About Extensions of Generalized Apostol-Type Polynomials

Existence and reduction of generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials

“…In a similar way, let α ∈ N ∪ {0} and the generalized Apostol-Genocchi polynomials of order α be defined as in (3), with λ = −1. We state the following:…”

“…We have seen in (4) and (5) that the polynomials B n (x; λ) (with λ = −1) can be expressed, up to a multiplicative constant, in terms of the generalized Apostol-Euler polynomials of order α. Moreover, the parameter α in (2) can be any α ∈ C (except when λ = −1), without the restriction α ∈ N ∪ {0} in (1) and (3). Then, we can take the generalized Apostol-Euler polynomials of order α ∈ C defined as 72…”

“…. (3) These are valid in a suitable neigbourhood of t = 0, where λ is (with some exceptions) any complex number. They are generalizations of the classical Bernoulli, Euler and Genocchi polynomials B n (x), E n (x) and G n (x), that correspond to the cases λ = 1 and α = 1 (moreover, the so-called Bernoulli, Euler and Genocchi numbers are B n = B n (0), E n = 2 n E n ( 1 2 ) and G n = G n (0)).…”

“…They are generalizations of the classical Bernoulli, Euler and Genocchi polynomials B n (x), E n (x) and G n (x), that correspond to the cases λ = 1 and α = 1 (moreover, the so-called Bernoulli, Euler and Genocchi numbers are B n = B n (0), E n = 2 n E n ( 1 2 ) and G n = G n (0)). In the mathematical literature, the parameters α and λ have been included independently (we give some historical details in Subsection 1.1); once the parameters have been used together, the definitions (1), (2) and (3) have been extended to α ∈ C. The goal of this paper is to clarify when this extension is possible (Subsection 1.2), and to reduce the above-mentioned definitions with complex λ and α to a smaller class of polynomials that suppresses the trivial relationships between them (Section 2, see 7, (8) and Definition 4). Except for multiplicative constants, our reduction covers every case of generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials without the necessity of adding extra parameters.…”

“…(of course, similar definitions are given for Euler or Genocchi polynomials), that are generalizations of the preexisting cases corresponding to λ = 1 and α = 1. A recent review that deals with many of these generalizations and unifications is [3]. Fortunately, many of the properties and relations that appear in all these papers or books are valid when the polynomials do exist, although this requires a suitable restriction of the parameters.…”