On the generalized Apostol-type Frobenius-Euler polynomials

Burak Kurt; Yilmaz Simsek

doi:10.1186/1687-1847-2013-1

Research
Open Access
Published: 04 January 2013

On the generalized Apostol-type Frobenius-Euler polynomials

Burak Kurt¹ &
Yilmaz Simsek²

Advances in Difference Equations volume 2013, Article number: 1 (2013) Cite this article

2628 Accesses
152 Citations
Metrics details

Abstract

The aim of this paper is to derive some new identities related to the Frobenius-Euler polynomials. We also give relation between the generalized Frobenius-Euler polynomials and the generalized Hurwitz-Lerch zeta function at negative integers. Furthermore, our results give generalized Carliz’s results which are associated with Frobenius-Euler polynomials.

MSC:05A10, 11B65, 28B99, 11B68.

1 Introduction, definitions and notations

Throughout this presentation, we use the following standard notions: $N = {1, 2, …}$ , $N 0 = {0, 1, 2, …} = N ∪ {0}$ , $Z − = {− 1, − 2, …}$ . Also, as usual ℤ denotes the set of integers, ℝ denotes the set of real numbers and ℂ denotes the set of complex numbers. Furthermore, $(λ) 0 = 1$ and

(λ) k = λ (λ + 1) (λ + 2) ⋯ (λ + k − 1),

where $k ∈ N$ , $λ ∈ C$ .

The classical Frobenius-Euler polynomial $H n (α) (x; u)$ of order α is defined by means of the following generating function:

(1 − u e t − u) α e x t = ∑ n = 0 ∞ H n (α) (x; u) t n n!,

(1)

where u is an algebraic number and $α ∈ Z$ .

Observe that $H n (1) (x; u) = H n (x; u)$ , which denotes the Frobenius-Euler polynomials and $H n (α) (0; u) = H n (α) (u)$ , which denotes the Frobenius-Euler numbers of order α. $H n (x; − 1) = E n (x)$ , which denotes the Euler polynomials (cf. [1–24]).

Definition 1.1 (for details, see [16, 17])

Let $a, b, c ∈ R +$ , $a ≠ b$ , $x ∈ R$ . The generalized Apostol-type Frobenius-Euler polynomials are defined by means of the following generating function:

(a t − u λ b t − u) α c x t = ∑ n = 0 ∞ H n (α) (x; u; a, b, c; λ) t n n! .

(2)

Remark 1.2 If we set $x = 0$ and $α = 1$ in (2), we get

a t − u λ b t − u = ∑ n = 0 ∞ H n (u; a, b, c; λ) t n n!,

(3)

where $H n (u; λ; a, b, c)$ denotes the generalized Apostol-type Frobenius-Euler numbers (cf. [17]).

2 New identities

In this section, we derive many new identities related to the generalized Apostol-type Frobenius-Euler numbers and polynomials of order α.

Theorem 2.1 Let $α, β ∈ Z$ . Each of the following relationships holds true:

(4)

(5)

(6)

and

H n (− α) (x; u 2; a 2, b 2, c 2; λ 2) = ∑ k = 0 n (n k) H k (− α) (x; u; a, b, c; λ) H n − k (− α) (x; − u; a, b, c; λ) .

(7)

Proof of (6) From (2),

∑ n = 0 ∞ H n (− α) (x; u; a, b, c; λ) t n n! ∑ n = 0 ∞ H n (α) (y; u; a, b, c; λ) t n n! = c (x + y) t .

(8)

Therefore,

∑ n = 0 ∞ (∑ k = 0 n (n k) H n − k (α) (y; u; a, b, c; λ) H k (− α) (x; u; a, b, c; λ)) t n n! = ∑ n = 0 ∞ (x ln c) n t n n! .

Thus, by using the Cauchy product in (8) and then equating the coefficients of $t n n!$ on both sides of the resulting equation, we obtain the desired result.

The proofs of (4), (5) and (7) are the same as that of (2), thus we omit them. □

Observe that in (6) we have

((x + y) ln c) n = (H (α) (y; u; a, b, c; λ) + H (− α) (x; u; a, b, c; λ)) n,

where $(H (α) (y; u; a, b, c; λ)) n$ is replaced by $H n (α) (y; u; a, b, c; λ)$ .

Theorem 2.2 Let $α ∈ N$ . Then we have

∑ k = 0 α (α k) (− u) α − k (x ln c + k ln a) n = ∑ p = 0 n ∑ k = 0 α (n p) (α k) (− u) α − k (k ln b) p H n − p (α) (x; u; a, b, c; λ) .

Proof By using (2), we get

∑ n = 0 ∞ (∑ k = 0 α (α k) (− u) α − k (x ln c + k ln a) n) t n n! = ∑ n = 0 ∞ (∑ p = 0 n ∑ k = 0 α (n p) (α k) (− u) α − k (k ln b) p H n − p (α) (x; u; a, b, c; λ)) t n n! .

By equating the coefficients of $t n n!$ on both sides of the resulting equation, we obtain the desired result. □

Theorem 2.3 The following relationship holds true:

(9)

Proof We set

(2 u − 1) a t − u λ b t − u c x t a t − (1 − u) λ b t − (1 − u) c y t = (a t − u) (a t − (1 − u)) c (x + y) t (1 λ b t − u − 1 λ b t − (1 − u)) .

From the above equation, we see that

(2 u − 1) (∑ n = 0 ∞ H n (x; u; a, b, c; λ) t n n!) (∑ n = 0 ∞ H n (y; 1 − u; a, b, c; λ) t n n!) = (a t − 1 + u) ∑ n = 0 ∞ H n (x + y; u; a, b, c; λ) t n n! − (a t − u) ∑ n = 0 ∞ H n (x + y; 1 − u; a, b, c; λ) t n n! .

Therefore,

(2 u − 1) ∑ n = 0 ∞ ∑ r = 0 n (n r) H r (x; u; a, b, c; λ) H n − r (y; 1 − u; a, b, c; λ) t n n! = (u − 1) ∑ n = 0 ∞ H n (x + y; u; a, b, c; λ) t n n! + u ∑ n = 0 ∞ H n (x + y; 1 − u; a, b, c; λ) t n n! + ∑ n = 0 ∞ ∑ r = 0 n (n r) (ln a) n − r H r (x + y; u; a, b, c; λ) t n n! − ∑ n = 0 ∞ ∑ r = 0 n (n r) (ln a) n − r H r (x + y; 1 − u; a, b, c; λ) t n n! .

Comparing the coefficients of $t n n!$ on both sides of the above equation, we arrive at the desired result. □

Remark 2.4 By substituting $a = 1$ , $b = c = e$ , $λ = 1$ into Theorem 2.3, we get Carlitz’s results (for details, see [[1], Eq. 2.19]) as follows:

(2 u − 1) ∑ r = 0 n (n r) H r (x; u) H n − r (y; 1 − u) = (u − 1) H n (x + y; u) + u H n (x + y; 1 − u) + H n (x + y; u) − H n (x + y; 1 − u) .

We give the following generating function of the polynomials $Y n (x; λ; a)$ :

t λ a t − 1 a x t = ∑ n = 0 ∞ Y n (x; λ; a) t n n! (a ≥ 1)

(10)

(cf. [16, 17]). We also note that

Y n (0; λ; a) = Y n (λ; a) .

If we substitute $x = 0$ and $a = 1$ into (10), we see that

Y n (λ; 1) = 1 λ − 1 .

Theorem 2.5 The generalized Apostol-type Frobenius-Euler polynomial holds true as follows:

(11)

Proof Substituting $c = b$ for $α = 1$ into (2) and taking derivative with respect to t, we obtain

∑ n = 0 ∞ H n + 1 (x; u; a, b, b; λ) t n n! = a t ln a a t − u a t − u λ b t − u b x t + ln b λ b t a t − u (a t − u λ b t − u) 2 b x t + ln (b x) a t − u λ b t − u b x t .

Using (10), we have

∑ n = 0 ∞ H n + 1 (x; u; a, b, b; λ) t n n! = ln (a 1 u) t ∑ n = 0 ∞ ∑ k = 0 n (n k) Y n − k (1; 1 u; a) H k (x; u; a, b, b; λ) t n n! + ln (b λ u) t ∑ n = 0 ∞ ∑ k = 0 n (n k) Y n − k (1 u; a) H k (2) (x; u; a, b, b; λ) t n n! + ln (b x) ∑ n = 0 ∞ H n (x; u; a, b, b; λ) t n n! .

Thus, after some elementary calculations, we arrive at (11). □

Theorem 2.6 Let $| u | < 1$ and $m ∈ N$ . Then we have

H (− m) (u; a, b, c; λ) = ∑ k = 0 n (n k) H k (− α) (− x; u; a, b, c; λ) H n − k (α − m) (x; u; a, b, c; λ) .

(12)

Proof In (2), we replace α by −α, then we set

(a t − u λ b t − u) − α c (− x) t ∑ n = 0 ∞ H n (α − m) (x; u; a, b, c; λ) t n n! = (a t − u λ b t − u) − m .

