AMApplied Mathematics2152-7385Scientific Research Publishing10.4236/am.2015.69134AM-58602ArticlesPhysics&Mathematics New Extension of Unified Family Apostol-Type of Polynomials and Numbers eihEl-Sayed El-Desouky1*RababSabry Gomaa1*Department of Mathematics, Mansoura University, Mansoura, Egypt* E-mail:b_desouky@yahoo.com(EEE);dr.rsg12@yahoo.com(RSG);0508201506091495150522 May 2015accepted 1 August 5 August 2015© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

The purpose of this paper is to introduce and investigate new unification of unified family of Apostol-type polynomials and numbers based on results given in  . Also, we derive some properties for these polynomials and obtain some relationships between the Jacobi polynomials, Laguerre polynomials, Hermite polynomials, Stirling numbers and some other types of generalized polynomials.

Euler Bernoulli and Genocchi Polynomials Stirling Numbers Laguerre Polynomials Hermite Polynomials
1. Introduction

The generalized Bernoulli polynomials of order and the generalized Euler polynomials are defined by (see  ):

and

where denotes the set of complex numbers.

Recently, Luo and Srivastava  introduced the generalized Apostol-Bernoulli polynomials and the generalized Apostol-Euler polynomials as follows.

Definition 1.1. (Luo and Srivastava  ) The generalized Apostol-Bernoulli polynomials of order are defined by the generating function

Definition 1.2. (Luo  ) The generalized Apostol-Euler polynomials of order are defined by the generating function

Natalini and Bernardini  defined the new generalization of Bernoulli polynomials in the following definition.

Definition 1.3. The generalized Bernoulli polynomials, , are defined, in a suitable neighbourhood of by means of generating function

Recently, Tremblay et al.  investigated a new class of generalized Apostol-Bernoulli polynomial as follows.

Definition 1.4. The generalized Apostol-Bernoulli polynomials of order, , are defined, in a suitable neighbourhood of by means of generating function

Also, Sirvastava et al.  introduced a new interesting class of Apostol-Bernoulli polynomials that are closely related to the new class that we present in this paper. They investigated the following form.

Definition 1.5. Let and. Then the generalized Bernoulli polynomials of order are defined by the following generating function:

This sequel to the work by Sirvastava et al.  introduced and investigated a similar generalization of the family of Euler polynomials defined as follows.

Definition 1.6. Let and. Then the generalized Euler polynomials of order are defined by the following generating function

It is easy to see that setting and in (1.8) would lead to Apostol-Euler polynomials defined by (1.4). The case where has been studied by Luo et al.  .

In Section 2, we introduce the new extension of unified family of Apostol-type polynomials and numbers that are defined in  . Also, we determine relations between some results given in      and our results. Moreover, we introduce some new identities for polynomials defined in  . In Section 3, we give some basic properties of the new unification of Apostol-type polynomials and numbers. Finally in Section 4, we introduce some relationships between the new unification of Apostol-type polynomials and other known polynomials.

2. Unification of Multiparameter Apostol-Type Polynomials and Numbers

Definition 2.1. Let, and. Then the new unification of Apostol-type polynomials are defined, in a suitable neighbourhood of by means of generating function

where is a sequence of complex numbers.

Remark 2.1. If we set in (2.1), then we obtain the new unification of multiparameter Apostol-type numbers, as

The generating function in (2.1) gives many types of polynomials as special cases, for example, see Table 1.

Remark 2.2. From NO. 13 in Table 1 and ( , Table 1), we can obtain the polynomials and the numbers given in  - .

Theorem 3.1. Let and. Then

Proof. For the first equation, from (2.1)

using Cauchy product rule, we can easily obtain (3.1).

For the second Equation (3.2), from (2.1)

Special cases
1setting , hence if in (2.1) (generalized Bernoulli polynomials of order r, see  )
2setting , hence if in (2.1) (generalized Euler polynomials of order r, see  )
3setting , hence if in (2.1) (unification of Apostol-type polynomials of order r, see  )
4setting , hence if in (2.1) (generalized Bernoulli polynomials of order r, see  )
5setting , hence if in (2.1) (generalized Euler polynomials of order r, see  )
6setting , hence if in (2.1) (generalized Bernoulli polynomials, see  )
7setting , hence if in (2.1) (generalized Euler polynomials, see  )
8setting in (2.1) (generalized Bernoulli polynomials of order r, see  )
9setting in (2.1) (generalized Euler polynomials of order r, see  )
10setting in (2.1) (generalized Genocchi polynomials of order r, see  )
11setting in (2.1) (generalized Apostol-Bernoulli polynomials of order r, see  )
12setting in (2.1) (generalized Apostol-Euler polynomials of order r, see  )
13setting in (2.1) (a new unified family of generalized Apostol-Euler, Bernoulli and Genocchi polynomials, see  )

Equating the coefficient of on both sides, yields (3.2).

