Summation Formulas for Certain Combinatorial Sequences

Abstract

In this work, we establish some characteristics for a sequence, Aα(n,k), including recurrence relations, generating function and inversion formula, etc. Based on the sequence, we derive, by means of the generating function approach, some transformation formulas concerning certain combinatorial numbers named after Lah, Stirling, harmonic, Cauchy and Catalan, as well as several closed finite sums. In addition, the relationship between Aα(n,k) and r-Whitney–Lah numbers is established, and some formulas for the r-Whitney–Lah numbers are obtained.

Combinatorial Identities Concerning Binomial Quotients

“…Among many methods, such as the generating function method [11][12][13], WZ method [14,15], Riordan array approach [16,17] and hypergeometric series technique [18], to prove combinatorial identities, the telescoping technique proposed by Zeilberger [19,20] is a quite efficient approach to computing binomial sums. This is the main method used in this paper, and thus we will give a brief introduction to it here.…”