Further combinatorial identities deriving from the nth power of a [formula omitted] matrix

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In this paper we use a formula for the nth power of a 2 × 2 matrix A (in terms of the entries in A) to derive various combinatorial identities. Three examples of our results follow. (1) We show that if m and n are positive integers and s ∈ { 0 , 1 , 2 ,..., ⌊ ( mn - 1 ) / 2 ⌋ } , then ∑ i , j , k , t 2 1 + 2 t - mn + n ( - 1 ) nk + i ( n + 1 ) 1 + δ ( m - 1 ) / 2 , i + k m - 1 - i i m - 1 - 2 i k n ( m - 1 - 2 ( i + k ) ) 2 j j t - n ( i + k ) n - 1 - s + t s - t = mn - 1 - s s . (2) The generalized Fibonacci polynomial f m ( x , s ) can be expressed as f m ( x , s ) = ∑ k = 0 ⌊ ( m - 1 ) / 2 ⌋ m - k - 1 k x m - 2 k - 1 s k . We prove that the following functional equation holds: f mn ( x , s ) = f m ( x , s ) × f n ( f m + 1 ( x , s ) + sf m - 1 ( x , s ) ,- ( - s ) m ) . (3) If an arithmetical function f is multiplicative and for each prime p there is a complex number g ( p ) such that f ( p n + 1 ) = f ( p ) f ( p n ) - g ( p ) f ( p n - 1 ) , n ⩾ 1 , then f is said to be specially multiplicative. We give another derivation of the following formula for a specially multiplicative function f evaluated at a prime power: f ( p k ) = ∑ j = 0 ⌊ k / 2 ⌋ ( - 1 ) j k - j j f ( p ) k - 2 j g ( p ) j . We also prove various other combinatorial identities.