itegral Trasfors Spec. Fuct. 19, Dere, R, Sisek, Y: Noralized polyoials ad their ultiplicatio forulas. Adv. Differ. Equ. 2013, doi: / Dere, R, Sisek, Y: Geocchi polyoials associated with the ubral algebra. Appl. Math. Coput. 218, Dere, R, Sisek, Y: Applicatios of ubral algebra to soe special polyoials. Adv. Stud. Cotep. Math. 22, Kurt, B, Sisek, Y: Frobeius-Euler type polyoials related to Herite-Beroulli polyoials, aalysis ad applied ath. AIP Cof. Proc. 1389, Srivastava, HM: Soe geeralizatios ad basic or q- extesios of the Beroulli, Euler ad Geocchi polyoials. Appl. Math. If. Sci. 5, Srivastava, HM, Kurt, B, Sisek, Y: Soe failies of Geocchi type polyoials ad their iterpolatio fuctios. Itegral Trasfors Spec. Fuct. 23, ; see also corrigedu, Itegral Trasfors Spec. Fuct. 23, Srivastava, HM, Ozarsla, MA, Kaauglu, C: Soe geeralized Lagrage-based Apostol-Beroulli, Apostol-Euler ad Apostol-Geocchi polyoials. Russ. J. Math. Phys. 20, Srivastava, HM, Choi, J: Zeta ad q-zeta Fuctios ad Associated Series ad Itegrals. Elsevier, Asterda 2012 doi: / Cite this article as: Lu ad Luo: Soe properties of the geeralized Apostol-type polyoials. Boudary Value Probles :64.