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The Fibonacci sequence has many interesting properties and studied by many mathematicians. The terms of this sequence appear in nature and is connected with combinatorics and other branches of mathematics. In this paper, we investigate the orbit of a special subgroup of the modular group. Taking Tc:= ..., we determined the orbit {Tr c (8): r = N}. Each rational number of this set is the form Pr(c)/Qr(c), where Pr(c) and Qr(c) are the polynomials in Z[c]. It is shown that Pr(1), and Qr(1) the sum of the coefficients of the polynomials Pr(c) and Qr(c) respectively, are the Fibonacci numbers, where Pr(c) = .... [ABSTRACT FROM AUTHOR]

In this paper, we prove three new Katugampola fractional Hermite-Hadamard type inequalities for harmonically convex functions by using the left and the right fractional integrals independently. One of our Katugampola fractional Hermite-Hadamard type inequalities is better than given in [17]. Also, we give two new Katugampola fractional identities for differentiable functions. By using these identities, we obtain some new trapezoidal type inequalities for harmonically convex functions. Our results generalize many results from earlier papers. [ABSTRACT FROM AUTHOR]