New Sums Identities In Weighted Catalan Triangle With The Powers Of Generalized Fibonacci And Lucas Numbers

In this paper, we consider a generalized Catalan triangle de…ned by km n 2n n k for positive integer m: Then we compute the weighted half binomial sums with the certain powers of generalized Fibonacci and Lucas numbers of the form n X k=0 2n n+ k km n X tk; where Xn either generalized Fibonacci or Lucas numbers, t and r are integers for 1 m 6: After we describe a general methodology to show how to compute the sums for further values of m. 1. Introduction Shapiro [6] derived the following triangle similar to Pascal’s triangle with entries given by Bn;k = k n 2n n k ; which called Catalan triangle because the Catalan numbers Cn = 1 n+1 2n n are the entries in the …rst column. Shapiro derived sums identities from the Catalan triangle. For example, he gave the following identities: n X p=1 (Bn;p) 2 = C2n 1 and n X p=1 Bn;pBn+1;p = C2n: We also refer to [5] and references therein for other examples. 2000 Mathematics Subject Classi…cation. 11B37. Key words and phrases. Catalan triangle, sums identites, partial binomial sum, recursions. 1 2 EMRAH KILIÇ AND AYNUR YALÇINER The authors [4] gave also an alternative proof of the identities above and established the following identity: n X p=1 (pBn;p) 2 = (3n 2)C2(n 1): In a somewhat di¤erent from the Catalan triangle, K¬l¬ç and Ionascu [2] derived the following result: for any a 2 C f0g ; n X p=1 2n n+ k a + a k = 1 an (a+ 1) 2n + (n+ 1)Cn: The authors also gave applications to the generalized Fibonacci and Lucas sequences, de…ned by Un = AUn 1 + Un 2; Vn = AVn 1 + Vn 2; where U0 = 0; U1 = 1; and V0 = 2; V1 = A; respectively. The Binet forms are Un = n n and Vn = n + n where ; = (A p )=2 and = A + 4: For example, we recall one result from [2]: n X