Convolutions of the bi-periodic Fibonacci numbers

Let q(n) be the bi-periodic Fibonacci numbers, defined by q(n) = c(n)q(n-1) + q(n-2) (n >= 2) with q(0) = 0 and q(1) = 1, where c(n) = a if n is even, c(n) = b if n is odd, where a and b are nonzero real numbers. When c(n) = a = b = 1, q(n) = F-n are Fibonacci numbers. In this paper, the convolution identities of order 2, 3 and 4 for the bi-periodic Fibonacci numbers q(n) are given with binomial (or multinomial) coefficients, by using the symmetric formulas.