A new class of Laguerre-based Apostol type polynomials · A new class of Laguerre-based Apostol type polynomials Waseem A. Khan1, Serkan Araci2* and Mehmet Acikgoz3 Abstract:

Khan et al., Cogent Mathematics (2016), 3: 1243839http://dx.doi.org/10.1080/23311835.2016.1243839

PURE MATHEMATICS | RESEARCH ARTICLE

A new class of Laguerre-based Apostol type polynomialsWaseem A. Khan1, Serkan Araci2* and Mehmet Acikgoz3

Abstract: In this paper, we introduce a generating function for a new gen-eralization of Laguerre-based Apostol-Bernoulli polynomials, Apostol-Euler and Apostol-Genocchi polynomials. By making use of the generating function method and some functional equations mentioned in the paper, we con-duct a further investigation in order to obtain symmetric identities of these polynomials.

Subjects: Science; Mathematics & Statistics; Applied Mathematics; Theory of Numbers

Keywords: Hermite polynomials; Laguerre polynomials; Apostol-type polynomials; functional equations; symmetric identities

2010 Mathematics subject classifications: Primary 11B68; 33E20; Secondary 33C05

1. IntroductionThroughout the paper, we make use of the following notations:

and

ℕ: ={1, 2, 3,…

}, ℕ0: =

{0, 1, 2, 3,…

}= ℕ ∪

{0}

ℤ−: =

{−1,−2,−3,…

}= ℤ

−

0�{0}.

*Corresponding author: Serkan Araci, Faculty of Economics, Department of Economics, Administrative and Social Sciences, Hasan Kalyoncu University, TR 27410 Gaziantep, Turkey E-mail: [email protected]

Reviewing editor:Hari M. Srivastava, University of Victoria, Canada

Additional information is available at the end of the article

ABOUT THE AUTHORSerkan Araci was born in Hatay, Turkey, on 1 October 1988. He received his BS and MS degrees in mathematics from the University of Gaziantep, Gaziantep, Turkey, in 2010 and 2013, respectively. Additionally, the title of his MS thesis is “Bernstein polynomials and their reflections in analytic number theory” and, for this thesis, he received the Best Thesis Award of 2013 from the University of Gaziantep. He has published more than 90 papers in reputed international journals. His research interests include p-adic analysis, analytic theory of numbers, q-series and q-polynomials, and theory of umbral calculus. Araci is an editor and a referee for some international journals.

PUBLIC INTEREST STATEMENTIn the paper, we have established the generating functions for the Laguerre-based Apostol-type polynomials and Laguerre-based Apostol-type Hermite polynomials by making use of Tricomi function of the generating function for Laguerre polynomials. The equivalent forms of these generating functions can be derived by using Equations. (1.1), (1.6), and (2.1). They can be viewed as the equivalent forms of the generating functions (2.3), (2.6), and (2.8), respectively. In the previous sections, we have used the concepts and the formalism associated with Laguerre polynomials to introduce the Laguerre-based Apostol-type polynomials and Laguerre-based Apostol-type Hermite polynomials and establish their properties. The approach presented here is general and we have established the summation rules, which can be used to derive the results for Laguerre-based Apostol-type polynomials from the results of the corresponding Appell polynomials.

Received: 01 April 2016Accepted: 27 September 2016First Published: 04 October 2016

© 2016 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license.

Page 1 of 17

Serkan Araci

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Page 2 of 17


Here, as convention, ℤ denotes the set of integers, ℝ denotes the set of real numbers and ℂ denotes the set of complex numbers.

The generating function of Laguerre polynomials are defined by means of the generating function (Srivastava & Manocha, 1984):

or equivalently by

where J0�2√x�

are called 0th order Bessel function, and nth order Bessel function Jn(x) are given by the series:

We recall that the Gould-Hopper generalized Hermite polynomials are defined as

where m is positive integer (see Srivastava & Manocha, 1984). These polynomials are specified by the generating function

(see Srivastava & Manocha, 1984).

In particular, we note that

where Hn(x, y) are called 2-variable Hermite-Kampé de Fériet polynomials (Srivastava & Manocha, 1984) that can be defined by the generating function:

and it reduces to the ordinary Hermite polynomials Hn(x) (see Srivastava & Manocha, 1984) when we take the values y = −1 and 2x instead of x in the Equation (1.4). Furthermore, we recall that the 3-variable Laguerre-Hermite polynomials (3VLHP) LHn(x, y, z) are defined by the series (Kurt, 2010)

The generating function of the Equation (1.5) is that

(1.1)1

1 − texp

(−xt

1 − t

)=

∞∑n=0

Ln(x)tn (|t| < 1)

(1.2)exp(yt)J0

�2√x�=

∞�n=0

ynLn

�x

y

�tn

(1.3)xn

2 Jn

�2√x�=

∞�r=0

(−1)rxr

r!(n + r)!

�n ∈ ℕ0

�.

gmn (x, y) = n!

[n

m

]∑r=0

xn−mryr

r!(n −mr)!

exp(xt + ytm

)=∑

gmn (x, y)tn

n!

g2n(x, y) = Hn(x, y)

(1.4)exp(xt + yt2

)=

∞∑n=0

Hn(x, y)tn

n!

(1.5)LHn(x, y, z) = n!

[n∕2]∑k=0

zkyn−2kLn−2k

(x

y

)

k!(n − 2k)!.

