A Note on the Truncated-Exponential Based Apostol-Type Polynomials

In this paper, we propose to investigate the truncated-exponential-based Apostol-type polynomials and derive their various properties. In particular, we establish the operational correspondence between this new family of polynomials and the familiar Apostol-type polynomials. We also obtain some implicit summation formulas and symmetric identities by using their generating functions. The results, which we have derived here, provide generalizations of the corresponding known formulas including identities involving generalized Hermite-Bernoulli polynomials.


Introduction
Operational techniques involving differential operators, which is a consequence of the monomiality principle, provide efficient tools in the theory of conventional polynomial systems and their various generalizations.Steffensen [1] suggested the concept of poweroid, which happens to be behind the idea of monomiality.The principle of monomiality was subsequently reformulated and developed by Dattoli [2].The strategy underlining this viewpoint is apparently simple, but the outcomes are remarkably deep.
In the theory of the monomiality principle, a polynomial set p n (x) (n ∈ N; x ∈ C) is quasi-monomial if there exist two operators M and P, which are named the multiplicative and the derivative operators, respectively, are defined as follows: M{p n (x)} = p n+1 (x) and P{p n (x)} = np n−1 (x), together with the initial condition given by The operators M and P satisfy the following commutation relation: Thus, clearly, these operators display a Weyl group structure.The properties of the polynomials p n (x) can be deduced from those of the operators M and P. If M and P possess a differential character, then the polynomials p n (x) satisfy the following differential equation: M P{p n (x)} = np n (x).
The polynomial family p n (x) can be explicitly constructed through the action of M n on p 0 (x) as follows: p n (x) = M n {p 0 (x)}.
Just as in (1), we shall always assume that p 0 (x) = 1.In view of the above identity (4), the exponential generating function of p n (x) can be written in the form: We now introduce the truncated-exponential polynomials e n (x) (see [3]) defined by the following series: that is, by the first n + 1 terms of the Taylor-Maclaurin series for the exponential function e x .These truncated-exponential polynomials play an important rôle in many problems in optics and quantum mechanics.However, their properties are apparently as widespread as they should be.
The truncated-exponential polynomials e n (x) have been used to evaluate several overlapping integrals associated with the optical mode evolution or for characterizing the structure of the flattened beams.
Their usefulness has led to the possibility of appropriately extending their definition.Actually, Dattoli et al. [4] systematically studied the properties of these polynomials.
The definition (6) does lead us to most (if not all) of the properties of the polynomials e n (x).We note the following representation: which follows readily from the classical gamma-function representation (see, for details, [3]).Consequently, we have the following generating function for the truncated-exponential polynomials e n (x) (see [4]): The definition ( 6) of e n (x) can thus be extended to a family of potentially useful truncated-exponential polynomials as follows (see [4]): [2]e n (x) = which obviously possesses a generating function in the form (see [4]): We also recall the higher-order truncated-exponential polynomials [r]e n (x), which are defined by the following series (see [4]): [r]e n (x) = and specified by the following generating function (see [4]): The special two-variable case of the polynomials in (11) (that is, the case when r = 2) are important for applications.Moreover, these polynomials help us derive several potentially useful identities in a simple way and in investigating other novel families of polynomial systems.Actually, Equation (12) enables us to give a new family of polynomials as has been given in Theorem 1.
A 2-variable extension of the truncated-exponential polynomials is given by (see [4]) and possesses the following generating function (see [4]): With a view to introducing a mixed family of polynomials related to the familiar Sheffer sequence, we first consider the 2-variable truncated-exponential polynomials (2VTEP) e n (x, y) of order r, which are expressed explicitly by (see [5]) and which are generated by From ( 8), ( 10), ( 12), ( 14) and ( 16), we can deduce several special cases of the 2VTEP e (r) n (x, y), For example, we have and e (1) As it is shown in [6,7], the 2VTEP e (r) n (x, y) are quasi-monomial (see also [1,2]) with respect to multiplicative and derivative operators given by M e (r) = (x + ry∂ y y∂ r−1 x ) and where Thus, if we apply the monomiality principle as well as the Equations ( 18) and ( 19), we have and The 2VTEP e (r) n (x, y) are quasi-monomial, so their properties can be derived from those of the multiplicative and derivative operators M e (r) and P e (r) , respectively.We thus find that M e (r) P e (r) {e (r) which satisfies a differential equation for e (r) n (x, y) as follows: Again, since e (r) n (x, y) can be explicitly constructed as follows: Equation ( 24) yields the following generating function of the 2VTEP e (r) n (x, y): We can easily verify the following relation between M e (r) and P e (r) : [ P e (r) , M e (r) ] = 1.
Denoting the classical Bernoulli, Euler and Genocchi polynomials by B n (x), E n (x) and G n (x), respectively, we now recall their familiar generalizations n (x) of order α, which are generated by (see, for details, [8][9][10][11][12][13][14]; see also [15] as well as the references cited therein): and Obviously, we have n (x) =: E n (x) and G (1) It is also known that n (0) =: E n and G (1) for the Bernoulli, Euler, and Genocchi numbers B n , E n and G n , respectively.

