Some Computational Formulas for D-Nِrlund Numbers

and Applied Analysis 3 It follows from 1.11 or 1.12 that t n, k t n − 2, k − 2 − 1 4 n − 2 t n − 2, k , 1.13 and that t n, 0 δn,0 n ∈ N0 : N ∪ {0} , t n, n 1 n ∈ N , t n, k 0 n k odd , t n, k 0 k > n or k < 0 , 1.14 where δm,n denotes the Kronecker symbol. By 1.13 , we have t 2n 1, 1 −1 n 2n ! 42n ( 2n n ) , t 2n 2, 2 −1 n n! 2 n ∈ N0 , 1.15 t 2n 2, 4 −1 n 1 n! 2 ( 1 1 22 1 32 · · · 1 n2 ) n ∈ N , 1.16 t 2n 1, 3 −1 n−1 2n ! 42n−1 ( 2n n )( 1 1 32 1 52 · · · 1 2n − 1 2 ) n ∈ N . 1.17 The main purpose of this paper is to prove some identities involving D numbers, Bernoulli numbers, and central factorial numbers of the first kind and obtain a generating function and several computational formulas for the D-Nörlund numbers. That is, we will prove the following main conclusion. Theorem 1.1. Let n ∈ N, k ∈ N \ {1}. Then D k 2n 2n − k 2 2n − k 1 k − 2 k − 1 D k−2 2n − 2n 2n − 1 k − 2 k − 1 D k−2 2n−2 . 1.18 Remark 1.2. By 1.18 , we may immediately deduce the following see 4, page 147 : D 2n 1 2n −1 n 2n ! 4n ( 2n n ) , D 2n 2 2n −1 4n 2n 1 n! . 1.19 Theorem 1.3. Let n ≥ k n, k ∈ N0 . Then D 2n 1 2n−2k 4n−k ( 2n 2k ) t 2n 1, 2k 1 , 1.20 D 2n 2n−2k 4n−k ( 2n−1 2k−1 ) t 2n, 2k k ≥ 1 . 1.21 4 Abstract and Applied Analysis Remark 1.4. By 1.20 and 1.17 , we may immediately deduce the following: D 2n 3 2n −1 n 2n ! 2 · 42n ( 2n 2 n 1 )( 1 1 32 1 52 · · · 1 2n 1 2 ) . 1.22 Theorem 1.5. Let n ∈ N0. Then