(1.6)1

1 − ztexp

(−xt

1 − zt+

yt2

1 − zt2

)=

∞∑n=0

LHn(x, y, z)tn

Page 3 of 17


and it also equals to

At the value z = −1

2 in the Equation (1.7), we have

and

where LHn(x, y) denotes 2-variable Laguerre-Hermite polynomials (2VLHP) (Magnus, Oberhettinger, & Soni, 1966) and LHn(x) denotes the Laguerre-Hermite polynomials (LHP) (Ozarslan, 2013).

The generalized Bernoulli polynomials B(�)n (x), generalized Euler polynomials E(�)n (x) and the generalized Genocchi polynomials G(�)

n (x) of (non-negative integer) order � are defined, respec-tively, by the following generating functions (see Luo, 2006, 2011; Luo & Srivastava, 2005, 2006, 2011a, 2011b):

The literature contains a large number of interesting properties and relationships involving these polynomials (Araci, Bagdasaryan, & Srivastava, 2014; Araci, Şen, Acikgoz, & Orucoglu 2015; Comtet, 1974; Khan, Al Saad, & Khan, 2010; Kurt, 2010; Luke, 1969; Luo, 2006, 2011; Luo & Srivastava, 2005, 2006, 2011a, 2011b; Magnus, Oberhettinger, & Soni, 1966; Ozarslan, 2013, 2011; Ozden, 2010, 2011; Ozden, Simsek, & Srivastava, 2010; Pathan, 2012; Pathan & Khan, 2015; Kilbas, Srivastava, & Trujillo, 2006; Srivastava & Manocha, 1984; Srivastava, 2014; Srivastava, Kurt, & Simsek, 2012; Srivastava, Garg, & Choudhary, 2011; Tuenter, 2001). Luo and Srivastava (2005, 2006, 2011b) introduced the generalized Apostol-Bernoulli polynomials B(�)n (x) of order �. Luo (2006) also investigated the gener-alized Apostol-Euler polynomials E(�)n (x) and the generalized Apostol-Genocchi polynomials G(�)

n (x) of (non-negative integer) order � (see also Luo, 2006, 2011a; Luo & Srivastava, 2011).

Let � be a non-negative integer. The generalized Apostol-Bernoulli polynomials B(�)n (x;�) of order �, the generalized Apostol-Euler polynomials E(�)n (x;�) of order �, the generalized Apostol-Genocchi polynomials G(�)

n (x;�) of order � are defined, respectively, by the following generating functions (see Luo & Srivastava, 2011b)

and

(1.7)exp(yt + zt2)J0

�2√xt�=

∞�n=0

LHn(x, y, z)tn

n!.

LHn

(x, y,−

1

2

)=

LHn(x, y)

LHn(x, 1,−1) =

LHn(x)

(1.8)

(t

et − 1

)𝛼

ext =

∞∑n=0

B(𝛼)

n(x)tn

n!, (|t| < 2𝜋; 1𝛼 : = 1)

(1.9)

(2

et + 1

)𝛼

ext =

∞∑n=0

E(𝛼)

n(x)tn

n!, (|t| < 𝜋; 1

𝛼: = 1)

(1.10)

(2t

et + 1

)𝛼

ext =

∞∑n=0

G(𝛼)

n(x)tn

n!, (|t| < 𝜋; 1

𝛼: = 1)

(1.11)

(t

𝜆et − 1

)𝛼

ext =

∞∑n=0

B(𝛼)

n(x;𝜆)

tn

n!, (|t| < 2𝜋 when 𝜆 = 1; |t| < ‖‖log 𝜆‖‖ when 𝜆 ≠ 1)

(1.12)

(2

𝜆et + 1

)𝛼

ext =

∞∑n=0

E(𝛼)n (x;𝜆)tn

n!, (|t| < ||log (−𝜆)||)

Page 4 of 17


It can be easily noted that

Recently, Kurt (2010) gave the following generalization of the Bernoulli polynomials of order �, which is recalled in Definition 1.

Definition 1 For arbitrary real or complex parameter �, the generalized Bernoulli polynomials B[�,m−1]n (x)(m ∈ ℕ) are defined in centered at t = 0 by means of the generating function:

Clearly, if we take m = 1 in (1.14), then the definition (1.14) becomes the definition (1.13).

More recently, Tremblay, Gaboury, and Fugère (2011) further gave the following generalization of Kurt’s definition (1.14) in the following form.

Definition 2 For arbitrary real or complex parameter � and � and the natural numbers m ∈ ℕ, the generalized Bernoulli polynomials B[�,m−1]

n (x;�) are defined in, centered at t = 0, with |t| < ||log 𝜆||, by means of the generating function:

Clearly, if we take m = 1 in (1.15), then the definition (1.15) becomes the definition (1.11).

We now give the following definition for the generalized Euler polynomials E(�)n (x).

Definition 3 For arbitrary real or complex parameter � and natural number m ∈ ℕ, the generalized Euler polynomials E[�,m−1]

n (x) are defined in centered at t = 0, with |t| < 𝜋, by means of the generating function:

Obviously, setting m = 1 in (1.16), we have E[�,0]n (x;1) = E(�)n (x).

Definition 4 For arbitrary real or complex parameter � and � and the natural number m, the gener-alized Euler polynomials E[�,m−1]

n (x) are defined in centered at t = 0, with |t| < ||log (−𝜆)||, by means of the generating function:

(1.13)(

2t

𝜆et + 1

)𝛼

ext =

∞∑n=0

G(𝛼)

n (x;𝜆)tn

n!, (|t| < ||log (−𝜆)||).

B(�)

n(x) = B

(�)

n(x;1), E(�)

n(x) = E

(�)

n(x;1) and G(�)

n(x) = G

(�)

n(x;1).

