A Generalization of the Kummer Identity and Its Application to Fibonacci-lucas Sequences

They found that transformation of (1) or (2) is related to Fibonacci numbers. Recently, Shapiro et al. [8] and Sprugnoli [10] introduced the theories of the Riordan array and the Riordan group, respectively, in an effort to answer the following question: What are the conditions under which a combinatorial sum can be evaluated by transforming the generating function? We think that the works of Gould and Haukkanen mentioned above can be extended by using the Riordan group or the Riordan array. We adopt the concept of the Riordan group in this paper because both theories are essentially the same. It is certain that the idea of the Riordan group can be traced back to Mullin and Rota [5], Rota [6], and Roman and Rota [7]. The reader is referred to [5][10] for more details. In the present paper we are concerned with the following identity, called the Kumrner identity: