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Abstract
In this paper, we define the generalized r-Whitney numbers of the first and second kind. Moreover, we drive the generalized Whitney numbers of the first and second kind. The recurrence relations and the generating functions of these numbers are derived. The relations between these numbers and generalized Stirling numbers of the first and second kind are deduced. Furthermore, some special cases are given. Finally, matrix representation of The relations between Whitney and Stirling numbers are given.
Highlights
The r-Whitney numbers of the first and second kind were introduced, respectively, by Mezö [1] as = mn ( x)n n ∑wm,r ( n, k )k (1)k =0 nn ∑ =Wm,r (n, k ) mk ( x)k . (2)
K =0 ( ) where −iα = 0, −α, − (n −1)α and w1, 0; − iα
∏k (i − j ) =(−1)k−i (k − i)! j!, =j 0, j ≠i given by Gould [15], we obtain the exponential generating function of r-Whitney numbers of the second kind, see
Summary
The r-Whitney numbers of the first and second kind were introduced, respectively, by Mezö [1] as. K =0 n (mx + r )n ∑ =Wm,r (n, k ) mk ( x)k Many properties of these numbers and their combinatorial interpretations can be seen in Mezö [2] and Cheon [3]. El-Desouky et al. This paper is organized as follows: In Sections 2 and 3 we derive the generalized r-Whitney numbers of the first and second kind.The recurrence relations and the generating functions of these numbers are derived.
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