(1.14)

⎛⎜⎜⎜⎜⎝

tm

et −m−1∑h=0

th

h!

⎞⎟⎟⎟⎟⎠

�

ext =

∞�n=0

B[�,m−1]n (x)

tn

n!.

(1.15)

⎛⎜⎜⎜⎜⎝

tm

�et −m−1∑h=0

th

h!

⎞⎟⎟⎟⎟⎠

�

ext =

∞�n=0

B[�,m−1]n (x;�)

tn

n!.

(1.16)

⎛⎜⎜⎜⎜⎝

2m

et +m−1∑h=0

th

h!

⎞⎟⎟⎟⎟⎠

�

ext =

∞�n=0

E[�,m−1]n (x)

tn

n!.

Page 5 of 17


It is easy to see that setting m = 1 in (1.17), we have E[�,0]n (x;�) = E(�)n (x;�). From (1.17) we have

Definition 5 For arbitrary real or complex parameter � and natural number m ∈ ℕ, the generalized Genocchi polynomials G[�,m−1]

n (x) are defined in centered at t = 0, with |t| < 𝜋, by means of the gen-erating function:

Obviously, setting m = 1 in (1.19), we have G[�,0]n (x;1) = G(�)

n (x).

Definition 6 For arbitrary real or complex parameter � and �, and the natural number m, the gener-alized Genocchi polynomials G[�,m−1]

n (x) are defined in centered at t = 0, with |t| < ||log (−𝜆)||, by means of the generating function

It is easy to see that setting m = 1 in (1.20), we have G[�,0]n (x;�) = G(�)

n (x;�).

In this paper, we introduce a new class of generalized Apostol-type polynomials, a countable set of polynomials LY

(�,m)

n,�(x, y;k, a, b) generalizing Apostol-type Laguerre-Bernoulli, Apostol-type

Laguerre-Euler and Apostol-type Laguerre-Genocchi polynomials and Laguerre polynomials of 2-variables Ln(x, y) specified by the generating relation (1.2) and Mittag-Leffler function.

In this paper, we develop some elementary properties and derive the implicit summation formu-lae for these generalized polynomials by using different analytical means on their respective gener-ating functions.

2. A new class of Laguerre-based Apostol-type polynomialsRecently, Ozden (2010, 2011), Ozden, Simsek, and Srivastava (2010) and Ozarslan (2011, 2013) in-troduced the unification of the Apostol-type polynomials including Bernoulli, Euler and Genocchi polynomials Y (�)

n,�(x;k,a, b) of higher order � which are defined by

Ozarslan (2011) gave the following precise conditions of convergence of the series involved in (2.1):

(1.17)⎛⎜⎜⎜⎜⎝

2m

�et +m−1∑h=0

th

h!

⎞⎟⎟⎟⎟⎠

�

ext =

∞�n=0

E[�,m−1]n (x;�)

tn

n!.

(1.18)E[�,m−1]

0(x;�) =

(2m

� + 1

)�

.

(1.19)

⎛⎜⎜⎜⎜⎝

2mtm

et +m−1∑h=0

th

h!

⎞⎟⎟⎟⎟⎠

�

ext =

∞�n=0

G[�,m−1]n (x)

tn

n!.

(1.20)

⎛⎜⎜⎜⎜⎝

2mtm

�et +m−1∑h=0

th

h!

⎞⎟⎟⎟⎟⎠

�

ext =

∞�n=0

G[�,m−1]n (x;�)

tn

n!.

(2.1)

(21−ktk

𝛽bet − ab

)𝛼

ext =

∞∑n=0

Y(𝛼)

n,𝛽(x;k,a, b)

tn

n!(|||||t + b log

(𝛽

a

)|||||< 2𝜋, x ∈ ℝ; 1𝛼 : = 1; k ∈ ℕ0;a, b ∈ ℝ ⧵ {0}; 𝛼, 𝛽 ∈ ℂ

).

Page 6 of 17


(i) if ab > 0 and k ∈ ℕ, then ||||t + b log

(𝛽

a

)|||| < 2𝜋, x ∈ ℝ; 1𝛼 : = 1; 𝛼, 𝛽 ∈ ℂ

(ii) if ab > 0 and k = 0, then 0 < Im(t + b log

(𝛽

a

))< 2𝜋, x ∈ ℝ; 1𝛼 : = 1; 𝛼, 𝛽 ∈ ℂ

(iii) if ab < 0 and k ∈ ℕ0, then ||||t + b log

(𝛽

a

)|||| < 𝜋, x ∈ ℝ; 1𝛼 : = 1; 𝛼, 𝛽 ∈ ℂ

Definition 7 The generalized Laguerre-based Apostol-type Bernoulli, Laguerre-based Apostol-type Euler and Laguerre-based Apostol-type Genocchi polynomials LY

(�,m)

n,�(x, y;k, a, b), m ≥ 1 for a real or

complex parameter � defined in a suitable neighborhood of t = 0 by means of the following generat-ing function

so that

For x = 0 in Equation (2.2), the result reduces to known result of Ozden (2010); 2011 and Ozden et al. (2010).

For k = a = b = 1 and � = � in (2.2), we state the following definition.

Definition 8 Let � and � be arbitrary real or complex parameters. The generalized Laguerre Apostol-type Bernoulli polynomials are defined by

At the value m = 1 in the Equation (2.3), the result reduces to the known result of Khan et al. (2010):

Setting k + 1 = −a = b = 1 and � = � in (2.2), we define the following.

