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`International Journal of AnalysisVolume 2013 (2013), Article ID 328032, 11 pageshttp://dx.doi.org/10.1155/2013/328032`
Research Article

## General-Appell Polynomials within the Context of Monomiality Principle

Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

Received 22 September 2012; Accepted 6 December 2012

Copyright © 2013 Subuhi Khan and Nusrat Raza. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A general class of the 2-variable polynomials is considered, and its properties are derived. Further, these polynomials are used to introduce the 2-variable general-Appell polynomials (2VgAP). The generating function for the 2VgAP is derived, and a correspondence between these polynomials and the Appell polynomials is established. The differential equation, recurrence relations, and other properties for the 2VgAP are obtained within the context of the monomiality principle. This paper is the first attempt in the direction of introducing a new family of special polynomials, which includes many other new special polynomial families as its particular cases.

#### 1. Introduction and Preliminaries

The Appell polynomials are very often found in different applications in pure and applied mathematics. The Appell polynomials [1] may be defined by either of the following equivalent conditions:   is an Appell set ( being of degree exactly ) if either,(i)   or (ii)there exists an exponential generating function of the form where has (at least the formal) expansion:

Roman [2] characterized Appell sequences in several ways. Properties of Appell sequences are naturally handled within the framework of modern classical umbral calculus by Roman [2]. We recall the following result [2, Theorem ], which can be viewed as an alternate definition of Appell sequences.

The sequence is Appell for , if and only if where

In view of (1) and (3), we have

The Appell class contains important sequences such as the Bernoulli and Euler polynomials and their generalized forms. Some known Appell polynomials are listed in Table 1.

Table 1: List of some Appell polynomials.

We recall that, according to the monomiality principle [15, 16], a polynomial set is “quasimonomial”, provided there exist two operators and playing, respectively, the role of multiplicative and derivative operators, for the family of polynomials. These operators satisfy the following identities, for all :

The operators and also satisfy the commutation relation and thus display the Weyl group structure. If the considered polynomial set is quasimonomial, its properties can easily be derived from those of the and operators. In fact, Combining the recurrences (6) and (7), we have which can be interpreted as the differential equation satisfied by , if and have a differential realization. Assuming here and in the sequel , then can be explicitly constructed as which yields the series definition for . Identity (10) implies that the exponential generating function of can be given in the form

We note that the Appell polynomials are quasimonomial with respect to the following multiplicative and derivative operators: or, equivalently, respectively.

The special polynomials of two variables are useful from the point of view of applications in physics. Also, these polynomials allow the derivation of a number of useful identities in a fairly straightforward way and help in introducing new families of special polynomials. For example, Bretti et al. [17] introduced general classes of two variables Appell polynomials by using properties of an iterated isomorphism, related to the Laguerre-type exponentials.

We consider the 2-variable general polynomial (2VgP) family defined by the generating function where has (at least the formal) series expansion

We recall that the 2-variable Hermite Kampé de Fériet polynomials (2VHKdFP) [18], the Gould-Hopper polynomials (GHP) [19], and the Hermite-Appell polynomials (HAP) [20] are defined by the generating functions respectively. Thus, in view of generating functions (14), (16), (17), and (18), we note that the 2VHKdFP , the GHP , and the HAP belong to 2VgP family.

In this paper, operational methods are used to introduce certain new families of special polynomials related to the Appell polynomials. In Section 2, some results for the 2-variable general polynomials (2VgP) are derived. Further, the 2-variable general-Appell polynomials (2VgAP) are introduced and framed within the context of monomiality principle. In Section 3, the Gould-Hopper-Appell polynomials (GHAP) are considered, and their properties are established. Some members belonging to the Gould-Hopper-Appell polynomial family are given.

#### 2. 2-Variable General-Appell Polynomials

In order to introduce the 2-variable general-Appell polynomials (2VgAP), we need to establish certain results for the 2VgP . Therefore, first we prove the following results for the 2VgP .

Lemma 1. The 2VgP defined by generating function (14), where is given by (15), are quasimonomial under the action of the following multiplicative and derivative operators: respectively.

Proof. Differentiating (14) partially with respect to , we have
If is an invertible series and has Taylor's series expansion in powers of , then in view of the identity we can write
Now, using (23) in the l.h.s. of (21), we find
Making use of generating function (14) in the l.h.s. of the above equation, we have which, on equating the coefficients of like powers of in both sides, gives
Thus, in view of monomiality principle equation (6), the above equation yields assertion (19) of Lemma 1. Again, using identity (22) in (14), we have
Equating the coefficients of like powers of in both sides of (27), we find which in view of monomiality principle equation (7) yields assertion (20) of Lemma 1.

