Abstract
Recently, many mathematicians have studied different kinds of the Euler, Bernoulli, and Genocchi numbers and polynomials. In this paper, we give another definition of polynomialsŨn(x). We observe an interesting phenomenon of “scattering” of the zeros of the polynomialsŨn(x)in complex plane. We find out some identities and properties related to polynomialsŨn(x). Finally, we also derive interesting relations between polynomialsŨn(x), Stirling numbers, central factorial numbers, and Euler numbers.
Highlights
Many mathematicians have studied in the areas of the Bernoulli numbers and polynomials, Euler numbers and polynomials, Genocchi numbers and polynomials, Stirling numbers, and central factorial numbers
From (1) and (2), we are aware of some properties of the Stirling numbers of the first kind s(n, k) as follows: s (n, k) = s (n − 1, k − 1) − (n − 1) s (n − 1, k), (3)
By using (5) and (6), we are aware of some properties of the central factorial numbers T(n, k) as follows: T (0, 0) = 1, T (n, 0) = 0 (n ∈ N), (7)
Summary
A Research on a Certain Family of Numbers and Polynomials Related to Stirling Numbers, Central Factorial Numbers, and Euler Numbers. Many mathematicians have studied different kinds of the Euler, Bernoulli, and Genocchi numbers and polynomials. We give another definition of polynomials Ũn(x). We observe an interesting phenomenon of “scattering” of the zeros of the polynomials Ũn(x) in complex plane. We find out some identities and properties related to polynomials Ũn(x). We derive interesting relations between polynomials Ũn(x), Stirling numbers, central factorial numbers, and Euler numbers
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