Derivation Operator And Combinatorial Identities

In this paper, based on the proof of combinatorial identities given by other academic authors,some new practical combinatorial identities are obtained by using partial fraction method and telescoping method. On this basis, we acquired some relevant q-analogue. The main contents are as follows:1. In this paper, by means of the partial fraction decomposition methods and other mathematic-cal tools, we explore the following algebraic factor:(?)Firstly, we through an ordinary factor to understand the entire calculation process, and then get the main factor of the results. As the general form of Weideman's identity, from which we can derive q-analog for several well-known results on harmonic number sums. Finally,the existence of the equation is verified by mathematical induction and Bell polynomial2. In terms of the telescoping method, a new binomial identity is established(?)There is a new form that molecular and denominator swapped by replacing some parameters. Then applying the derivative operators to it, we derive several interesting harmonic number identities. By taking some special parameter values, some simple and practical harmonic number identities are obtained.3. Through the in-depth understanding of the above-mentioned content 2, it is found that there are many interesting points. So the corresponding q-analogues are done for the above-mentioned harmonic number identities and the partial combined identities obtained in 2.