Abstract
The main purpose of this paper is to give many new formulas involving the Fibonacci numbers, the golden ratio, the Lucas numbers, and other special numbers. By using generating functions for the special numbers with their functional equations method, we also give many new relations among the Fibonacci numbers, the Lucas numbers, the golden ratio, the Stirling numbers, and other special numbers. Moreover, some applications of the Fibonacci numbers and the golden ratio in chemistry are given.
Highlights
In almost all the intellectual achievements of humankind, the discovery of different sets of numbers has played many important roles
It is well known that the Fibonacci numbers and other special numbers are used in mathematical models that frequently appear in theory of chemistry theory involving the structure of the elements, the periodic table of the elements, and their symmetrical and crystalline structures
By using generating functions and well-known formulas involving the Fibonacci numbers and the Lucas numbers, we derive some new formulas related to the golden ratio, which is given by the Equation (13)
Summary
In almost all the intellectual achievements of humankind, the discovery of different sets of numbers has played many important roles. Have studied chemistry and its application using number theory involving the Fibonacci numbers, the Lucas numbers, and the golden ratio. The motivation of this paper is to provide many new formulas and relations involving the Fibonacci numbers, the Lucas numbers, the golden ratio, the Stirling numbers, and other special numbers with the aid of the generating functions for the special numbers. We give interesting relations and formulas between the numbers Fn , Ln , α (golden ratio), and β These results have the potential to be used in number theory, chemistry, biology, physics, and other areas. With the aid of generating functions, we give some new identities and relations involving the numbers Fn , Ln , `n (v), and golden ratio α.
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