Abstract
In this paper, we introduce the extended 2D Bernoulli polynomials by and the extended 2D Euler polynomials by where . By using the concepts of the monomiality principle and factorization method, we obtain the differential, integro-differential and partial differential equations for these polynomials. Note that the above mentioned differential equations for the extended 2D Bernoulli polynomials reduce to the results obtained in (Bretti and Ricci in Taiwanese J. Math. 8(3): 415–428, 2004), in the special case , . On the other hand, all the results for the second family are believed to be new, even in the case , . Finally, we give some open problems related with the extensions of the above mentioned polynomials. MSC:11B68, 33C05.
Highlights
A polynomial set {Pn(x)}∞ n= is quasi-monomial if and only if there exist two operators Pand M, independent of n, such thatP Pn(x) = nPn– (x) and M Pn(x) = Pn+ (x).Here, Mand Pplay the role of multiplicative and derivative operators, respectively
In this paper, we introduce the extended 2D Bernoulli polynomials byα cxt+ytj
Owing to the fact that every polynomial set is quasi-monomial [ ], by using the monomiality principle, new results were obtained for Hermite, Laguerre, Legendre and Appell polynomials in [ – ]
Summary
A polynomial set {Pn(x)}∞ n= is quasi-monomial if and only if there exist two operators Pand M , independent of n, such that. P Pn(x) = nPn– (x) and M Pn(x) = Pn+ (x). Mand Pplay the role of multiplicative and derivative operators, respectively. Owing to the fact that every polynomial set is quasi-monomial [ ], by using the monomiality principle, new results were obtained for Hermite, Laguerre, Legendre and Appell polynomials in [ – ]. We recall some basic definitions and properties of the polynomial families that we discuss throughout the paper. The celebrated Appell polynomials can be defined by the following generating relation: GA(x, t) = A(t)ext =
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