Abstract
In this paper, we study some properties of the generalized Apostoltype polynomials (see (Luo and Srivastava in Appl. Math. Comput. 217:57025728, 2011)), including the recurrence relations, the differential equations and some other connected problems, which extend some known results. We also deduce some properties of the generalized ApostolEuler polynomials, the generalized ApostolBernoulli polynomials, and ApostolGenocchi polynomials of high order.
MSC: 11B68, 33C65.
Keywords:
generalized Apostol type polynomials; recurrence relations; differential equations; connected problems; quasimonomial1 Introduction, definitions and motivation
The classical Bernoulli polynomials
and
So that, obviously, the classical Bernoulli polynomials
For the classical Bernoulli numbers
respectively.
Some interesting analogues of the classical Bernoulli polynomials and numbers were first investigated by Apostol (see [4], p.165, Eq. (3.1)) and (more recently) by Srivastava (see [5], pp.8384). We begin by recalling here Apostol’s definitions as follows.
Definition 1.1 (Apostol [4]; see also Srivastava [5])
The ApostolBernoulli polynomials
with, of course,
where
Recently, Luo and Srivastava [6] further extended the ApostolBernoulli polynomials as the socalled ApostolBernoulli polynomials of order α.
Definition 1.2 (Luo and Srivastava [6])
The ApostolBernoulli polynomials
with, of course,
where
On the other hand, Luo [7], gave an analogous extension of the generalized Euler polynomials as the socalled ApostolEuler polynomials of order α.
Definition 1.3 (Luo [7])
The ApostolEuler polynomials
with, of course,
where
On the subject of the Genocchi polynomials
Definition 1.4 The ApostolGenocchi polynomials
with, of course,
where
Recently, Luo and Srivastava [15] introduced a unification (and generalization) of the abovementioned three families of the generalized Apostol type polynomials.
Definition 1.5 (Luo and Srivastava [15])
The generalized Apostol type polynomials
where
denote the socalled Apostol type numbers of order α.
So that, by comparing Definition 1.5 with Definitions 1.2, 1.3 and 1.4, we have
A polynomial
which can be combined to get the identity
The Appell polynomials [17] can be defined by considering the following generating function:
where
is analytic function at
From [18], we know that the multiplicative and derivative operators of
where
By using (1.21), we have the following lemma.
Lemma 1.6 ([18])
The Appell polynomials
where the numerical coefficients
Let
where
In [19], the authors investigated the connection coefficients between two polynomials. And there is a result about connection coefficients between two σAppell polynomial sets.
Lemma 1.7 ([19])
Let
where
In recent years, several authors obtained many interesting results involving the related Bernoulli polynomials and Euler polynomials [5,2040]. And in [29], the authors studied some series identities involving the generalized Apostol type and related polynomials.
In this paper, we study some other properties of the generalized Apostol type polynomials
2 Recursion formulas and differential equations
From the generating function (1.14), we have
A recurrence relation for the generalized Apostol type polynomials is given by the following theorem.
Theorem 2.1For any integral
Proof Differentiating both sides of (1.14) with respect to t, and using some elementary algebra and the identity principle of power series, recursion (2.2) easily follows. □
By setting
Corollary 2.2For any integral
By setting
Corollary 2.3For any integral
By setting
Corollary 2.4For any integral
From (1.14) and (1.22), we know that the generalized Appostol type polynomials
From the Eq. (23) of [15], we know that
By using (1.24) and (1.26), we can obtain the multiplicative and derivative operators
of the generalized Appostol type polynomials
From (2.1), we can obtain
Then by using (1.20), (2.8) and (2.10), we obtain the following result.
Theorem 2.5For any integral
By setting
Corollary 2.6For any integral
Furthermore, applying Lemma 1.7 to
Theorem 2.7The generalized Apostol type polynomials
Specially, by setting
Corollary 2.8The generalized ApostolEuler polynomials
3 Connection problems
From (1.14) and (1.28), we know that the generalized Apostol type polynomials
From Table 1 in [19], we know that the derivative operators of monomials
Applying Lemma 1.7 to
Theorem 3.1
where
By setting
Corollary 3.2
which is just Eq. (3.1) of[23].
By setting
Corollary 3.3
By setting
Corollary 3.4
which is just Eq. (24) of[15].
Applying Lemma 1.7 to
Theorem 3.5
where
By setting
Corollary 3.6
which is just Eq. (3.2) of[23].
By setting
Corollary 3.7
By setting
Corollary 3.8
Applying Lemma 1.7 to
Theorem 3.9
By setting
Corollary 3.10
which is just Eq. (3.3) of[23].
By setting
Corollary 3.11
By setting
Corollary 3.12
When
Theorem 3.13If
By setting
Corollary 3.14
By setting
Corollary 3.15
which is just the case of
When
Theorem 3.16If
By setting
Corollary 3.17
When
By setting
Corollary 3.18
which is equal to (3.8).
If
Corollary 3.19
4 Hermitebased generalized Apostol type polynomials
Finally, we give a generation of the generalized Apostol type polynomials.
The twovariable HermiteKampé de Fériet polynomials (2VHKdFP)
with the following generating function:
And the 2VHKdFP
Acting the operator
we define the Hermitebased generalized Apostol type polynomials
Clearly, we have
From the generating function (4.5), we easily obtain
and
which can be combined to get the identity
Acting with the operator
where
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally in writing this paper, and read and approved the final manuscript.
Acknowledgements
Dedicated to Professor Hari M Srivastava.
The present investigation was supported, in part, by the National Natural Science Foundation of China under Grant 11226281, Fund of Science and Innovation of Yangzhou University, China under Grant 2012CXJ005, Research Project of Science and Technology of Chongqing Education Commission, China under Grant KJ120625 and Fund of Chongqing Normal University, China under Grant 10XLR017 and 2011XLZ07.
References