Corollary 3.1. If in (3.1), we have

Theorem 3.2. The following identity holds true, when and in (2.1)

Proof. From (2.1)

Hence, we can easily obtain (3.5).

Remark 3.1. If we put, and in (3.5), then it gives [ , Equation (34)],

where is the unification of the Apostol-type polynomials.

Theorem 3.3. The unification of Apostol-type numbers satisfy

Proof. When in (2.1), we have

Using Cauchy product rule, we obtain (3.6).

Theorem 3.4. The following relationship holds true

where and and,.

Proof. Starting with (2.1), we get

Using Cauchy product rule on the right hand side of the last equation and equating the coefficients of on both sides, yields (3.7).

Using No. 13 in Table 1, we obtain Nörlund’s results, see  and Carlitz’s generalizations, see  by our approach in Theorem 3.5 and Theorem 3.6 as follows

Theorem 3.5. For, we have

Proof. For the first equation and starting with (2.1), we get

Equating the coefficients of on both sides, yields (3.8).

For the second equation and starting with (2.1), we get

then, we have

Equating coefficients of on both sides, yields (3.9).

Theorem 3.6. For and we have

Proof. For the first equation and starting with (2.1), we get

Equating the coefficients of on both sides, yields (3.10).

Also, It is not difficult to prove (3.11).

4. Some Relations between <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/1-7402763x150.png" xlink:type="simple"/></inline-formula> and Other Polynomials and Numbers

In this section, we give some relationships between the polynomials and Laguerre polynomials, Jacobi polynomials, Hermite polynomials, generalized Stirling numbers of second kind, Stirling numbers and Bleimann-Butzer-hahn basic.

Theorem 4.1. For, and, we have relationship

between the new unification of Apostol-type polynomials and generalized Stirling numbers of second kind, see  .

Proof. Using (3.4) and from definition of generalized Stirling numbers of second kind, we easily obtain (4.1).

Theorem 4.2. For, and, we have the relationship

between the new unification of Apostol-type polynomials and Stirling numbers of second kind.

Proof. Using (3.4) and from definition of Stirling numbers of second kind (see  ), we easily obtain (4.2).

Theorem 4.3. The relationship

holds between the new unification of multiparameter Apostol-type polynomials and generalized Laguerre polynomials (see  , No. (3), Table 1).

Proof. From (3.4) and substitute

then we get (4.3).

Theorem 4.4. For. The relationship

holds between the new unification of Apostol-type polynomials and Jacobi polynomials (see  , p. 49, Equation (35)).

Proof. From (3.4) and substitute

then we get (4.4).

Theorem 4.5. The relationship

holds between the new unification of Apostol-type polynomials and Hermite polynomials (see  , No. (1) Table 1).

Proof. From (3.4) and substitute

then we get (4.5).

Theorem 4.6. When, , and in (9) and for,

, , and, ,

, , we have the following relationship

between the new unified family of generalized Apostol-Euler, Bernoulli and Genocchi polynomials, and

(the generalized Lah numbers) (see  ).

Proof. From  , Equation (2.1),

Equating the coefficients of on both sides, yields (4.6).

Using No. 13 in Table 1 (see  ) and the definition of the unified Bernstein and Bleimann-Butzer-Hahn basis (see  ),

where, , , we obtain the following theorem.

Theorem 4.7. For we have relationship

between the unified Bernstein and Bleimann-Butzer-Hahn basis, the new unified family of generalized Apostol-Bernoulli, Euler and Genocchi polynomials (see  ) and generalized Stirling numbers of first kind (see  ).

Proof. From (2.1) and (4.7) and with some elementary calculation, we easily obtain (4.8).

Cite this paper

Beih El-SayedEl-Desouky,Rabab SabryGomaa, (2015) New Extension of Unified Family Apostol-Type of Polynomials and Numbers. Applied Mathematics,06,1495-1505. doi: 10.4236/am.2015.69134

