On the generalized Apostol-type Frobenius-Euler polynomials

Burak Kurt; Yilmaz Simsek

doi:10.1186/1687-1847-2013-1

Research
Open Access

On the generalized Apostol-type Frobenius-Euler polynomials

Burak Kurt¹ and
Yilmaz Simsek²Email author

Advances in Difference Equations20132013:1

https://doi.org/10.1186/1687-1847-2013-1

Received: 8 November 2012
Accepted: 13 December 2012
Published: 4 January 2013

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.

Keywords

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

1 Introduction, definitions and notations

Throughout this presentation, we use the following standard notions:

N = {1, 2, \dots}

,

N_{0} = {0, 1, 2, \dots} = N \cup {0}

,

Z^{-} = {- 1, - 2, \dots}

. 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) \dots (λ + k - 1),

where $k \in N$ , $λ \in C$ .

The classical Frobenius-Euler polynomial

H_{n}^{(α)} (x; u)

of order α is defined by means of the following generating function:

{(\frac{1 - u}{e^{t} - u})}^{α} e^{x t} = \sum_{n = 0}^{\infty} H_{n}^{(α)} (x; u) \frac{t^{n}}{n!},

(1)

where u is an algebraic number and $α \in Z$ .

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 \in R^{+}

,

a \neq b

,

x \in R

. The generalized Apostol-type Frobenius-Euler polynomials are defined by means of the following generating function:

{(\frac{a^{t} - u}{λ b^{t} - u})}^{α} c^{x t} = \sum_{n = 0}^{\infty} H_{n}^{(α)} (x; u; a, b, c; λ) \frac{t^{n}}{n!} .

(2)

Remark 1.2 If we set

x = 0

and

α = 1

in (2), we get

\frac{a^{t} - u}{λ b^{t} - u} = \sum_{n = 0}^{\infty} H_{n} (u; a, b, c; λ) \frac{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 \in 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}) = \sum_{k = 0}^{n} (\binom{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),

\sum_{n = 0}^{\infty} H_{n}^{(- α)} (x; u; a, b, c; λ) \frac{t^{n}}{n!} \sum_{n = 0}^{\infty} H_{n}^{(α)} (y; u; a, b, c; λ) \frac{t^{n}}{n!} = c^{(x + y) t} .

(8)

Therefore,

\sum_{n = 0}^{\infty} (\sum_{k = 0}^{n} (\binom{n}{k}) H_{n - k}^{(α)} (y; u; a, b, c; λ) H_{k}^{(- α)} (x; u; a, b, c; λ)) \frac{t^{n}}{n!} = \sum_{n = 0}^{\infty} {(x ln c)}^{n} \frac{t^{n}}{n!} .

Thus, by using the Cauchy product in (8) and then equating the coefficients of $\frac{t^{n}}{n!}$ 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 \in N

. Then we have

\sum_{k = 0}^{α} (\binom{α}{k}) {(- u)}^{α - k} {(x ln c + k ln a)}^{n} = \sum_{p = 0}^{n} \sum_{k = 0}^{α} (\binom{n}{p}) (\binom{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

\begin{matrix} \sum_{n = 0}^{\infty} (\sum_{k = 0}^{α} (\binom{α}{k}) {(- u)}^{α - k} {(x ln c + k ln a)}^{n}) \frac{t^{n}}{n!} \\ = \sum_{n = 0}^{\infty} (\sum_{p = 0}^{n} \sum_{k = 0}^{α} (\binom{n}{p}) (\binom{α}{k}) {(- u)}^{α - k} {(k ln b)}^{p} H_{n - p}^{(α)} (x; u; a, b, c; λ)) \frac{t^{n}}{n!} . \end{matrix}

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

Theorem 2.3 The following relationship holds true:

(9)

Proof We set

\begin{matrix} (2 u - 1) \frac{a^{t} - u}{λ b^{t} - u} c^{x t} \frac{a^{t} - (1 - u)}{λ b^{t} - (1 - u)} c^{y t} \\ = (a^{t} - u) (a^{t} - (1 - u)) c^{(x + y) t} (\frac{1}{λ b^{t} - u} - \frac{1}{λ b^{t} - (1 - u)}) . \end{matrix}

From the above equation, we see that

\begin{matrix} (2 u - 1) (\sum_{n = 0}^{\infty} H_{n} (x; u; a, b, c; λ) \frac{t^{n}}{n!}) (\sum_{n = 0}^{\infty} H_{n} (y; 1 - u; a, b, c; λ) \frac{t^{n}}{n!}) \\ = (a^{t} - 1 + u) \sum_{n = 0}^{\infty} H_{n} (x + y; u; a, b, c; λ) \frac{t^{n}}{n!} - (a^{t} - u) \sum_{n = 0}^{\infty} H_{n} (x + y; 1 - u; a, b, c; λ) \frac{t^{n}}{n!} . \end{matrix}