Sándor, J, Crstici, B: Handbook of Number Theory, Kluwer Academic, Dordrecht (2004)

Srivastava, HM, Choi, J: Series Associated with the Zeta and Related Functions, Kluwer Academic, Dordrecht (2001)

Srivastava, HM, Pintér, Á: Remarks on some relationships between the Bernoulli and Euler polynomials. Appl. Math. Lett.. 17, 375–380 (2004). Publisher Full Text

Apostol, TM: On the Lerch zeta function. Pac. J. Math.. 1, 161–167 (1951). Publisher Full Text

Srivastava, HM: Some formulas for the Bernoulli and Euler polynomials at rational arguments. Math. Proc. Camb. Philos. Soc.. 129, 77–84 (2000). Publisher Full Text

Luo, QM, Srivastava, HM: Some generalizations of the ApostolBernoulli and ApostolEuler polynomials. J. Math. Anal. Appl.. 308, 290–320 (2005). Publisher Full Text

Luo, QM: ApostolEuler polynomials of higher order and Gaussian hypergeometric functions. Taiwan. J. Math.. 10, 917–925 (2006)

Horadam, AF: Genocchi polynomials. Proceedings of the Fourth International Conference on Fibonacci Numbers and Their Applications, pp. 145–166. Kluwer Academic, Dordrecht (1991)

Horadam, AF: Negative order Genocchi polynomials. Fibonacci Q.. 30, 21–34 (1992)

Horadam, AF: Generation of Genocchi polynomials of first order by recurrence relations. Fibonacci Q.. 30, 239–243 (1992)

Jang, LC, Kim, T: On the distribution of the qEuler polynomials and the qGenocchi polynomials of higher order. J. Inequal. Appl.. 2008, 1–9 (2008). PubMed Abstract

Luo, QM: Fourier expansions and integral representations for the Genocchi polynomials. J. Integer Seq.. 12, 1–9 (2009)

Luo, QM: qextensions for the ApostolGenocchi polynomials. Gen. Math.. 17, 113–125 (2009)

Luo, QM: Extension for the Genocchi polynomials and its Fourier expansions and integral representations. Osaka J. Math.. 48, 291–309 (2011)

Luo, QM, Srivastava, HM: Some generalizations of the ApostolGenocchi polynomials and the Stirling numbers of the second kind. Appl. Math. Comput.. 217, 5702–5728 (2011). Publisher Full Text

Dattoli, G, Srivastava, HM, Zhukovsky, K: Orthogonality properties of the Hermite and related polynomials. J. Comput. Appl. Math.. 182, 165–172 (2005). Publisher Full Text

Appell, P: Sur une classe de polynomials. Ann. Sci. Éc. Norm. Super.. 9, 119–144 (1880). PubMed Abstract  Publisher Full Text

He, MX, Ricci, PE: Differential equation of Appell polynomials via the factorization method. J. Comput. Appl. Math.. 139, 231–237 (2002). Publisher Full Text