Definition 9 Let � and � be arbitrary real or complex parameters. The generalized Laguerre Apostol-type Euler polynomials are defined by

For x = 0 in the Equation (2.6), Further taking m = 1, the result reduces to the known result of Khan et al. (2010):

(2.2)

⎛⎜⎜⎜⎜⎝

21−ktk

�bet − ab

m−1∑h=0

th

h!

⎞⎟⎟⎟⎟⎠

�

exp(yt)J0

�2√xt�=

∞�n=0L

Y (�,m)

n,�(x, y;k, a, b)

tn

n!

LY(�,m)

n,�(x, y;k, a, b) =

n∑r=0

(n

r

)Y (�,m)

n−r,�(k,a, b)yrLr

(x

y

).

(2.3)

⎛⎜⎜⎜⎜⎝

t

�et −m−1∑h=0

th

h!

⎞⎟⎟⎟⎟⎠

�

exp(yt)J0

�2√xt�=

∞�n=0

LBn[�,m−1]

(x, y;�)tn

n!.

(2.4)

�t

𝜆et − 1

�𝛼

exp(yt)J0

�2√xt�=

∞�n=0

LBn(𝛼)(x, y;𝜆)

tn

n!

(�t� < 2𝜋 when 𝜆 = 1; �t� < ��log(−𝜆)�� when 𝜆 ≠ 1.

(2.5)

⎛⎜⎜⎜⎜⎝

2

�et +m−1∑h=0

th

h!

⎞⎟⎟⎟⎟⎠

�

exp(yt)J0

�2√xt�=

∞�n=0

LEn[�,m−1]

(x, y;�)tn

n!.

Page 7 of 17


Setting k + 1 = −2a = b = 1 and 2� = � in (2.2), we define the following.

Definition 10 Let � and � be arbitrary real or complex parameters. The generalized Laguerre Apos-tol-type Genocchi polynomials are introduced by

For x = 0 in Equation (2.6), Further taking �,m, � = 1, the result reduces to the known result of Khan et al. (2010):

Definition 11 The generalized Laguerre-based Apostol-type Hermite-Bernoulli, Laguerre-based Apostol-type Hermite-Euler and Laguerre-based Apostol-type Hermite-Genocchi polynomials

L�n(x, y, z), for a real or complex parameter � defined in a suitable neighborhood of t = 0 by means of the following generating function:

where �n(x, y, z) =H Y(�,m)

n,�(x, y, z;k, a, b) contain as its special cases both generalized Apostol-type

polynomials (2.1), Y (�,m)

n,�(x;k, a, b) , (1.15) to (1.20) and Kampé de Fériet generalization of the Hermite

polynomials Hn(x, y) (cf. Equation (1.4)).

By substituting x = y = z = 0 in (2.9), we obtain the corresponding unification of the generalized Apostol-type Bernoulli, Apostol-type Euler and Apostol-type Genocchi numbers Y (�,m)

n,�(k,a, b) (m ≥ 1)

are defined for a real or complex parameter � by means of the generating function

Then by (2.9) and (1.7), we have the representation

For � = 0, in Equation (2.9), the result reduces to Equation (1.7).

Setting x = 0, m = 1 and replacing y by x and z by y, respectively, in (2.9), we get a recent result of Pathan and Khan (2015). For k = � = a = b = 1, x = 0 and replacing y by x and z by y, respectively,

(2.6)�

2

𝜆et + 1

�𝛼

exp(yt)J0

�2√xt�=

∞�n=0

LEn(𝛼)(x, y;𝜆)

tn

n!

(�t� < 𝜋, when 𝜆 = 1; �t� < ��log(−𝜆)��, when 𝜆 ≠ 1).

(2.7)

⎛⎜⎜⎜⎜⎝

2t

�et +m−1∑h=0

th

h!

⎞⎟⎟⎟⎟⎠

�

exp(yt)J0

�2√xt�=

∞�n=0

LGn[�,m−1]

(x, y;�)tn

n!.

(2.8)�

2t

𝜆et + 1

�exp(yt)J0

�2√xt�=

∞�n=0

LGn(x, y;𝜆)tn

n!

(�t� < 𝜋, when𝜆 = 1; �t� < ��log(−𝜆)��, when 𝜆 ≠ 1).

(2.9)

⎛⎜⎜⎜⎜⎝

21−ktk

�bet − ab

m−1∑h=0

th

h!

⎞⎟⎟⎟⎟⎠

�

eyt+zt2

J0

�2√xt�=

∞�n=0

L�n(x, y, z)tn

n!

(2.10)

⎛⎜⎜⎜⎜⎝

21−ktk

�bet − ab

m−1∑h=0

th

h!

⎞⎟⎟⎟⎟⎠

�

=

∞�n=0

Y (�,m)

n,�(k, a, b)

tn

n!.

L�n(x, y, z;k, a, b) =

n∑r=0

(n

r

)Y (�,m)

n−r,�(k,a, b)LHr(x, y, z).

Page 8 of 17


in (2.9), the result reduces to the known result of Pathan and Khan (2015). Further if � = 1 the result reduces to known result of Pathan (2012]:

Besides by (2.10), we can also obtain the generalized Hermite-Euler polynomials E(�)n (x, y) and the generalized Hermite-Genocchi polynomials G(�)

n (x, y) each of order � and degree n, respectively, de-fined by the following generating functions

and

It may be seen that for y = 0, (2.11) to (2.13) are, respectively, the generalizations of (1.8) to (1.10).

