Higher-order Euler-type polynomials and their applications

Abstract

In this paper, we construct generating functions for higher-order Euler-type polynomials and numbers. By using the generating functions, we obtain functional equations related to a generalized partial Hecke operator and Euler-type polynomials and numbers. A special case of higher-order Euler-type polynomials is eigenfunctions for the generalized partial Hecke operators. Moreover, we give not only some properties, but also applications for these polynomials and numbers.

Bivariate general Appell interpolation problem

“…To the authors knowledge the bivariate sequence {r k } b k∈IN defined in (38) or, equivalently in (41), does not appear in the literature. We call it bivariate natural Euler polynomial of order 2 and denote it by E (2) n b k∈IN :…”

“…It is known [11] that in this case the elementary Appell sequence is {p n } b n∈IN , with p n (x, y) = H (2) n (x, y)…”

“…We will give the unique solution expressed in the basis of the so-called bivariate, general Appell polynomial sequences [21]. It can be so formulated: let X be the linear space of bivariate, real continuous functions having continuous partial derivatives of all necessary orders, defined in a domain D ⊂ IR 2 . Usually, for simplicity, D = [0, 1] × [0, 1].…”

“…(2) n (x) being the Bernoulli polynomial of order 2 [8,13,22,26] defined by the generating function (2) n (x) t n n! .…”

“…(2) n n∈IN is a univariate Appell sequence; therefore, from (29), according to Theorem 6.13 in [10, p. 90], we get r n (x, y) = B (2) n (x + y).…”