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
G. Bretti, P. Natalini, and P. E. Ricci, “Generalizations of the Bernoulli and Appell polynomials,” Abstract and Applied Analysis, vol. 2004, no. 7, pp. 613–623, 2004.
Q.-M. Luo and H. M. Srivastava, “Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials,” Journal of Mathematical Analysis and Applications, vol. 308, no. 1, pp. 290–302, 2005.
Q.-M. Luo, “Apostol-Euler polynomials of higher order and Gaussian hypergeometric functions,” Taiwanese Journal of Mathematics, vol. 10, no. 4, pp. 917–925, 2006.
K. Douak, “The relation of the -orthogonal polynomials to the Appell polynomials,” Journal of Computational and Applied Mathematics, vol. 70, no. 2, pp. 279–295, 1996.
G. Dattoli, S. Lorenzutta, and D. Sacchetti, “Integral representations of new families of polynomials,” Italian Journal of Pure and Applied Mathematics, no. 15, pp. 19–28, 2004.
W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, vol. 52 of Die Grundlehren der mathematischen Wissenschaften, Springer, New York, NY, USA, 3rd edition, 1966.
G. Dattoli, M. Migliorati, and H. M. Srivastava, “Sheffer polynomials, monomiality principle, algebraic methods and the theory of classical polynomials,” Mathematical and Computer Modelling, vol. 45, no. 9-10, pp. 1033–1041, 2007.
G. Dattoli, “Hermite-Bessel and Laguerre-Bessel functions: a by-product of the monomiality principle,” in Advanced Special Functions and Applications (Melfi, 1999), vol. 1 of Proc. Melfi Sch. Adv. Top. Math. Phys., pp. 147–164, Aracne, Rome, Italy, 2000.
G. Bretti, C. Cesarano, and P. E. Ricci, “Laguerre-type exponentials and generalized Appell polynomials,” Computers & Mathematics with Applications, vol. 48, no. 5-6, pp. 833–839, 2004.
H. W. Gould and A. T. Hopper, “Operational formulas connected with two generalizations of Hermite polynomials,” Duke Mathematical Journal, vol. 29, pp. 51–63, 1962.
S. Khan, G. Yasmin, R. Khan, and N. A. M. Hassan, “Hermite-based Appell polynomials: properties and applications,” Journal of Mathematical Analysis and Applications, vol. 351, no. 2, pp. 756–764, 2009.
G. Dattoli, C. Cesarano, and S. Lorenzutta, “Bernoulli numbers and polynomials from a more general point of view,” Rendiconti di Matematica e delle sue Applicazioni, vol. 22, pp. 193–202, 2002.
G. Dattoli, S. Lorenzutta, and C. Cesarano, “Finite sums and generalized forms of Bernoulli polynomials,” Rendiconti di Matematica e delle sue Applicazioni, vol. 19, no. 3, pp. 385–391, 1999.