一些与Dyck路有关的数的恒等式(英文)

Identities Involoving Some Numbers Related to Dyck Paths

中文关键词:  恒等式  Dyck路  Catalan数  Motzkin数

英文关键词:Identities  Dyck path  Catalan numbers  Motzkin numbers

中文摘要:

      本文通过Cauchy留数定理和算子方法导出了一些形如∑_i=0ⁿ (-1)^n-i(n i)U_{m+k+i, k+i} =f(n) 和∑_i=0²ⁿ(-1 )ⁱ(2n i) U_{m+k+i, k+i} = g(n)的差分恒等式,这里U_{n, κ}表示Dyck路在不同条件下的计数公式,f(n),g(n)与m(n)只和n有关的函数.

英文摘要:

      In the paper we derive many identities of forms ∑_i=0ⁿ (-1)^n-i(n i)U_{m+k+i, k+i} =f(n) and ∑_i=0²ⁿ(-1 )ⁱ(2n i) U_{m+k+i, k+i} = g(n) by the Cauchy Residue Theorem and an operator method, where U_{n, κ} are numbers of Dyck paths counted under different conditions, and f(n), g(n) and m are functions depending only on about n.