We continue with another basic example of (2.9) by taking m, k = 2 and � = 1. Thus we have

where Φn(x, y) =H Y(1,2)

n,�(x, y, z;2, a, b). We have

Replace n by n − p + 2, p ≤ n − 2

This formula gives a representation of LHn(x, y, z) in terms of sums of Φ. This is the key to the next conclusion for finding another representation of LHn(x, y, z) in terms of sums of Ψ where Ψn(x, y, z) = HYn,�

(1,1)(x, y, z;1, a, b). For this taking � = m = k = 1 in (2.9), we have

Comparing the coefficients of tn, we have

When investigating the connection between Hermite polynomials LHn(x, y, z) and generalized Apostol-type polynomials L�n(x, y, z), the following theorem is of great importance.

(2.11)

(t

et − 1

)ext+yt

2

=

∞∑n=0

HBn(x, y)tn

n!.

(2

et + 1

)𝛼

ext+yt2

=

∞∑n=0

HEn(𝛼)(x, y)

tn

n!(|t| < 𝜋; 1

𝛼 = 1)

(2t

et + 1

)𝛼

ext+yt2

=

∞∑n=0

HGn(𝛼)(x, y)

tn

n!(|t| < 𝜋; 1

𝛼 = 1).

2−1

(t2

�bet − ab(1 + t)

) ∞∑n=0

LHn(x, y, z)tn

n!=

∞∑n=0

LΦn(x, y, z)tn

n!

2−1

∞∑n=0

LHn(x, y, z)tn

n!

= �b

∞∑p=0

tp

p!

∞∑n=0

LΦn(x, y, z)tn−2

n!− ab

∞∑n=0

LΦn(x, y, z)tn−2

n!− ab

∞∑n=0

LΦn(x, y, z)tn−1

n!.

1

2LHn(x, y, z)

n!= �

b

n−2∑p=0

LΦn−p+2(x, y, z)

p!(n − p + 2)!− ab

(LΦn+2(x, y, z)

(n + 2)!−

LΦn+1(x, y)

(n + 1)!

).

�t

�bet − ab

�eyt+zt

2

J0

�2√xt�=

∞�n=0

LΨn(x, y, z)tn

n!

∞�n=0

LHn(x, y, z)tn

n!=1

t

��bet − ab

� ∞�n=0

LΨn(x, y, z)tn

n!

LHn(x, y, z)

n!= �

b

n+1∑p=0

LΨn−p+1(x, y, z)

p!(n − p + 1)!− ab

(LΨn+1(x, y, z)

(n + 1)!

).

(2.12)

(2.13)

Page 9 of 17


Theorem 1 The following holds true

where �n(x, y, z) =H Y(1,1)

n+k,�(x, y, z;k, a, b).

Proof We begin with the definition 11 and write

Then using the definition of Kampé de Fériet generalization of the Laguerre-Hermite polynomials

LHn(x, y) and (2.9), we have

Finally, comparing the coefficients of tn, we complete the proof of the theorem. ✷

For m = k = 1 in Theorem 1 yields the following result for LHn(x, y, z).

Corollary 1 The following formula holds:

where Ψn(x, y, z) =H Y(1,1)

n,�(x, y;1, a, b) .

Theorem 2 The following formula involving Laguerre-Apostol-type polynomials LY(�,m)

n (x, y;k, a, b) holds true:

Proof By Definition 7, we easily get the proof of the theorem. So we omit it. ✷

Theorem 3 The following formula involving Laguerre-Apostol-type polynomials LY(�,m)

n (x, y;k, a, b) holds true:

Proof The proof of this theorem follows from Definition 7. So we omit the proof. ✷

3. Implicit formulae involving Laguerre-based Apostol-type polynomialsThis section is devoted to employing the definition of the Laguere-based Apostol-type polynomials

LY(�,m)

n,�(x, y;k, a, b). First we prove the following results involving Laguerre-based Apostol-type

polynomials LY(�,m)

n,�(x, y;k, a, b).

Theorem 4 The following implicit summation formulae for Laguerre-based Apostol-type polynomials

LY(�,m)

n,�(x, y;k, a, b) holds true:

LHn(x, y) =n!

21−k(n + k)![�bL�n+k(x, y + 1, z;k, a, b) − a

bL�n+k(x, y, z;k, a, b)]

eyt+zt2

J0

�2√xt�=

1

21−ktk

�21−ktk

�bet − ab

�(�bet − ab)eyt+zt

2

J0(2

√xt)

=1

21−ktk

��21−ktk

�bet − ab

��be(y+1)t+zt

2

J0

�2√xt�− (

21−ktk

�bet − ab

)abeyt+zt2

J0

�2√xt��.

∞∑n=0L

Hn(x, y, z)tn

n!=

∞∑n=0

n!

21−k(n + k)![�bL�n+k(x, y + 1, z;k, a, b) − a

bL�n+k(x, y, z;k, a, b)]

tn

n!.

LHn(x, y, z) =1

(n + 1)[�b LΨn+1(x, y + 1, z;1, a, b) − a

bLΨn+1(x, y, z;1, a, b)]

LY(�+� ,m)

n,�(x, y + z;k, a, b) =

n∑r=0

(n

r

)Y (�,m)

n−r (y;k, a, b)LY(� ,m)

r (x, z;k, a, b).

LY(�,m)

n (x, y;k, a, b) =

n∑r=0

Y (�−1,m)

n−r,�(k,a, b)LY

(m)

r,�(x, y;k, a, b).

