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In this paper, we obtain a closed form for ${F_{\sum\nolimits_{i = 1}^k {} }}$, ${P_{\sum\nolimits_{i = 1}^k {} }}$and ${J_{\sum\nolimits_{i = 1}^k {} }}$ for some positive integers k where Fr, Pr and Jr are the rth Fibonacci, Pell and Jacobsthal numbers, respectively. We also give three open problems for the general cases ${F_{\sum\nolimits_{i = 1}^n {} }}$, ${P_{\sum\nolimits_{i = 1}^n {} }}$ and ${J_{\sum\nolimits_{i = 1}^n {} }}$for any arbitrary positive integer n.

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The hyperbolic Fibonacci and hyperbolic Lucas functions have been introduced before and have been improved to functions of the symmetrical form. In this paper, we generalize the mentioned definitions, which will be called Horadam hyperbolic sine function (HSF ) and Horadam hyperbolic cosine function (HCF ). Further, we present many identities and hyperbolic properties of our new definitions.

No abstract