Abstract
We prove some finite sum identities involving reciprocals of the binomial and central binomial coefficients, as well as harmonic, Fibonacci and Lucas numbers, some of which recover previously known results, while the others are new.
Highlights
In the last hundred years or so, there has been an explosion of new results and applications of special functions in various areas of mathematics and related fields, such as engineering, quantum physics, astronomy and combinatorics
We offer a study dealing with binomial, central binomial coefficients, and harmonic numbers that include gamma and digamma functions and the classical Fibonacci and Lucas numbers
In 2002, Mansour [8] studied some finite sums involving reciprocal of the binomial coefficients and deduced a generalization of (1) together with many other results
Summary
In the last hundred years or so, there has been an explosion of new results and applications of special functions in various areas of mathematics and related fields, such as engineering, quantum physics, astronomy and combinatorics. We offer a study dealing with binomial, central binomial coefficients, and harmonic numbers that include gamma and digamma functions and the classical Fibonacci and Lucas numbers. In 2002, Mansour [8] studied some finite sums involving reciprocal of the binomial coefficients and deduced a generalization of (1) together with many other results. He proved that n k =0 ak bn−k (n + 1)( ab)n+1. Differentiating our formula with respect to s and x, and setting particular values for s and x, we discover many interesting identities involving the reciprocals of the binomial coefficients and harmonic numbers, as well as Fibonacci and Lucas numbers. We continue with the following lemma, which plays a key role in the proofs of our main results
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