Definition 3. The Apostol-Genocchi polynomials G
(α) n (x) of order α are defined by (|t| with G (α) where G n (x; λ) should obviously be constrained to take on nonnegative integer values (see, for details, [14]).A similar remark would apply also to the order α in all other analogous situations considered in this paper.
Our main objective in this article is to first appropriately combine the 2-variable truncated-exponential polynomials and the Apostol-type polynomials by means of operational techniques.This leads us to the truncated-exponential-based Apostol-type polynomials.By framing these polynomials within the context of the monomiality principle, we then establish their potentially useful properties.We also derive some other properties and investigate several implicit summation formulas for this general family of polynomials by making use of several different analytical techniques on their generating functions.We choose to point out some relevant connections between the truncated-exponential polynomials and the Apostol-type polynomials and thereby derive extensions of several symmetric identities.

Two-Variable Truncated-Exponential-Based Apostol-Type Polynomials
We now start with the following theorem arising from the generating functions for the truncated-exponential-based Apostol-type polynomials (TEATP), which are denoted by Theorem 1.The generating function for the 2-variable truncated-exponential-based Apostol-type polynomials Proof.Replacing x in the left-hand side and the right-hand side of (39) by the multiplicative operator Using Equation ( 25) in the left-hand side and Equation ( 18) in the right-hand side of Equation ( 46), we see that Now, using Equation ( 16) in the left-hand side and denoting the resulting 2-variable truncated-exponential-based Apostol-type polynomials (2VTEATP) in the right-hand side by which yields the assertion (45) of Theorem 1.To frame the 2VTEATP e (r) Y n,β (x, y; k, a, b) within the context of monomiality principle, we state the following result. and Proof.Let us consider the following expression: Differentiating both sides of Equation (45) partially with respect to t, we see that possesses a power-series expansion in t.Thus, using (51), Equation (52) becomes x + ry∂ y y∂ r−1 x Again, by using the generating function (45) in left-hand side of Equation ( 53) and rearranging the resulting summation, we have Comparing the coefficients of t n n! in the Equation (54), we get which, in view of the monomiality principle exhibited in Equation ( 20) for e (r) Y (α) n,β (x, y; k, a, b), yields the assertion (49) of Theorem 2.
We now prove the assertion (50) of Theorem 2. For this purpose, we start with the following identity arising from Equations (45) and (51): Rearranging the summation in the left-hand side of Equation (56), and then equating the coefficients of the same powers of t in both sides of the resulting equation, we find that which, in view of the monomiality principle exhibited in Equation ( 21) for e (r) Y (α) n,β (x, y; k, a, b)), yields the assertion (50) of Theorem 2. Our demonstration of Theorem 2 is thus completed.
We note that the properties of quasi-monomials can be derived by means of the actions of the multiplicative and derivative operators.We derive the differential equation for the 2VTEATP Proof.Theorem 3 can be easily proved by combining (49) and (50) with the monomiality principle exhibited in (22).
Remark 3. When r = 2, the 2VTEP e (r) (x, y) of order r reduces to the 2VTEP [2] e n (x, y).Therefore, if we set r = 2 in Equation (45), we get the following generating function for the 2-variable truncated-exponential Apostol-type polynomials (2VTEATP) [2] e (r) Y (α) n,β (x, y; k, a, b) : The series definition and other results for the 2VTEATP [2] e (r) Y n,β (x, y; k, a, b) can be obtained by taking r = 2 in Theorems 1 and 2. Table 1 shown the special cases of the 2VTEATP .e (r) Y n (x, y; k, a, b).Remark 4. For the case y = 1, the polynomials [2] e n (x, 1) reduce to the truncated-exponential polynomials [2] e n (x).Therefore, by taking y = 1 in Equation (59), we get the following generating function for the truncated-exponential Apostol-type polynomials (TEATP) [2]   III.