Page 10 of 17


Proof We replace t by t + u and rewrite the generating function (2.2) as

Replacing y by z in the above equation and equating the resulting equation to the above equation, we get

On expanding the exponential function in the above gives

which, on using series manipulation formula

in the left-hand side, becomes

Now replacing q by q − n, l by l − p and using the lemma (Srivastava & Manocha, 1984) in the left-hand side of (3.1), we get

Finally on equating the coefficients of the like powers of tq and ul in the above equation, we get the required result. ✷

For k = a = b = 1 and � = � in Theorem 4, we get the following corollary.

Corollary 2 The following implicit summation formulae for Laguerre-based Apostol-type Bernoulli polynomials LB

[�,m−1]n (x, y;�) holds true:

For k + 1 = −a = b = 1 and � = � in Theorem 4, we get the corollary.

LY(�,m)

q+l,�(x, z;k, a, b) =

q∑n=0

l∑p=0

(q

n

)(l

p

)(z − y)

n+p

LY (�,m)

q+l−p−n,�(x, y;k, a, b).

⎛⎜⎜⎜⎜⎝

21−k(t + u)k

�bet+u − ab

m−1∑h=0

(t+u)h

h!

⎞⎟⎟⎟⎟⎠

�

J0(2

√x(t + u)) = e−y(t+u)

∞�q,l=0

LY(�,m)

q+l,�(x, y;k, a, b)

tq

q!

ul

l!.

e(z−y)(t+u)∞∑q,l=0

L�q+l,� (x, y;k, a, b)tq

q!

ul

l!=

∞∑q,l=0

LY(�,m)

q+l,�(x, z;k, a, b)

tq

q!

ul

l!.

∞∑N=0

[(z − y)(t + u)]N

N!

∞∑q,l=0

LY(�,m)

q+l,�(x, y;k, a, b)

tq

q!

ul

l!=

∞∑q,l=0

LY(�,m)

q+l,�(x, z;k, a, b)

tq

q!

ul

l!

∞∑N=0

f (N)(x + y)N

N!=

∞∑n,m=0

f (n +m)xn

n!

ym

m!

(3.1)∞∑

n,p=0

(z − y)n+ptnup

n!p!

∞∑q,l=0

LY(�,m)

q+l,�(x, y;k, a, b)

tq

q!

ul

l!=

∞∑q,l=0

LY(�,m)

q+l,�(x, z;k, a, b)

tq

q!

ul

l!.

∞∑q,l=0

q,l∑n,p=0

(z − y)n+p

n!p! LY(�,m)

q+l−n−p,�(x, y;k, a, b)

tq

(q − n)!

ul

(l − p)!

=

∞∑q,l=0

LY(�,m)

q+l,�(x, z;k, a, b)

tq

q!

ul

l!.

LB[�,m−1]

q+l(x, z;�) =

q∑n=0

l∑p=0

(q

n

)(l

p

)(z − y)

n+p

LB[�,m−1]

q+l−p−n(x, y;�).

Page 11 of 17


Corollary 3 The following implicit summation formulae for Laguerre-based Apostol-type Euler polynomials LE


Letting k = −2a = b = 1 and 2� = � in Theorem 4, we get the corollary.

Corollary 4 The following implicit summation formulae for Laguerre-based Apostol-type Genoc-chi polynomials LG


Theorem 5 The following implicit summation formula involving Laguerre-based Apostol-type polynomials LY

(�,m)


Proof When we replace y by y + z in (2.2), use (1.2) and rewrite the generating function, we conclude the proof of this theorem. ✷





Corollary 6 The following implicit summation formulae for Laguerre-based Apostol-type Euler polynomials HE

(�,m)

n (x, y;�) holds true:




Theorem 6 The following implicit summation formulae for Laguerre-based Apostol-type polynomials LY

(�,m)


LE(�,m)

q+l(x, z;�) =

q∑n=0

l∑p=0

(q

n

)(l

p

)(z − y)n+p LE

[�,m−1]

q+l−p−n(x, y;�).

LG[�,m−1]

q+l(x, z;�) =

q∑n=0

l∑p=0

(q

n

)(l

p

)(z − y)n+p LG

[�,m−1]

q+l−p−n(x, y;�).

LY(�,m)

n,�(x, y + z;k, a, b) =

n∑s=0

(n

s

)Y (�,m)

n−s,�(z;k, a, b)ysLs

(x

y

).

LB(�,m)

n (x, y + z;�) =

n∑s=0

(n

s

)B[�,m−1]n−s (z;�)ysLs

(x

y

).

LE[�,m−1]n (x, y + z;�) =

n∑s=0

(n

s

)E[�,m−1]n−s (z;�)ysLs

(x

y

).

LG[�,m−1]n (x, y + z;�) =

n∑s=0

(n

s

)G[�,m−1]n−s (z;�)ysLs

(x

y

).

LY(�,m)

n,�(x, y;k, a, b) =

n∑r=0

(n

r

)Y (�,m)

n−r,�(y − u;k, a, b)xrLr

(x

u

).

Page 12 of 17


Proof By exploiting the generating function (1.2), we can write Equation (2.2) as

Now replacing n by n − r in the right-hand side and using the lemma (Srivastava & Manocha, 1984) in the right-hand side of Equation (3.2), we complete the proof of the theorem. ✷










Theorem 7 The following implicit summation formulae for Laguerre-based Apostol-type polynomi-als LY

(�,m)


Proof By using the generating function (2.2), it is easy to prove this theorem. ✷





(3.2)

⎛⎜⎜⎜⎜⎝

21−ktk

�bet − ab

m−1∑h=0

th

h!

⎞⎟⎟⎟⎟⎠

�

e(y−u)teutJ0

�2√xt�=

∞�n=0

Y (�,m)

n,�(y − u;k, a, b)

tn

n!

