James Mc Laughlin

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In this paper we give a new formula for the n-th power of a 2 × 2 matrix. More precisely, we prove the following: Let A = a b c d be an arbitrary 2 × 2 matrix, T = a + d its trace, D = ad − bc its determinant and define y n : = n/2 i=0 n − i i T n−2i (−D) i. A n = y n − d y n−1 b y n−1 c y n−1 y n − a y n−1. We use this formula together with an existing(More)
In this paper we use a formula for the n-th 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 2 1+2t−mn+n (−1) nk+i(n+1) 1 + δ (m−1)/2, i+k m − 1 − i i m − 1 − 2i k × n(m − 1 − 2(i + k)) 2j j t − n(i + k) n − 1 − s(More)
It is shown that there are no non-trivial fifth-, seventh-, eleventh-, thirteenth-or seventeenth powers in the Fibonacci sequence. For eleventh, thirteenth-and seventeenth powers an alternative (to the usual exhaustive check of products of powers of fundamental units) method is used to overcome the problem of having a large number of independent units and(More)
In some recent papers, the authors considered regular continued fractions of the form where a 0 ≥ 0, a ≥ 2 and m ≥ 1 are integers. The limits of such continued fractions, for general a and in the cases m = 1 and m = 2, were given as ratios of certain infinite series. However, these formulae can be derived from known facts about two continued fractions of(More)
Let (αn(a, k), βn(a, k)) be a WP-Bailey pair. Assuming the limits exist, let (α * n (a), β * n (a)) = lim k→1 (αn(a, k), β n (a, k) 1 − k) be the derived WP-Bailey pair. By considering a particular limiting case of a transformation due to George Andrews, we derive some transformation and summation formulae for derived WP-Bailey pairs. We then use the(More)