Implicit Formulas Involving the 2-Variable Truncated-Exponential Based Apostol-Type Polynomials
In this section, we employ the definition of the 2-variable truncated-exponential-based Apostol-type polynomials e (r) Y (α) n,β (x, y; k, a, b) that help in proving the generalizations of the previous works of Khan et al. [33] and Pathan and Khan (see [34][35][36]).For the derivation of implicit formulas involving the 2-variable truncated-exponential-based Apostol-type polynomials e (r) Y (α) n,β (x, y; k, a, b), the same considerations as developed for the ordinary Hermite and related polynomials in the works by Khan et al. [33] and Pathan et al. (see [34][35][36]) apply as well.We first prove the following results involving the 2-variable truncated-exponential-based Apostol-type polynomials e (r) Y Proof.We replace t by t + u and rewrite (45) as follows: Replacing x by z in the Equation (62) and equating the resulting equation to the above equation, we get Upon expanding the exponential function (63), we get which, by appealing to the following series manipulation formula: in the left-hand side of (64), becomes Now, replacing q by q − n and l by l − p, and using a lemma in [37] in the left-hand side of (66), we get Finally, on equating the coefficients of the like powers of t and u in the equation (67), we get the required result (61) asserted by Theorem 4.
Proof.By the definition (45), we have Proof.We first replace x by x + z in (45).Then, by using ( 16), we rewrite the generating function (45) as follows: Furthermore, upon replacing n by n − s in l.h.s and comparing the coefficients of t n , we complete the proof of Theorem 6.
n (x, y; λ) holds true: n (x, y; λ) holds true: n (x, y; λ) holds true: Proof.Let us rewrite Equation (45) as follows: Replacing n by n − r and using (45), and then equating the coefficients of the of t n , we complete the proof of Theorem 7.
n (x, y; λ) holds true: Proof.Using the generating function (45), we find that which, upon equating the coefficients of t n , yields the assertion (86) of Theorem 8.
Remark 5. Several corollaries and consequences of Theorem 11 can be deduced by using many of the aforementioned specializations of the various parameters involved in Theorem 8.
Proof.Let us first consider the following expression: Comparing the coefficients of t n on the right-hand sides of Equations ( 88) and (89), we arrive at the desired result (87).Proof.Let us first consider the following application: On the other hand, we have Proof.The proof of Theorem 11 is analogous to that of Theorem 10, so we omit the details involved in the proof of Theorem 11.
Remark 7. Several corollaries and consequences of Theorem 11 can be derived by applying many of the aforementioned specializations of the various parameters involved in Theorem 11.
We conclude our present investigation by proving the following symmetric identity involving the number S k (n, λ), which is defined by (44).
Finally, after a suitable manipulation with the summation index in (98) followed by a comparison of the coefficients of t n , the proof of Theorem 12 is completed.

Conclusions
Özden ( [29]) defined the unified polynomials Y We have also presented a further investigation to obtain some implicit summation formulas and symmetric identities by means of their generating functions.
In our next investigation, we propose to study an appropriate combination of the operational approach with that involving integral transforms with a view to studying integral representations related to the truncated-exponential-based Apostol-type polynomials which we have introduced and studied in this article.

Theorem 2 .
The 2VTEATP e (r) Y (α) n,β (x, y; k, a, b) are quasi-monomial with respect to the following multiplicative and derivative operators:

For k = a = b = 1 and β = λ in Theorem 6 ,Corollary 7 .
we get the following corollary.The following implicit summation formula for the 2-variable truncated-exponential-based Bernoulli polynomials e (r) B

Corollary 9 .
Letting k = −2a = b = 1 and 2β = λ in Theorem 6, we get the following corollary.The following implicit summation formula for the 2-variable truncated-exponential-based Genocchi polynomials e (r) G

For k = a = b = 1 and β = λ in Theorem 7 ,Corollary 10 .
we get the following corollary.The following implicit summation formula for the 2-variable truncated-exponential-based Apostol-type Bernoulli polynomials e (r) B

1 1 −
Y(cdt) r , which shows that the function g(t) is symmetric in the parameters a and b.Then, by expanding g(t) into series in two different ways, we get g(t) = (dx, d r y; k, a, b) ,β (dx, d r y; k, a, b) e (r) Y (α) m,β (cX, c r Y; k, a, b)t n n,β (cx, c r y; k, a, b) m,β (dX, d r Y; k, a, b) n−m,β (cx, c r y; k, a, b) e (r) Y (α) m,β (dX, d r Y; k, a, b)t n .(89)

Remark 6 .Theorem 11 .
α) n−m,β cx + c d i + j, c r y; k, a, b e (r) Y (α) m,β (dX, d r Y; k, a, b) t n .(95) By comparing the coefficients of t n on the right-hand sides of (94) and (95), we arrive at the desired result (93) asserted by Theorem 10.Several corollaries and consequences of Theorem 11 can be derived by making use of many of the aforementioned specializations of the various parameters involved in Theorem 10.For each pair of integers a and b and all integers n ∈ N 0 , the following identity holds true: r Y; k, a, b).

k t k β b e t − a b α e xt 1 1 −
(x; k, a, b) of order α by means of the following generating function (see also Remark 1 above):2 1−k t k β b e t − a b α < 2π when β = a; |t| < b log( β a) when β = a;1 α := 1; k ∈ N 0 ; a, b ∈ R \ {0}; α, β ∈ C .Basing our investigation upon this generating function, we have introduced generating function for the 2-variable truncated-exponential-based Apostol-type polynomials denoted by e (r) Y (α) n,β (x, y; k, a, b) as follows: yt r , which we have found to be instrumental in deriving quasi-monomiality with respect to the following multiplicative and derivative operators:M e (r)Y = x + ry∂ y y∂ r−1 x + αk(β b e t − a b ) − αβ b ∂ x e ∂ x ∂ x (β b e t − a b )and P e (r)Y = ∂ x .

Table 1 .
Some special cases of the 2VTEATP .