Remark 2. The operators given by (19) and (20) satisfy commutation relation (8). Also, using expressions (19) and (20) in (9), we get the following differential equation satisfied by 2VgP :

Remark 3. Since , therefore, in view of monomiality principle equation (10), we have
Also, in view of (11), (14), and (19), we have
Now, we proceed to introduce the 2-variable general-Appell polynomials (2VgAP). In order to derive the generating functions for the 2VgAP, we take the 2VgP as the base in the Appell polynomials generating function (1). Thus, replacing by the multiplicative operator of the 2VgP in the l.h.s. of (1) and denoting the resultant 2VgAP by , we have
Now, using (31) in the exponential term in the l.h.s. of (32), we get the generating function for as
In view of (5), generating function (33) can be expressed equivalently as
Now, we frame the 2VgAP within the context of monomiality principle formalism. We prove the following results.

Theorem 4. The 2VgAP are quasimonomial with respect to the following multiplicative and derivative operators: or, equivalently, respectively.

Proof. Differentiating (33) partially with respect to , we find
Since and are invertible series of , therefore, and possess power series expansions of . Thus, in view of identity (22), we have which, on using generating function (33), becomes or, equivalently,
Now, equating the coefficients of like powers of in the above equation, we find which in view of (6) yields assertion (35a) of Theorem 4. Also, in view of relation (5), assertion (35a) can be expressed equivalently as (35b).
Again, in view of identity (22), we have which on using generating function (33) becomes
Equating the coefficients of like powers of in the above equation, we find which, in view of (7), yields assertion (36) of Theorem 4.

Theorem 5. The 2VgAP satisfy the following differential equations or, equivalently,

Proof. Using (35a) and (36) in (9), we get assertion (45a). Further, using (35b) and (36) in (9), we get assertion (45b).

Note 1. With the help of the results derived above and by taking (or ) of the Appell polynomials listed in Table 1, we can derive the generating function and other results for the members belonging to 2VgAP family.

#### 3. Examples

We consider examples of certain members belonging to the 2VgAP family.

Taking (that is, when the 2VgP reduces to the GHP ) in generating function (33), we find that the Gould-Hopper-Appell polynomials (GHAP) are defined by the following generating function: or, equivalently,

Using (1) in (46) (or (3) in (47)), we get the following series definition for in terms of the Appell polynomials :

In view of (35a), (35b), and (36), we note that the GHAP are quasimonomial under the action of the following multiplicative and derivative operators: or, equivalently,

respectively. Also, in view of (45a) and (45b), we find that the GHAP satisfy the following differential equation: or, equivalently,

Remark 6. In view of (16) and (17), we note that, for , the GHAP reduce to the Hermite-Appell polynomials (HAP) . Therefore, taking in (46), (47), (48), (49a), (49b), (50), (51a), and (51b), we get the corresponding results for the HAP .

Remark 7. In view of (16), we note that the 2VHKdFP are related to the classical Hermite polynomials [11] or as Therefore, taking and replacing by (or taking ) in (46)–(51b), we get the corresponding results for the classical Hermite-Appell polynomials (or ).

There are several members belonging to 2VgP family. Thus the results for the corresponding 2VgAP can be obtained by taking other examples. We present the list of some members belonging to the GHAP family in Table 2.

Table 2: List of some Gould-Hopper-Appell polynomials .

Note 2. Since, for , the GHAP reduce to the HAP therefore, for , Table 2 gives the list of the corresponding HAP .

The results established in this paper are general and include new families of special polynomials, consequently introducing many new special polynomials.

#### Appendix

New classes of Bernoulli numbers and polynomials are introduced, which are used to evaluate partial sums involving other special polynomials, see, for example, [21, 22]. Here, we consider the Gould-Hopper-Bernoulli polynomials (GHBP), Gould-Hopper-Euler polynomials (GHEP), Hermite-Bernoulli polynomials (HBP), Hermite-Euler polynomials (HEP), classical Hermite-Bernoulli polynomials (cHBP), and classical Hermite-Euler polynomials (cHEP). We give the surface plots of these polynomials for suitable values of the parameters and indices. Also, we give the graphs of the corresponding single-variable polynomials.

The GHBP , GHEP , HBP , and HEP are defined by the following series: respectively. Taking in (A.3) and (A.4), we get the series definitions for the cHBP and cHEP as respectively.

To draw the surface plots of these polynomials, we use the values of the Bernoulli polynomials and the Euler polynomials for . We give the list of the first few Bernoulli and the Euler polynomials in Table 3.

Table 3: List of the first few Bernoulli and the Euler polynomials.

Now, we consider the GHBP , GHEP , HBP , HEP , cHBP , and cHEP , for and , so that we have respectively. Using the particular values of and given in Table 3, we find respectively.

In view of equations (A.13)–(A.18), we get Figure 1.

Figure 1

#### Acknowledgments

The authors are thankful to the anonymous referee for useful suggestions towards the improvement of the paper. This work has been done under the Senior Research Fellowship (Office Memo no. Acad/D-1562/MR) sanctioned to the second author by the University Grants Commission, Government of India, New Delhi.

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