Therefore,

\begin{matrix} (2 u - 1) \sum_{n = 0}^{\infty} \sum_{r = 0}^{n} (\binom{n}{r}) H_{r} (x; u; a, b, c; λ) H_{n - r} (y; 1 - u; a, b, c; λ) \frac{t^{n}}{n!} \\ = (u - 1) \sum_{n = 0}^{\infty} H_{n} (x + y; u; a, b, c; λ) \frac{t^{n}}{n!} + u \sum_{n = 0}^{\infty} H_{n} (x + y; 1 - u; a, b, c; λ) \frac{t^{n}}{n!} \\ + \sum_{n = 0}^{\infty} \sum_{r = 0}^{n} (\binom{n}{r}) {(ln a)}^{n - r} H_{r} (x + y; u; a, b, c; λ) \frac{t^{n}}{n!} \\ - \sum_{n = 0}^{\infty} \sum_{r = 0}^{n} (\binom{n}{r}) {(ln a)}^{n - r} H_{r} (x + y; 1 - u; a, b, c; λ) \frac{t^{n}}{n!} . \end{matrix}

Comparing the coefficients of $\frac{t^{n}}{n!}$ 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:

\begin{matrix} (2 u - 1) \sum_{r = 0}^{n} (\binom{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) . \end{matrix}

We give the following generating function of the polynomials

Y_{n} (x; λ; a)

:

\frac{t}{λ a^{t} - 1} a^{x t} = \sum_{n = 0}^{\infty} Y_{n} (x; λ; a) \frac{t^{n}}{n!} (a \geq 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) = \frac{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

\begin{matrix} \sum_{n = 0}^{\infty} H_{n + 1} (x; u; a, b, b; λ) \frac{t^{n}}{n!} \\ = \frac{a^{t} ln a}{a^{t} - u} \frac{a^{t} - u}{λ b^{t} - u} b^{x t} + \frac{ln b λ b^{t}}{a^{t} - u} {(\frac{a^{t} - u}{λ b^{t} - u})}^{2} b^{x t} + ln (b^{x}) \frac{a^{t} - u}{λ b^{t} - u} b^{x t} . \end{matrix}

Using (10), we have

\begin{array}{rcl} \sum_{n = 0}^{\infty} H_{n + 1} (x; u; a, b, b; λ) \frac{t^{n}}{n!} & = & \frac{ln (a^{\frac{1}{u}})}{t} \sum_{n = 0}^{\infty} \sum_{k = 0}^{n} (\binom{n}{k}) Y_{n - k} (1; \frac{1}{u}; a) H_{k} (x; u; a, b, b; λ) \frac{t^{n}}{n!} \\ + \frac{ln (b^{\frac{λ}{u}})}{t} \sum_{n = 0}^{\infty} \sum_{k = 0}^{n} (\binom{n}{k}) Y_{n - k} (\frac{1}{u}; a) H_{k}^{(2)} (x; u; a, b, b; λ) \frac{t^{n}}{n!} \\ + ln (b^{x}) \sum_{n = 0}^{\infty} H_{n} (x; u; a, b, b; λ) \frac{t^{n}}{n!} . \end{array}

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

Theorem 2.6 Let

| u | < 1

and

m \in N

. Then we have

H^{(- m)} (u; a, b, c; λ) = \sum_{k = 0}^{n} (\binom{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

{(\frac{a^{t} - u}{λ b^{t} - u})}^{- α} c^{(- x) t} \sum_{n = 0}^{\infty} H_{n}^{(α - m)} (x; u; a, b, c; λ) \frac{t^{n}}{n!} = {(\frac{a^{t} - u}{λ b^{t} - u})}^{- m} .

By using (2), we get

\sum_{n = 0}^{\infty} H_{n}^{(- α)} (- x; u; a, b, c; λ) \frac{t^{n}}{n!} \sum_{n = 0}^{\infty} H_{n}^{(α - m)} (x; u; a, b, c; λ) \frac{t^{n}}{n!} = \sum_{n = 0}^{\infty} H_{n}^{(- m)} (u; a, b, c; λ) \frac{t^{n}}{n!} .

Therefore,

\sum_{n = 0}^{\infty} \sum_{k = 0}^{n} (\binom{n}{k}) H_{k}^{(- α)} (- x; u; a, b, c; λ) H_{n - k}^{(α - m)} (x; u; a, b, c; λ) \frac{t^{n}}{n!} = \sum_{n = 0}^{\infty} H_{n}^{(- m)} (u; a, b, c; λ) \frac{t^{n}}{n!} .