Ben Cheikh, Y, Chaggara, H: Connection problems via lowering operators. J. Comput. Appl. Math.. 178, 45–61 (2005). Publisher Full Text

Garg, M, Jain, K, Srivastava, HM: Some relationships between the generalized ApostolBernoulli polynomials and HurwitzLerch zeta functions. Integral Transforms Spec. Funct.. 17, 803–815 (2006). Publisher Full Text

Lin, SD, Srivastava, HM, Wang, PY: Some expansion formulas for a class of generalized HurwitzLerch zeta functions. Integral Transforms Spec. Funct.. 17, 817–827 (2006). Publisher Full Text

Liu, HM, Wang, WP: Some identities on the Bernoulli, Euler and Genocchi polynomials via power sums and alternate power sums. Discrete Math.. 309, 3346–3363 (2009). Publisher Full Text

Lu, DQ: Some properties of Bernoulli polynomials and their generalizations. Appl. Math. Lett.. 24, 746–751 (2011). Publisher Full Text

Luo, QM: The multiplication formulas for the ApostolBernoulli and ApostolEuler polynomials of higher order. Integral Transforms Spec. Funct.. 20, 377–391 (2009). Publisher Full Text

Luo, QM, Srivastava, HM: Some relationships between the ApostolBernoulli and ApostolEuler polynomials. Comput. Math. Appl.. 51, 631–642 (2006). Publisher Full Text

Ozden, H, Simsek, Y, Srivastava, HM: A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials. Comput. Math. Appl.. 60, 2779–2787 (2010). Publisher Full Text

Yang, SL: An identity of symmetry for the Bernoulli polynomials. Discrete Math.. 308, 550–554 (2008). Publisher Full Text

Yang, SL, Qiao, ZK: Some symmetry identities for the Euler polynomials. J. Math. Res. Expo.. 30(3), 457–464 (2010)

Lu, DQ, Srivastava, HM: Some series identities involving the generalized Apostol type and related polynomials. Comput. Math. Appl.. 62, 3591–3602 (2011). Publisher Full Text

Gould, HW, Hopper, AT: Operational formulas connected with two generalizations of Hermite polynomials. Duke Math. J.. 29, 51–63 (1962). Publisher Full Text

Appell, P, Kampé de Fériet, J: Functions hypergéométriques et hypersphériques: Polynómes d’Hermite, GauthierVillars, Paris (1926)

Dattoli, G, Khan, S, Ricci, PE: On CroftonGlaisher type relations and derivation of generating functions for Hermite polynomials including the multiindex case. Integral Transforms Spec. Funct.. 19, 1–9 (2008). Publisher Full Text

Dere, R, Simsek, Y: Normalized polynomials and their multiplication formulas. Adv. Differ. Equ.. 2013, (2013) Article ID 31. doi:10.1186/16871847201331

Dere, R, Simsek, Y: Genocchi polynomials associated with the umbral algebra. Appl. Math. Comput.. 218, 756–761 (2011). Publisher Full Text

Dere, R, Simsek, Y: Applications of umbral algebra to some special polynomials. Adv. Stud. Contemp. Math.. 22, 433–438 (2012)

Kurt, B, Simsek, Y: FrobeniusEuler type polynomials related to HermiteBernoulli polynomials, analysis and applied math. AIP Conf. Proc.. 1389, 385–388 (2011)

Srivastava, HM: Some generalizations and basic (or q) extensions of the Bernoulli, Euler and Genocchi polynomials. Appl. Math. Inf. Sci.. 5, 390–444 (2011)

Srivastava, HM, Kurt, B, Simsek, Y: Some families of Genocchi type polynomials and their interpolation functions. Integral Transforms Spec. Funct.. 23, 919–938 (2012) see also corrigendum, Integral Transforms Spec. Funct. 23, 939940 (2012)
see also corrigendum, Integral Transforms Spec. Funct. 23, 939940 (2012)
Publisher Full Text 
Srivastava, HM, Ozarslan, MA, Kaanuglu, C: Some generalized Lagrangebased ApostolBernoulli, ApostolEuler and ApostolGenocchi polynomials. Russ. J. Math. Phys.. 20, 110–120 (2013)

Srivastava, HM, Choi, J: Zeta and qZeta Functions and Associated Series and Integrals, Elsevier, Amsterdam (2012)