General-Appell Polynomials within the Context of Monomiality Principle

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.


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: {  ()}( ∈ N 0 ) is an Appell set (  being of degree exactly ) if either,  The generalized Bernoulli polynomials [4] (IV) ! The new generalized Bernoulli polynomials [6] Table 1: Continued.
In fact, (i) Combining the recurrences ( 6) and (7), we have which can be interpreted as the differential equation satisfied by   (), if M and P have a differential realization.(ii) Assuming here and in the sequel  0 () = 1, then   () can be explicitly constructed as which yields the series definition for   ().(iii) 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.

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 ! The Gould Hopper-Euler polynomials (III) !The Gould-Hopper-generalized Bernoulli polynomials (IV)  ; (XIV) If (, ) is an invertible series and   (, )/(, ) has Taylor's series expansion in powers of , then in view of the identity   {   (, )} =  (   (, )) (22) 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  0 (, ) = 1, 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 M 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, 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) 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.

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

Table 1 :
List of some Appell polynomials.
S. No.

Table 3 :
List of the first few Bernoulli and the Euler polynomials.International Journal of Analysis 11 In view of equations (A.13)-(A.18),we get Figure 1.