Abstract
Simsek [Y. Simsek, New families of special numbers for computing negative order Euler numbers and related numbers and polynomials, Appl. Anal. Discrete Math. 12(2018), 1-35.] conjectured that B(d,k)=(kd + x1kd-1 + x2kd-2 +...+ xd-1k)2k-d; where x1,x2,..., xd-1; d are positive integers, and proposed the following open problem: (I) How can we compute the coefficients x1, x2, ... , xd-1? (II) Is it possible to find the function fd(x) = ??,k=1 B(d,k)xk? By using the familiar Stirling numbers of the first and second kind, we solve this problem. We further obtain a general result on the generalized numbers of B(d,k).
Highlights
Simsek [12] constructed a new family of special numbers denote by y1(n, k; λ): (1)
By the generating function (30) we find the recurrence relation of fd(x; λ)
The Eulerian polynomials are closely related to the Apostol-Bernoulli numbers [2]: (1 − x)n+1 (35)
Summary
We further obtain a general result on the generalized numbers of B(d, k). Faaa di Bruno’s formula, Eulerian polynomial, Generating function, Explicit expression. Simsek [12] obtained the following recurrence Xd−1, d are positive integers, and proposed the following open problem: (I) How can we compute the coefficients x1, x2, .
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