A new class of Laguerre-based Apostol type polynomials

ABOUT THE AUTHOR Serkan 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 STATEMENT In 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 2016 Accepted: 27 September 2016 First Published: 04 October 2016


Introduction
Throughout the paper, we make use of the following notations: and ℕ: = 1, 2, 3, … , ℕ 0 : = 0, 1, 2, 3, … = ℕ ∪ 0 ℤ − : = −1, −2, −3, … = ℤ − 0 � 0 . 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 STATEMENT
In 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. ∑ k=0 z k y n−2k L n−2k x y k!(n − 2k)! .
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) 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 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 | | , 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). (1.14) (1.15) (1.16) It is easy to see that setting m = n (x) are defined in centered at t = 0, with |t| < , by means of the generating function: Definition 6 For arbitrary real or complex parameter and , and the natural number m, the gener- In this paper, we introduce a new class of generalized Apostol-type polynomials, a countable set Laguerre-Euler and Apostol-type Laguerre-Genocchi polynomials and Laguerre polynomials of 2-variables L n (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 formulae for these generalized polynomials by using different analytical means on their respective generating functions.

A new class of Laguerre-based Apostol-type polynomials
Recently, Ozden (2010), Ozden, Simsek, and Srivastava (2010 and Ozarslan (2011Ozarslan ( , 2013 introduced 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) (1.20) (2.1) The generalized Laguerre-based Apostol-type Bernoulli, Laguerre-based Apostol-type Euler and Laguerre-based Apostol-type Genocchi polynomials L Y ( ,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 generating function so that 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 Apostol-type Genocchi polynomials are introduced by 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: 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 H n (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, 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, (2.7) (2.9) 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, defined 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 Φ n (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 L H n (x, y, z) in terms of sums of Φ. This is the key to the next conclusion for finding another representation of L H n (x, y, z) in terms of sums of Ψ where Ψ n (x, y, z) = H Y n, (1,1) (x, y, z;1, a, b). For this taking = m = k = 1 in (2.9), we have Comparing the coefficients of t n , we have When investigating the connection between Hermite polynomials L H n (x, y, z) and generalized Apostol-type polynomials L n (x, y, z), the following theorem is of great importance.
(2.11)    (x, y;k, a, b) Proof We replace t by t + u and rewrite the generating function (2.2) as

Implicit formulae involving Laguerre-based Apostol-type polynomials
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 t q and u l 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 L B [ ,m−1] n (x, y; ) holds true: For k + 1 = −a = b = 1 and = in Theorem 4, we get the corollary. (x, y;k, a, b). 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. ✷ For k = a = b = 1 and = in Theorem 5, we get the following corollary.
Corollary 7 The following implicit summation formulae for Laguerre-based Apostol-type Genocchi polynomials L G [ ,m−1] n (x, y; ) holds true: Theorem 6 The following implicit summation formulae for Laguerre-based Apostol-type polynomials L Y ( ,m) n, (x, y;k, a, b) holds true: , y; ). 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. ✷ For k = a = b = 1 and = in Theorem 6, we get the following corollary.
Corollary 10 The following implicit summation formulae for Laguerre-based Apostol-type Genocchi polynomials L G [ ,m−1] n (x, y; ) holds true: Theorem 7 The following implicit summation formulae for Laguerre-based Apostol-type polynomials L Y ( ,m) n, (x, y;k, a, b) holds true: Proof By using the generating function (2.2), it is easy to prove this theorem. ✷ For k = a = b = 1 and = in Theorem 7, we get the following corollary.

Symmetry identities for the Laguerre-based Apostol-type polynomials
In this section, we give general symmetry identities for the generalized Laguerre-based Apostoltype polynomials L Y ( ,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 t n on the right-hand sides of the last two equations we arrive at the desired result. ✷ For k = a = b = 1 and = in Theorem 8, we get the following corollary.
Corollary 14 We have the following symmetry identity for the Laguerre-based generalized Apostol-Bernoulli polynomials For k + 1 = −a = b = 1 and = in Theorem 8, we get the corollary.
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.
Corollary 16 We have for each pair of positive even integers c and d or for each pair of positive odd integers c and d.
Theorem 9 The following identity holds true:

Proof Let
From this formula and using same technique as in Theorem 9, we arrive at the desired result. ✷ For k = a = b = 1 and = in Theorem 8, we get the following corollary.  (dw, dz;k, a, b).
For k + 1 = −a = b = 1 and = in Theorem 9, we get the corollary.
Corollary 18 We have for each pair of positive even integers and d or for each pair of positive odd integers c and d Letting k = −2a = b = 1 and 2 = in Theorem 9, we get the corollary.

Corollary 19
We have for each pair of positive even integers c and d or for each pair of positive odd integers c and d.
Theorem 10 The following identity holds true: Proof The proof is analogous to Theorem 9. So we omit the proof of this theorem. ✷ For k = a = b = 1 and = in Theorem 10, we get the following corollary.
Corollary 20 We have the following symmetry identity for the Laguerre-based generalized Apostol-Bernoulli polynomials For k + 1 = −a = b = 1 and = in Theorem 10, we state the following corollary.