∞�r=0

xrLr(x

u)tr

r!.

LB[�,m−1]n (x, y;�) =

n∑r=0

(n

r

)B[�,m−1]n−r (y − u;�)xrLr

(x

u

).

LE[�,m−1]n (x, y;�) =

n∑r=0

(n

r

)E[�,m−1]n−r (y − u;�)xrLr

(x

u

).

LG[�,m−1]n (x, y;�) =

n∑r=0

(n

r

)G[�,m−1]n−r (y − u;�)xrLr

(x

u

).

LY(�,m)

n,�(x, y + 1;k, a, b) =

n∑r=0

(n

r

)

L

Y (�,m)

n−r,�(x, y;k, a, b).

LB[�,m−1]

n(x, y + 1;�) =

n∑r=0

(n

r

)LB[�,m−1]

n−r(x, y;�).

Page 13 of 17







4. Symmetry identities for the Laguerre-based Apostol-type polynomialsIn this section, we give general symmetry identities for the generalized Laguerre-based Apostol-type polynomials LY

(�,m)

n,�(x, y;k, a, b) by applying the generating function (2.1) and (2.2).

Theorem 8 The following identity holds true:

Proof We start to prove by the following expression:

Then the expression for g(t) is symmetric in a and b and we can expand g(t) into series in two ways to obtain

On similar lines we can show that

by comparing the coefficients of tn on the right-hand sides of the last two equations we arrive at the desired result. ✷


LE[�,m−1]

n(x, y + 1;�) =

n∑r=0

(n

r

)LE[�,m−1]

n−r(x, y;�).

LG[�,m−1]

n(x, y + 1;�) =

n∑r=0

(n

r

)LG[�,m−1]

n−r(x, y;�).

n∑r=0

(n

r

)drcn−r LY

(�,m)

n−r,�(dx,dy;k, a, b) LY

(�,m)

r,�(cw, cz;k, a, b)

=

n∑r=0

(n

r

)crdn−r LY

(�,m)

n−r,�(cx, cy;k, a, b) LY

(�,m)

r,�(dw,dz;k, a, b).

g(t) =

⎛⎜⎜⎜⎜⎜⎝

ckdk22(1−k)t2k��bect − ab

m−1∑h=0

th

h!

��bedt − ab

m−1∑h=0

th

h!

�

⎞⎟⎟⎟⎟⎟⎠

�

exp(cd(x + z)t)J0

�2√cdyt

�J0

�2√cdwt

�.

g(t) =

∞∑n=0L

Y (�,m)

n,�(dx,dy;k, a, b)

(ct)n

n!

∞∑r=0 L

Y (�,m)


(dt)r

r!

=

∞∑n=0

n∑r=0 L

Y (�,m)

n−r,�(dx,dy;k, a, b)

(c)n−r

(n − r)! LY (�,m)


(d)r

r!tn.

g(t) =

∞∑n=0L

Y (�,m)

n,�(cx, cy;k, a, b)

(dt)n

n!

∞∑r=0 L

Y (�,m)

r,�(dw,dz;k, a, b)

(ct)r

r!

=

∞∑n=0

n∑r=0 L

Y (�,m)

n−r,�(cx, cy;k, a, b)

(d)n−r

(n − r)! LY (�,m)

r,�(dw,dz;k, a, b)

(c)r

r!tn

Page 14 of 17


Corollary 14 We have the following symmetry identity for the Laguerre-based generalized Apostol-Bernoulli polynomials


Corollary 15 We have for each pair of positive even integers c and d or for each pair of positive odd integers c and d.




Proof Let

From this formula and using same technique as in Theorem 9, we arrive at the desired result. ✷


Corollary 17 We have the following symmetry identity for the Laguerre-based generalized Apos-tol-Bernoulli polynomials

n∑r=0

(n

r

)drcn−r LB

[�,m−1]

n−r (dx,dy;�) LB[�,m−1]

r (cw, cz;�)

=

n∑r=0

(n

r

)crdn−r LB

[�,m−1]

n−r (cx, cy;�) LB[�,m−1]

r (dw,dz;�).

n∑r=0

(n

r

)drcn−r LE

[�,m−1]

n−r (dx,dy;�) LE[�,m−1]

r (cw, cz;�)

=

n∑r=0

(n

r

)crdn−r LE

[�,m−1]

n−r (cx, cy;�) LE[�,m−1]

r (dw,dz;�).

n∑r=0

(n

r

)drcn−r LG

[�,m−1]

n−r (dx,dy;�) LG[�,m−1]

r (cw, cz;�)

=

n∑r=0

(n

r

)crdn−r LG

[�,m−1]

n−r (cx, cy;�) LG[�,m−1]

r (dw,dz;�).

n∑r=0

(n

r

) c−1∑i=0

d−1∑j=0

cn−rdr LY(�,m)

n−r,�

(dx +

d

ci + j,dy;k, a, b

)LY

(�,m)


=

n∑r=0

(n

r

) d−1∑i=0

c−1∑j=0

crdn−r LY(�,m)

n−r,�

(cx +

c

di + j, cy;k, a, b

)LY

(�,m)

r,�(dw,dz;k, a, b).

g(t) =

⎛⎜⎜⎜⎜⎜⎝

22(1−k)ckdkt2k��bect − ab

m−1∑h=0

th

h!

��bedt − ab

m−1∑h=0

th

h!

�

⎞⎟⎟⎟⎟⎟⎠

�

(ecdt − 1)2

(ect − 1)(edt − 1)exp(cd(x + z)t)J0

�2√cdyt

�J0

�2√cdwt

�.

