The Gelin-Cesaro identity in some conditional sequences

Abstract

In this paper, we deal with two families of conditional sequences. The first family consists of generalizations of the Fibonacci sequence. We show that the Gelin-Ces`aro identity is satisfied. Also, we define a family of conditional sequences ${u_n}$by the recurrence relation ${u_n}$ = a${u_n}$−1 +${bu_n}$−2 if n is even, ${u_n}$ = ${CU_n}$−1+${du_n}$−2 if n is odd, with initial conditions ${u_o}$ = 0 and ${u_1}$ = 1, where a, b, c and d are non-zero numbers. Many sequences in the literature are special cases of this sequence. We find the generating function of the sequence and Binet’s formula for odd and even subscripted sequences. Then we show that the Catalan and Gelin-Cesaro identities are satisfied by the indices of this generalized sequence.