By using (2), we get

∑ n = 0 ∞ H n (− α) (− x; u; a, b, c; λ) t n n! ∑ n = 0 ∞ H n (α − m) (x; u; a, b, c; λ) t n n! = ∑ n = 0 ∞ H n (− m) (u; a, b, c; λ) t n n! .

Therefore,

∑ n = 0 ∞ ∑ k = 0 n (n k) H k (− α) (− x; u; a, b, c; λ) H n − k (α − m) (x; u; a, b, c; λ) t n n! = ∑ n = 0 ∞ H n (− m) (u; a, b, c; λ) t n n! .

Comparing the coefficients of $t n n!$ on both sides of the above equation, we arrive at (12). □

3 Interpolation function

In this section, we give a recurrence relation between the generalized Frobenius-Euler polynomials and the Hurwitz-Lerch zeta function. Recently, many authors have studied not only the Hurwitz-Lerch zeta function, but also its generalizations, for example (among others), Srivastava [19], Srivastava and Choi [24] and also Garg et al. [6]. The generalization of the Hurwitz-Lerch zeta function $Φ (z, s, a)$ is given as follows:

Φ μ, ν (ρ, σ) (z, s, a) : = ∑ n = 0 ∞ (μ) ρ n (ν) σ n z n (n + a) s

( $μ ∈ C$ , $a, υ ∈ C ∖ Z 0 −$ , $ρ, σ ∈ R +$ , $ρ < σ$ when $s, z ∈ C$ ( $| z | < 1$ ); $ρ = σ$ and $ℜ (s − μ + ν) > 0$ when $| z | = 1$ ). It is obvious that

Φ μ, 1 (1, 1) (z, s, a) = Φ μ ∗ (z, s, a) = ∑ n = 0 ∞ (μ) n n! z n (n + a) s

(13)

and

Φ n ∗ (z, s, a) = ∑ n = 0 ∞ (n) n n! z n (n + a) s = Φ (z, s, a),

where $Φ (z, s, a)$ denotes the Lerch-Zeta function (cf. [6, 19, 21, 24]).

Relation between the generalized Apostol-type Frobenius-Euler polynomials and the Hurwitz-Lerch zeta function is given as follows.

Theorem 3.1 Let $| λ u | < 1$ . We have

H n (α) (x; u; a, b, c; λ) = ∑ k = 0 α (α k) (− u) α − k − 1 G (− n; x, λ u; a, b, c; α, k),

(14)

where

G (s; x, β; a, b, c; α, j) = ∑ m = 0 ∞ (m + α − 1 m) β m (x ln c + j ln a + m ln b) s, | β | < 1 .

Proof From (2), we have

∑ n = 0 ∞ H n (α) (x; u; a, b, c; λ) t n n! = ∑ j = 0 α (α j) (− u) α − j − 1 ∑ m = 0 ∞ (m + α − 1 m) (λ u) m e α (x ln c + k ln a + m ln b) .

Therefore,

∑ n = 0 ∞ H n (α) (x; u; a, b, c; λ) t n n! = ∑ n = 0 ∞ ∑ k = 0 α (α k) (− u) α − k − 1 ∑ m = 0 ∞ (m + α − 1 m) (λ u) m (x ln c + k ln a + m ln b) n t n n! .

Comparing the coefficients of $t n n!$ on both sides of the above equation, we have arrive at (14). □

Remark 3.2 By substituting $a = 1$ , $b = c = e$ into (14), we have

H n (α) (x; u; λ) = − (1 − u) α u G (− n; x, λ u; 1, e, e; α, 1) = − (1 − u) α u Φ (λ u, − n, x),

where

G (− n; x, λ u; 1, e, e; α, 1) = Φ (λ u, − n, x) .

Remark 3.3 The function $G (s; x, β; a, b, c; α, j)$ is an interpolation function of the generalized Apostol-type Frobenius-Euler polynomials of order α at negative integers, which is given by the analytic continuation of the $G (s; x, β; a, b, c; α, j)$ for $s = − n$ , $n ∈ N$ .

4 Relations between Array-type polynomials, Apostol-Bernoulli polynomials and generalized Apostol-type Frobenius-Euler polynomial

In [17], Simsek constructed the generalized λ-Stirling type numbers of the second kind $S (n, v; a, b; λ)$ by means of the following generating function:

f S, v (t; a, b; λ) = (λ b t − a t) v v! = ∑ n = 0 ∞ S (n, v; a, b; λ) t n n! .