ReferencesSrivastava, H.M., Garg, M. and Choudhary, S. (2010) A New Generalization of the Bernoulli and Related Polynomials, Russian. Journal of Mathematical Physics, 17, 251-261. http://dx.doi.org/10.1134/S1061920810020093Srivastava, H.M., Garg, M. and Choudhary, S. (2011) Some New Families of Generalized Euler and Genocchi Polynomials. Taiwanese Journal of Mathematics, 15, 283-305.Srivastava, H.M. and Pintér, á. (2004) Remarks on Some Relationships between the Bernoulli and Euler Polynomials. Applied Mathematics Letters, 17, 375-380. http://dx.doi.org/10.1016/S0893-9659(04)90077-8Luo, Q.-M. and Srivastava, H.M. (2005) Some Generalizations of the Apostol Bernoulli and Apostol Euler Polynomials. Journal of Mathematical Analysis and Applications, 308, 290-302. http://dx.doi.org/10.1016/j.jmaa.2005.01.020Luo Q.-M. ,et al. (2006)Apostol-Euler Polynomials of Higher Order and Gaussian Hypergeometric Functions Taiwanese Journal of Mathematics 10, 917-925.Natalini, P. and Bernardini, A. (2003) A Generalization of the Bernoulli Polynomials. Journal of Applied Mathematics, 153-163. http://dx.doi.org/10.1155/s1110757x03204101Tremblay, R., Gaboury, S. and Fugère, B.-J. (2011) A New Class of Generalized Apostol-Bernoulli Polynomials and Some Analogues of the Srivastava-Pintér Addition Theorem. Applied Mathematics Letters, 24, 1888-1893. http://dx.doi.org/10.1016/j.aml.2011.05.012&ouml;zarslan, M.A. and Bozer, M. (2013) Unified Bernstein and Bleimann-Butzer-Hahn Basis and Its Properties. Advances in Difference Equations, 2013, 55. http://dx.doi.org/10.1186/1687-1847-2013-55Charalambides, C.A. (2005) Generalized Stirling and Lah Numbers. In: Charalambides, C.A., Ed., Combinatorial Methods in Discrete Distributions, John Wiley & Sons, Inc., Hoboken, 121-158.Srivastava, H.M. and Choi, J. (2001) Series Associated with the Zeta and Related Functions. Kluwer Academic, Dordrecht. http://dx.doi.org/10.1007/978-94-015-9672-5Gould, H.W. (1960) Stirling Number Representation Problems. Proceedings of the American Mathematical Society, 11, 447-451. http://dx.doi.org/10.1090/S0002-9939-1960-0114767-8Comtet L. ,et al. (1972)Nombers de Stirling generaux et fonctions symetriques Comptes Rendus de l’Académie des Sciences (Series A) 275, 747-750.Carlitz L. ,et al. (1962)Some Generalized Multiplication Formulae for the Bernoulli Polynomials and Related Functions Monatshefte für Mathematik 66, 1-8.N&ouml;rlund, N.E. (1924) V&ouml;rlesunge über differezerechnung. Springer-Verlag, Berlin. http://dx.doi.org/10.1007/978-3-642-50824-0Luo, Q.-M. (2004) On the Apostol Bernoulli Polynomials. Central European Journal of Mathematics, 2, 509-515. http://dx.doi.org/10.2478/BF02475959Karande, B.K. and Thakare, N.K. (1975) On the Unification of Bernoulli and Euler Polynomials. Indian Journal of Pure and Applied Mathematics, 6, 98-107.Dere, R., Simsek, Y. and Srivastava, H.M. (2013) A Unified Presentation of Three Families of Generalized Apostol Type Polynomials Based upon the Theory of the Umbral Calculus and the Umbral Algebra. Journal of Number Theory, 133, 3245-3263. http://dx.doi.org/10.1016/j.jnt.2013.03.004Apostol, T.M. (1951) On the Lerch Zeta Function. Pacific Journal of Mathematics, 1, 161-167. http://dx.doi.org/10.2140/pjm.1951.1.161Ozden, H. and Simsek, Y. (2014) Modification and Unification of the Apostol-Type Numbers and Polynomials and Their Applications. Applied Mathematics and Computation, 235, 338-351. http://dx.doi.org/10.1016/j.amc.2014.03.004Kurt B. ,et al. (2013)Some Relationships between the Generalized Apostol-Bernoulli and Apostol-Euler Polynomials Turkish Journal of Analysis and Number Theory 1, 54-58.Kurt B. ,et al. (2010)A Further Generalization of Bernoulli Polynomials and on 2D-Bernoulli Polynomials Applied Mathematical Sciences 47, 2315-2322.El-Desouky, B.S. and Gomaa, R.S. (2014) A New Unified Family of Generalized Apostol-Euler, Bernoulli and Genocchi Polynomials. Applied Mathematics and Computation, 247, 695-702. http://dx.doi.org/10.1016/j.amc.2014.09.002Luo, Q.-M., Guo, B.-N., Qui, F. and Debnath, L. (2003) Generalizations of Bernoulli Numbers and Polynomials. International Journal of Mathematics and Mathematical Sciences, 59, 3769-3776. http://dx.doi.org/10.1155/S0161171203112070