Page 15 of 17



Corollary 18 We have for each pair of positive even integers and d or for each pair of positive odd integers c and d




Proof The proof is analogous to Theorem 9. So we omit the proof of this theorem. ✷


Corollary 20 We have the following symmetry identity for the Laguerre-based generalized Apos-tol-Bernoulli polynomials

For k + 1 = −a = b = 1 and � = � in Theorem 10, we state the following corollary.

Corollary 21 We have for each pair of positive even integers c and d or for each pair of positive odd integers c and d

n∑r=0

(n

r

) c−1∑i=0

d−1∑j=0

cn−rdr LB[�,m−1]

n−r

(dx +

d

ci + j,dy;�

)LB

[�,m−1]

r (cw, cz;�)

=

n∑r=0

(n

r

) d−1∑i=0

c−1∑j=0

crdn−r LB[�,m−1]

n−r

(cx +

c

di + j, cy;�

)LB

[�,m−1]

r (dw,dz;�).

n∑r=0

(n

r

) c−1∑i=0

d−1∑j=0

cn−rdr LE[�,m−1]

n−r

(dx +

d

ci + j,dy;�

)LE

[�,m−1]

r (cw, cz;�)

=

n∑r=0

(n

r

) d−1∑i=0

c−1∑j=0

crdn−r LE[�,m−1]

n−r

(cx +

c

di + j, cy;�

)LE

[�,m−1]

r (dw,dz;�).

n∑r=0

(n

r

)c−1∑i=0

d−1∑j=0

cn−rdrLG[�,m−1]n−r

(dx +

d

ci + j,dy;�

)

L

G[�,m−1]r (cw, cz;�)

=

n∑r=0

(n

r

)d−1∑i=0

c−1∑j=0

crdn−rL G[�,m−1]n−r

(cx +

c

di + j, cy;�

)LG[�,m−1]r (dw,dz;�).

n∑r=0

(n

r

) c−1∑i=0

d−1∑j=0

cn−rdr LY(�,m)

n−r,�

(dx +

d

ci,dy;k, a, b

)LY

(�,m)

r,�(cw +

c

dj, cz;k, a, b)

=

n∑r=0

(n

r

) d−1∑i=0

c−1∑j=0

crdn−r LY(�,m)

n−r,�

(cx +

c

di, cy;k, a, b

)LY

(�,m)

r,�(dw +

d

cj,dz;k, a, b).

n∑r=0

(n

r

) c−1∑i=0

d−1∑j=0

cn−rdr LB[�,m−1]

n−r

(dx +

d

ci,dy;�

)LB

[�,m−1]

r (cw +c

dj, cz;�)

=

n∑r=0

(n

r

) d−1∑i=0

c−1∑j=0

crdn−r LB[�,m−1]

n−r

(cx +

c

di, cy;�

)LB

[�,m−1]

r (dw +d

cj,dz;�).

Page 16 of 17



Corollary 22 We have for each pair of positive even integers c and d or for each pair of positive odd integers c and d

5. ConclusionIn Section 2, we have established the generating functions for the Laguerre-based Apostol-type polynomials and Laguerre-based Apostol-type Hermite polynomials by making use of Tricomi func-tion of the generating function for Laguerre polynomials. The equivalent forms of these generating functions can be derived by using Equations (1.1), (1.6) and (2.1). They can be viewed as the equiva-lent forms of the generating functions (2.3), (2.6) and (2.8), respectively. In the previous sections, we have used the concepts and the formalism associated with Laguerre polynomials to introduce the Laguerre-based Apostol-type polynomials and Laguerre-based Apostol-type Hermite polynomials and establish their properties. The approach presented here is general and we have established the summation rules, which can be used to derive the results for Laguerre-based Apostol-type polyno-mials from the results of the corresponding Appell polynomials.

n∑m=0

(n

r

) c−1∑i=0

d−1∑j=0

cn−rdr LE[�,m−1]

n−r

(dx +

d

ci,dy;�

)LE

[�,m−1]

r (cw +c

dj, cz;�)

=

n∑r=0

(n

r

) d−1∑i=0

c−1∑j=0

crdn−r LE[�,m−1]

n−r

(cx +

c

di, cy;�

)LE

[�,m−1]

r (dw +d

cj,dz;�).

n∑r=0

(n

r

) c−1∑i=0

d−1∑j=0

cn−rdr LG[�,m−1]

n−r

(dx +

d

ci,dy;�

)LG

[�,m−1]

r (cw +c

dj, cz;�)

=

n∑r=0

(n

r

) d−1∑i=0

c−1∑j=0

crdn−r LG[�,m−1]

n−r

(cx +

c

di, cy;�

)G[�,m−1]

r (dw +d

cj,dz;�).

FundingThe authors received no direct funding for this research.

Author detailsWaseem A. Khan1

E-mail: [email protected] Araci2

E-mail: [email protected] Acikgoz3

E-mail: [email protected] Department of Mathematics, Integral University, Lucknow

226026, India.2 Faculty of Economics, Department of Economics,

Administrative and Social Sciences, Hasan Kalyoncu University, TR 27410 Gaziantep, Turkey.

3 Faculty of Arts and Science, Department of Mathematics, University of Gaziantep, TR 27310 Gaziantep, Turkey.

Citation informationCite this article as: A new class of Laguerre-based Apostol type polynomials, Waseem A. Khan, Serkan Araci & Mehmet Acikgoz, Cogent Mathematics (2016), 3: 1243839.

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http://dx.doi.org/10.1186/1687-1847-213-116

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