Abstract
Attention is focused to particular families of Sheffer polynomials which are different from the classical ones because they satisfy non-standard differential equations, including some of fractional type. In particular Sheffer polynomial families are considered whose characteristic elements are based on powers or exponential functions.
Highlights
In recent articles [1, 2], new sets of Sheffer [3] and Brenke [4] polynomials, based on higher order Bell numbers [5,6,7,8,9], have been studied
In this article we focus our attention on Sheffer polynomial families whose characteristic elements are based on powers or exponential functions, deriving the relevant differential equations, which are frequently of fractional type
When g(t) 1, the Sheffer sequence corresponding to the pair (1, f(t)) is called the associated Sheffer sequence {rn (x)} for f(t), and its exponential generating function is given by exp
Summary
In recent articles [1, 2], new sets of Sheffer [3] and Brenke [4] polynomials, based on higher order Bell numbers [5,6,7,8,9], have been studied. Several integer sequences [10] associated with the considered polynomials sets both of exponential [11, 12] and logarithmic type have been introduced [2]. We recall that the exponential and logarithmic polynomials have been recently studied even in the multidimensional case [13,14,15]. In this article we focus our attention on Sheffer polynomial families whose characteristic elements are based on powers or exponential functions, deriving the relevant differential equations, which are frequently of fractional type
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