Comparing the coefficients of $\frac{t^{n}}{n!}$ 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) : = \sum_{n = 0}^{\infty} \frac{{(μ)}_{ρ n}}{{(ν)}_{σ n}} \frac{z^{n}}{{(n + a)}^{s}}

(

μ \in C

,

a, υ \in C ∖ Z_{0}^{-}

,

ρ, σ \in R^{+}

,

ρ < σ

when

s, z \in 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) = \sum_{n = 0}^{\infty} \frac{{(μ)}_{n}}{n!} \frac{z^{n}}{{(n + a)}^{s}}

(13)

and

Φ_{n}^{*} (z, s, a) = \sum_{n = 0}^{\infty} \frac{{(n)}_{n}}{n!} \frac{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

| \frac{λ}{u} | < 1

. We have

H_{n}^{(α)} (x; u; a, b, c; λ) = \sum_{k = 0}^{α} (\binom{α}{k}) {(- u)}^{α - k - 1} G (- n; x, \frac{λ}{u}; a, b, c; α, k),

(14)

where

G (s; x, β; a, b, c; α, j) = \sum_{m = 0}^{\infty} (\binom{m + α - 1}{m}) \frac{β^{m}}{{(x ln c + j ln a + m ln b)}^{s}}, | β | < 1 .

Proof From (2), we have

\sum_{n = 0}^{\infty} H_{n}^{(α)} (x; u; a, b, c; λ) \frac{t^{n}}{n!} = \sum_{j = 0}^{α} (\binom{α}{j}) {(- u)}^{α - j - 1} \sum_{m = 0}^{\infty} (\binom{m + α - 1}{m}) {(\frac{λ}{u})}^{m} e^{α (x ln c + k ln a + m ln b)} .

Therefore,

\begin{matrix} \sum_{n = 0}^{\infty} H_{n}^{(α)} (x; u; a, b, c; λ) \frac{t^{n}}{n!} \\ = \sum_{n = 0}^{\infty} \sum_{k = 0}^{α} (\binom{α}{k}) {(- u)}^{α - k - 1} \sum_{m = 0}^{\infty} (\binom{m + α - 1}{m}) {(\frac{λ}{u})}^{m} {(x ln c + k ln a + m ln b)}^{n} \frac{t^{n}}{n!} . \end{matrix}

Comparing the coefficients of $\frac{t^{n}}{n!}$ 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) = - \frac{{(1 - u)}^{α}}{u} G (- n; x, \frac{λ}{u}; 1, e, e; α, 1) = - \frac{{(1 - u)}^{α}}{u} Φ (\frac{λ}{u}, - n, x),

where

G (- n; x, \frac{λ}{u}; 1, e, e; α, 1) = Φ (\frac{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 $s = - n$ , $n \in N$ .

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) = \frac{{(λ b^{t} - a^{t})}^{v}}{v!} = \sum_{n = 0}^{\infty} S (n, v; a, b; λ) \frac{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) = \frac{1}{v!} {(λ b^{t} - a^{t})}^{v} b^{x t} = \sum_{n = 0}^{\infty} S_{v}^{n} (x; a, b; λ) \frac{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 \in R^{+}

with

a \neq b

,

x \in R

and

n \in N_{0}

. Then the generalized Bernoulli polynomials

B_{n}^{(α)} (x; λ; a, b, c)

of order

α \in Z

are defined by means of the following generating functions:

f_{B} (x, a, b, c; λ; α) = {(\frac{t}{λ b^{t} - a^{t}})}^{α} c^{x t} = \sum_{n = 0}^{\infty} B_{n}^{(α)} (x; λ; a, b, c) \frac{t^{n}}{n!},

(17)

where

| t ln (\frac{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) = \frac{ν!}{u^{2 ν} {(n)}_{v}} \sum_{k = 0}^{n} (\binom{n}{k}) S_{v}^{n} (x, 1, b; \frac{λ}{u}) Y_{n - k}^{(ν)} (\frac{1}{u}; a) .

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

\sum_{n = 0}^{\infty} H_{n}^{(- v)} (x; u; a, b, b; λ) \frac{t^{n + v}}{n!} = \frac{ν!}{u^{2 ν}} \sum_{n = 0}^{\infty} S_{ν}^{n} (x, 1, b; \frac{λ}{u}) \frac{t^{n}}{n!} \sum_{n = 0}^{\infty} Y_{n}^{(ν)} (\frac{1}{u}; a) \frac{t^{n}}{n!} .

Comparing the coefficients of $\frac{t^{n}}{n!}$ 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) = \frac{ν!}{u^{2 ν} {(n)}_{α}} \sum_{k = 0}^{n} (\binom{n}{k}) S (k, ν, 1, b; \frac{λ}{u}) B_{n - k} (x, a, b; \frac{λ}{u}) .

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

\sum_{n = 0}^{\infty} H_{n - v}^{(- v)} (x; u; a, b, b; λ) \frac{t^{n + v}}{n!} = \frac{ν!}{u^{2 ν}} \sum_{n = 0}^{\infty} S (n, ν, 1, b; \frac{λ}{u}) \frac{t^{n}}{n!} \sum_{n = 0}^{\infty} B_{n} (x, a, b; \frac{λ}{u}) \frac{t^{n}}{n!} .

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

Declarations

Acknowledgements

Dedicated to Professor Hari M. Srivastava.

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

Authors’ Affiliations

(1)

Department of Mathematics, Faculty of Education, Akdeniz University, Antalya, 07058, Turkey

(2)

Department of Mathematics, Faculty of Science, Akdeniz University, Antalya, 07058, Turkey

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