/ / Article
Special Issue

## Completely Monotonic and Related Functions: Their Applications

View this Special Issue

Research Article | Open Access

Volume 2014 |Article ID 868167 | 8 pages | https://doi.org/10.1155/2014/868167

# -Extensions for the Apostol Type Polynomials

Accepted20 May 2014
Published02 Jun 2014

#### Abstract

The aim of this work is to introduce an extension for -standard notations. The -Apostol type polynomials and study some of their properties. Besides, some relations between the mentioned polynomials and some other known polynomials are obtained.

#### 1. Introduction, Preliminaries, and Definitions

Throughout this research we always apply the following notations. indicates the set of natural numbers, indicates the set of nonnegative integers, indicates the set of all real numbers, and denotes the set of complex numbers. We refer the readers to  for all the following -standard notations. The -shifted factorial is defined as The -numbers and -factorials are defined by respectively. The -polynomial coefficient is defined by The -analogue of the function is defined by The -binomial formula is known as In the standard approach to the -calculus, two exponential functions are used: As an immediate result of these two definitions, we have .

Recently, Luo and Srivastava  introduced and studied the generalized Apostol-Bernoulli polynomials and the generalized Apostol-Euler polynomials . Kurt  gave the generalization of the Bernoulli polynomials of order and studied their properties. They also studied these polynomials systematically; see [2, 49]. There are numerous recent investigations on this subject by many other authors; see [3, 1020]. More recently, Tremblay et al.  further gave the definition of and studied their properties. On the other hand, Mahmudov and Keleshteri [21, 22] studied various two dimensional -polynomials. Motivated by these papers, we define generalized Apostol type -polynomials as follows.

Definition 1. Let , , and. The generalized -Apostol-Bernoulli numbers and polynomials in of order are defined, in a suitable neighborhood of , by means of the generating functions: where .

Definition 2. Let , , and . The generalized -Apostol-Euler numbers and polynomials in of order are defined, in a suitable neighborhood of , by means of the generating functions:

Definition 3. Let , and . The generalized -Apostol-Genocchi numbers and polynomials in of order are defined, in a suitable neighborhood of , by means of the generating functions:

Clearly, for , one has For and , one has For , one has

#### 2. Properties of the Apostol Type -Polynomials

In this section, we show some basic properties of the generalized -polynomials. We only prove the facts for one of them. Obviously, by applying the similar technique, other ones can be proved.

Proposition 4. The generalized -polynomials , , and satisfy the following relations:

Proof. We only prove the second identity. By using Definition 2, we have Comparing the coefficients of the term in both sides gives the result.

Corollary 5. The generalized -polynomials , , and satisfy the following relations:

Proposition 6. The generalized -polynomials , , and satisfy the following relations:

Proof. We only prove (18). By using Definition 2 and starting from the left hand side of the relation (18), we have Comparing the coefficients of the term in both sides gives the result.

#### 3. -Analogue of the Luo-Srivastava Addition Theorem

In this section, we state and prove a -generalization of the Luo-Srivastava addition theorem.

Theorem 7. The following relation holds between generalized -Apostol-Euler and -Apostol-Bernoulli polynomials:

Proof. We take aid of the following identity to prove (21): Therefore, we can write From that we can conclude the following: That is, Substituting (25) into the right hand side of (16), we obtain Thus, from one hand, we can write As we know that we can continue as On the other hand, for , we can write and, as , we have Adding to we obtain Consequently,

Taking in Theorem 7, we get a -generalization of the Luo-Srivastava addition theorem .

Corollary 8. The following relation holds between generalized -Apostol-Euler and -Apostol-Bernoulli polynomials:

Letting , we get the Luo-Srivastava addition theorem (see ):

Next theorem gives relationship between and .

Theorem 9. The following relation holds between generalized -Apostol-Euler and -Apostol-Genocchi polynomials:

Proof. The proof follows from the following identity:

Theorem 10. The following relation holds between generalized -Apostol-Euler and -Stirling polynomials of the second kind:

Proof. The -Stirling polynomials of the second kind are defined by means of the following generating function: where ; see . Replacing identity (39) in the right hand side of (16), we have

Theorem 11. The relationship holds between the polynomials and the -Hermite polynomials defined by (see )

