Pell and Pell–Lucas Sequences of Fractional Order - Jagan Mohan Jonnalagadda; Marius-F. Danca (Fractal and Fractional)
The purpose of this paper is to introduce the fractional Pell numbers, together with several properties, via a Grünwald–Letnikov fractional operator of orders <semantics>q∈(0,1)</semantics> and <semantics>q∈(1,2)</semantics>. This paper also explores the fractional Pell–Lucas numbers and their properties.Due to the long-term memory property, fractional Pell sequences and fractional Pell–Lucas sequences present potential applications in modeling and computation. The closed form is deduced, and the numerical schemes are determined. The fractional characteristic equation is introduced, and it is shown that its solutions include a fractional silver ratio depending on the fractional order. In addition, the tiling problem and the concept of the fractional silver spiral are considered.A MATLAB program for applying the use of the fractional silver ratio is presented.