(15)

The generating function for these polynomials $S v n (x; a, b; λ)$ is given by

g v (x, t; a, b; λ) = 1 v! (λ b t − a t) v b x t = ∑ n = 0 ∞ S v n (x; a, b; λ) t n n!

(16)

(cf. [17]).

The generalized Apostol-Bernoulli polynomials were defined by Srivastava et al. [[22], p.254, Eq. (20)] as follows.

Let $a, b, c ∈ R +$ with $a ≠ b$ , $x ∈ R$ and $n ∈ N 0$ . Then the generalized Bernoulli polynomials $B n (α) (x; λ; a, b, c)$ of order $α ∈ Z$ are defined by means of the following generating functions:

f B (x, a, b, c; λ; α) = (t λ b t − a t) α c x t = ∑ n = 0 ∞ B n (α) (x; λ; a, b, c) t n n!,

(17)

where

| t ln (a b) + ln λ | < 2 π .

We note that $B n (1) (x; λ; a, b, c) = B n (x; λ; a, b, c)$ and also $B n (x; λ; 1, e, e) = B n (x; λ)$ , which denotes the Apostol-Bernoulli polynomials (cf. [1–24]).

Theorem 4.1 Let v be an integer. Then we have

H n − v (− ν) (x; u; a, b, c; λ) = ν! u 2 ν (n) v ∑ k = 0 n (n k) S v n (x, 1, b; λ u) Y n − k (ν) (1 u; a) .

Proof Replacing c by b in (2) and after some calculations, we have

∑ n = 0 ∞ H n (− v) (x; u; a, b, b; λ) t n + v n! = ν! u 2 ν ∑ n = 0 ∞ S ν n (x, 1, b; λ u) t n n! ∑ n = 0 ∞ Y n (ν) (1 u; a) t n n! .

Comparing the coefficients of $t n n!$ on both sides of the above equation, we arrive at the desired result. □

Corollary 4.2

H n − v (− ν) (x; u; a, b, c; λ) = ν! u 2 ν (n) α ∑ k = 0 n (n k) S (k, ν, 1, b; λ u) B n − k (x, a, b; λ u) .

Proof Replacing c by b in (2) and after some calculations, we have

∑ n = 0 ∞ H n − v (− v) (x; u; a, b, b; λ) t n + v n! = ν! u 2 ν ∑ n = 0 ∞ S (n, ν, 1, b; λ u) t n n! ∑ n = 0 ∞ B n (x, a, b; λ u) t n n! .

Comparing the coefficients of $t n n!$ on both sides of the above equation, we arrive at the desired result. □

