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- J. Mc Laughlin
- 2004

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)

- James Mc Laughlin, Nancy J. Wyshinski
- Discrete Applied Mathematics
- 2006

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)

In this we paper we prove several new identities of the Rogers-Ramanujan-Slater type. These identities were found as the result of computer searches. The proofs involve a variety of techniques, including constant-term methods, Bailey pairs, a theorem of Watson on basic hypergeometric series and the method of q-difference equations.

If k is set equal to aq in the definition of a WP Bailey pair, β n (a, k) = n X j=0 (k/a)n−j(k)n+j (q) n−j (aq) n+j α j (a, k), this equation reduces to βn = P n j=0 αj. This seemingly trivial relation connecting the α n 's with the β n 's has some interesting consequences, including several basic hypergeometric summation formulae, a connection to the… (More)

We consider a special case of a WP-Bailey chain of George Andrews, and use it to derive a number of curious transformations of basic hypergeometric series. We also derive two new WP-Bailey pairs, and use them to derive some new transformations for basic hypergeometric series. Finally, we briefly consider the implications of WP-Bailey pairs (αn(a, k), βn(a,… (More)

- J. MC LAUGHLIN
- 2001

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)

We derive closed-form expressions for several new classes of Hurwitzian-and Tasoevian continued fractions, including [0; p − 1, 1, u(a + 2nb) − 1, p − 1, 1, v(a + (2n + 1)b) − 1 ] ∞ n=0 , [0; c + dm n ] ∞ n=1 and [0; eu n , f v n ] ∞ n=1. One of the constructions used to produce some of these continued fractions can be iterated to produce both… (More)

- C B Easley, J M Laughlin, R E Gold, R M Hill
- Bulletin of environmental contamination and…
- 1982