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In this paper, we give general formulas for some weighted binomial sums with the powers of terms of certain binary recurrences. As applications of our results, we show that the weighted binomial sums with the generalized Fibonacci and Lucas numbers can be expressed via the second kinds of both Chebyshev polynomials and Stirling numbers. 1. INTRODUCTION For n > 1; dene the binary recurrences fUng and fVng by Un = pUn 1 Un 2 and Vn = pVn 1 Vn 2 where U0 = 0; U1 = 1 and V0 = 2; V1 = p; respectively. The Fibonacci subsequence fF2ng and the Pell subsequence fP2ng are the special cases of the sequence fUng for the cases p = 3 and p = 6; respectively. It is also known that the natural numbers are special cases of the sequence fUng for p = 2: The Binet formulas of fUng and fVng are as follows: Un = n n and Vn = n + ; (1.1) where ; = p p p2 4 =2: From [8], we have that for k 0 and n > 1; Ukn = VkUk(n 1) Uk(n 2); Vkn = VkVk(n 1) Vk(n 2): Wiemann and Cooper [5] mentioned about some conjectures of Melham for the sum: n X h=1 F 2m+1 2h ; where Fn stands for the nth Fibonacci number. Ozeki [2] considered Melhams sum and then he gave an explicit expansion for Melhams sum as a polynomial in F2n+1. In general, Prodinger [3] derive the general formula for the sum: n X

Formulas for weighted binomial sums using the powers of terms of binary recurrences