Hoggattjr 1976 reciprocal series fibonacci numbers subscripts

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theorem_p455: For a fixed index k, the sum over n of the reciprocals of the Fibonacci numbers with subscripts 2^n k equals (2L_k - F_{2k} sqrt 5 + 5F_k^2)/(2F_{2k}) when k is odd and (2 - F_k sqrt 5 + L_k)/(2F_k) when k is even.

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Hoggatt, Jr., V. E. and Bicknell, Marjorie, A reciprocal series of Fibonacci numbers with subscripts $2^nk$. Fibonacci Quart. 14 (1976), no. 5, 453-455.

The note sums the series of 1/F_{2^n k} over n >= 0 for every fixed k, generalizing the Millin-Good evaluation of the k = 1 case as (7 - sqrt 5)/2. The main closed form (stated on p. 455, the paper gives no theorem numbers) is (2L_k - F_{2k} sqrt 5 + 5F_k^2)/(2F_{2k}) for k odd and (2 - F_k sqrt 5 + L_k)/(2F_k) for k even. The method telescopes the identity F_{2k} = F_k L_k together with the Lucas identities L_{m+p} + L_{m-p} = L_m L_p (p even) and L_k^2 = L_{2k} + 2(-1)^k, sums the resulting Lucas series with a summation formula of Siler, and takes the limit using powers of alpha and beta. The paper also records the relation B = C + 1/F_{2s+1} between the odd case k = 2s+1 and the even case k = 2(2s+1). The index sequences k, 2k, 4k, ... have ratio exactly 2; in each closed form the coefficient of sqrt 5 is -1/2 and the other terms are rational. The paper evaluates the sums and does not discuss their irrationality.

Source: https://www.fq.math.ca/14-5.html. No notice is printed on the file's three pages; the journal's issue page, which lists the article, carries the footer "Copyright © 2010 The Fibonacci Association. All rights reserved." (https://www.fq.math.ca/14-5.html), every other right reserved.

Results.

• Theorem (p. 455, unnumbered): the closed form of the sum over n >= 0 of 1/F_{2^n k}, for k odd and for k even, with the k = 1 evaluation (7 - sqrt 5)/2 (p. 454) and the relation B = C + 1/F_{2s+1} (p. 455).

Read status. Claims checked: the closed form was read clause by clause on p. 455 and the k = 1 evaluation on p. 454; the derivation was read for structure only.

Bears on. #267 (the theorem evaluates the sum over the index sequences 2^j k, of ratio exactly 2, in closed form; the paper does not discuss irrationality or other index sequences)

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.