Proof. Indeed,

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

1. G. E. Andrews, R. Askey, and R. Roy, Special Functions, vol. 71 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, UK, 1999. View at: Publisher Site | MathSciNet
2. Q.-M. Luo and H. M. Srivastava, “Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials,” Journal of Mathematical Analysis and Applications, vol. 308, no. 1, pp. 290–302, 2005.
3. B. Kurt, “A further generalization of the Bernoulli polynomials and on the $2D$-Bernoulli polynomials ${B}_{n}^{2}\left(x,y\right)$,” Applied Mathematics, vol. 233, pp. 3005–3017, 2010. View at: Google Scholar
4. Q.-M. Luo, “$q$-extensions for the Apostol-Genocchi polynomials,” General Mathematics, vol. 17, no. 2, pp. 113–125, 2009.
5. Q.-M. Luo, “Some results for the $q$-Bernoulli and $q$-Euler polynomials,” Journal of Mathematical Analysis and Applications, vol. 363, no. 1, pp. 7–18, 2010. View at: Publisher Site | Google Scholar | MathSciNet
6. Q.-M. Luo, “$q$-analogues of some results for the Apostol-Euler polynomials,” Advanced Studies in Contemporary Mathematics, vol. 20, no. 1, pp. 103–113, 2010.
7. Q.-M. Luo and H. M. Srivastava, “Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials,” Computers & Mathematics with Applications, vol. 51, no. 3-4, pp. 631–642, 2006.
8. Q.-M. Luo and H. M. Srivastava, “Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5702–5728, 2011.
9. Q.-M. Luo, “An explicit relationship between the generalized Apostol-Bernoulli and Apostol-Euler polynomials associated with $\lambda$-Stirling numbers of the second kind,” Houston Journal of Mathematics, vol. 36, no. 4, pp. 1159–1171, 2010. View at: Google Scholar | MathSciNet
10. R. Tremblay, S. Gaboury, and B.-J. Fugère, “A new class of generalized Apostol-Bernoulli polynomials and some analogues of the Srivastava-Pintér addition theorem,” Applied Mathematics Letters of Rapid Publication, vol. 24, no. 11, pp. 1888–1893, 2011.
11. H. Ozden, Y. Simsek, and H. M. Srivastava, “A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials,” Computers & Mathematics with Applications, vol. 60, no. 10, pp. 2779–2787, 2010.
12. D.-Q. Lu and H. M. Srivastava, “Some series identities involving the generalized Apostol type and related polynomials,” Computers & Mathematics with Applications, vol. 62, no. 9, pp. 3591–3602, 2011.
13. H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic, Dodrecht, The Netherlands, 2001. View at: Publisher Site | MathSciNet
14. P. Natalini and A. Bernardini, “A generalization of the Bernoulli polynomials,” Journal of Applied Mathematics, vol. 2003, no. 3, pp. 155–163, 2003.
15. W. Wang, C. Jia, and T. Wang, “Some results on the Apostol-Bernoulli and Apostol-Euler polynomials,” Computers & Mathematics with Applications, vol. 55, no. 6, pp. 1322–1332, 2008.
16. J. Choi, D. S. Jang, and H. M. Srivastava, “A generalization of the Hurwitz-Lerch Zeta function,” Integral Transforms and Special Functions, vol. 19, no. 1-2, pp. 65–79, 2008.
17. M. Garg, K. Jain, and H. M. Srivastava, “Some relationships between the generalized Apostol-Bernoulli polynomials and Hurwitz-Lerch Zeta functions,” Integral Transforms and Special Functions, vol. 17, no. 11, pp. 803–815, 2006.
18. M. Ali Özarslan, “Unified Apostol-Bernoulli, Euler and Genocchi polynomials,” Computers & Mathematics with Applications, vol. 62, no. 6, pp. 2452–2462, 2011.
19. R. Dere and Y. Simsek, “Applications of umbral algebra to some special polynomials,” Advanced Studies in Contemporary Mathematics, vol. 22, no. 3, pp. 433–438, 2012.
20. B. Kurt and Y. Simsek, “Frobenious-Euler type polynomials related to Hermite-Bernoulli polynomials, analysis and applied math,” AIP Conference Proceedings, vol. 1389, pp. 385–388, 2011. View at: Google Scholar
21. N. I. Mahmudov, “On a class of $q$-Bernoulli and $q$-Euler polynomials,” Advances in Difference Equations, vol. 2013, article 108, p. 11, 2013. View at: Publisher Site | Google Scholar | MathSciNet
22. N. I. Mahmudov and M. E. Keleshteri, “On a class of generalized $q$-Bernoulli and $q$-Euler polynomials,” Advances in Difference Equations, vol. 2013, article 115, p. 10, 2013. View at: Publisher Site | Google Scholar | MathSciNet
23. T. N. T. Goodman, H. Oruç, and G. M. Phillips, “Convexity and generalized Bernstein polynomials,” Proceedings of the Edinburgh Mathematical Society II, vol. 42, no. 1, pp. 179–190, 1999.
24. N. I. Mahmudov, “Difference equations of q-Appell polynomials,” http://arxiv.org/abs/1403.0189. View at: Google Scholar

We are committed to sharing findings related to COVID-19 as quickly and safely as possible. Any author submitting a COVID-19 paper should notify us at help@hindawi.com to ensure their research is fast-tracked and made available on a preprint server as soon as possible. We will be providing unlimited waivers of publication charges for accepted articles related to COVID-19. 