Gap-Sums via Quasi-Arithmetic Means with Applications to Fibonacci and Lucas Sequences

Abstract

We develop a unified framework for studying the integers missing between consecutive terms of an increasing integer sequence, extending Barry's arithmetic gap-sum to geometric and harmonic analogues via the theory of quasi-arithmetic means. All three gap-sums admit a common interpretation: each equals the gap size multiplied by the appropriate mean of the missing integers. Building on this, we prove a general sparse summation theorem expressing the sum of a strictly monotonic function over a sparse integer sequence as the full range sum corrected by the gap-sums of the missing portions. Specializing on the three Pythagorean means recovers a classical formula of al-K\=ash\={\i} from the fifteenth century in the arithmetic case, and yields explicit formulas in the geometric and harmonic cases. As a concrete application of the geometric case, we derive a product identity involving the Fuss--Catalan numbers. Applying the harmonic case to the Fibonacci and Lucas sequences, we establish that the harmonic gap-sum converges to $\ln(\alpha)$ exponentially, where $\alpha$ is the golden ratio, and derive explicit two-term asymptotic expansions for the tails of the reciprocal Fibonacci and Lucas series with closed-form coefficients, and establish the asymptotic formula $H_{u_n} \sim n\ln(\alpha)$ for both $u_n = F_n$ and $u_n = L_n$, with explicit $O(1)$ error terms that differ due to their distinct initial conditions. As a further consequence, by comparing the gap-sum expansions with the classical Hardy--Wright expansion of harmonic numbers, we derive exact series identities expressing Euler's constant $\gamma$ in terms of harmonic numbers at Fibonacci and Lucas indices, and obtain a new identity relating the reciprocal Fibonacci constant $\psi$ and the reciprocal Lucas constant $\psi_L$.