References

1.
Carlitz L: Eulerian numbers and polynomials. Math. Mag. 1959, 32: 247–260. 10.2307/3029225
MathSciNet Article Google Scholar
2.
Choi J, Jang SD, Srivastava HM: A generalization of the Hurwitz-Lerch zeta function. Integral Transforms Spec. Funct. 2008, 19: 65–79.
MathSciNet Article Google Scholar
3.
Choi J, Srivastava HM: The multiple Hurwitz-Lerch zeta function and the multiple Hurwitz-Euler eta function. Taiwan. J. Math. 2011, 15: 501–522.
MathSciNet Google Scholar
4.
Choi J, Kim DS, Kim T, Kim YH: A note on some identities of Frobenius-Euler numbers and polynomials. Int. J. Math. Math. Sci. 2012. doi:10.1155/2012/861797
Google Scholar
5.
Ding D, Yang J: Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials. Adv. Stud. Contemp. Math. 2010, 20: 7–21.
MathSciNet Google Scholar
6.
Garg M, Jain K, Srivastava HM: Some relationships between the generalized Apostol-Bernoulli polynomials and Hurwitz-Lerch zeta functions. Integral Transforms Spec. Funct. 2006, 17: 803–815. 10.1080/10652460600926907
MathSciNet Article Google Scholar
7.
Gould HW: The q -series generalization of a formula of Sparre Andersen. Math. Scand. 1961, 9: 90–94.
MathSciNet Google Scholar
8.
Kim T: An identity of the symmetry for the Frobenius-Euler polynomials associated with the fermionic p -adic invariant q -integrals on $Z p$ . Rocky Mt. J. Math. 2011, 41: 239–247. 10.1216/RMJ-2011-41-1-239
Article Google Scholar
9.
Kim T: Identities involving Frobenius-Euler polynomials arising from non-linear differential equations. J. Number Theory 2012, 132: 2854–2865. arXiv:1201.5088v1 10.1016/j.jnt.2012.05.033
MathSciNet Article Google Scholar
10.
Kim T, Choi J: A note on the product of Frobenius-Euler polynomials arising from the p -adic integral on $Z p$ . Adv. Stud. Contemp. Math. 2012, 22: 215–223.
Google Scholar
11.
Kurt B, Simsek Y: Frobenious-Euler type polynomials related to Hermite-Bernoulli polynomials. AIP Conf. Proc. 2011, 1389: 385–388.
Article Google Scholar
12.
Lin S-D, Srivastava HM, Wang P-Y: Some expansion formulas for a class of generalized Hurwitz-Lerch zeta functions. Integral Transforms Spec. Funct. 2006, 17: 817–827. 10.1080/10652460600926923
MathSciNet Article Google Scholar
13.
Luo Q-M: q -analogues of some results for the Apostol-Euler polynomials. Adv. Stud. Contemp. Math. 2010, 20: 103–113.
Google Scholar
14.
Luo Q-M, Srivastava HM: Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind. Appl. Math. Comput. 2011, 217: 5702–5728. 10.1016/j.amc.2010.12.048
MathSciNet Article Google Scholar
15.
Srivastava HM, Saxena RK, Pogany TK, Saxena R: Integral and computational representation of the extended Hurwitz-Lerch zeta function. Integral Transforms Spec. Funct. 2011, 22: 487–506. 10.1080/10652469.2010.530128
MathSciNet Article Google Scholar
16.
Simsek Y: Generating functions for q -Apostol type Frobenius-Euler numbers and polynomials. Axioms 2012, 1: 395–403. doi:10.3390/axioms1030395 10.3390/axioms1030395
Article Google Scholar
17.
Simsek, Y: Generating functions for generalized Stirling type numbers, array type polynomials, Eulerian type polynomials and their application. arXiv:1111.3848v2
18.
Simsek Y, Kim T, Park DW, Ro YS, Jang LC, Rim SH: An explicit formula for the multiple Frobenius-Euler numbers and polynomials. JP J. Algebra Number Theory Appl. 2004, 4: 519–529.
MathSciNet Google Scholar
19.
Srivastava HM: Some generalizations and basic (or q -) extensions of the Bernoulli, Euler and Genocchi polynomials. Appl. Math. Inform. Sci. 2011, 5: 390–444.
Google Scholar
20.
Srivastava HM, Kim T, Simsek Y: q -Bernoulli numbers and polynomials associated with multiple q -zeta functions and basic L -series. Russ. J. Math. Phys. 2005, 12: 241–268.
MathSciNet Google Scholar
21.
Srivastava HM, Choi J: Series Associated with the Zeta and Related Functions. Kluwer Academic, Dordrecht; 2001.
Google Scholar
22.
Srivastava HM, Garg M, Choudhary S: A new generalization of the Bernoulli and related polynomials. Russ. J. Math. Phys. 2010, 17: 251–261. 10.1134/S1061920810020093
MathSciNet Article Google Scholar
23.
Srivastava HM, Garg M, Choudhary S: Some new families of the generalized Euler and Genocchi polynomials. Taiwan. J. Math. 2011, 15: 283–305.
MathSciNet Google Scholar
24.
Srivastava HM, Choi J: Zeta and q-Zeta Functions and Associated Series and Integrals. Elsevier, Amsterdam; 2012.
Google Scholar

Download references

Acknowledgements

Dedicated to Professor Hari M. Srivastava.

All authors are partially supported by Research Project Offices Akdeniz Universities.

Author information

Affiliations

Department of Mathematics, Faculty of Education, Akdeniz University, Antalya, 07058, Turkey
Burak Kurt
Department of Mathematics, Faculty of Science, Akdeniz University, Antalya, 07058, Turkey
Yilmaz Simsek

Authors

Burak Kurt
View author publications
You can also search for this author in PubMed Google Scholar
Yilmaz Simsek
View author publications
You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yilmaz Simsek.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors completed the paper together. All authors read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Kurt, B., Simsek, Y. On the generalized Apostol-type Frobenius-Euler polynomials. Adv Differ Equ 2013, 1 (2013). https://doi.org/10.1186/1687-1847-2013-1

Download citation

Received: 08 November 2012
Accepted: 13 December 2012
Published: 04 January 2013
DOI: https://doi.org/10.1186/1687-1847-2013-1

Keywords

Frobenius-Euler polynomials
Hermite-based Frobenius-Euler polynomials
Hermite-based Apostol-Euler polynomials
Apostol-Euler polynomials
Hurwitz-Lerch zeta function