General Solution of a Fibonacci-like Recursion Relation and Applications

In a series of papers published over the past few years, [1]? [2]s a method called the combinatorics function techniques or CFT5 was perfected to obtain the solution of any linear partial difference equation subject to a set of initial values. Fibonacci-like recursion relations are a special case of difference equations that could be solved by the CFT method,, Although many applications of the CFT have been published elsewhere, [3], [4], the study here leads to original results and provides a natural generalization of the problem investigated by Hock and McQuistan in "Occupational Degeneracy for X-Bell Particles on a Saturated XxN Lattice Space" [5]. For the reader who is not familiar with the CFT method, we summarize briefly the results of [2]. Consider a function B depending on n variables {m1^m15 ms$ ...5 mn) . The evaluation points Ms in the associated n-dimensional space whose coordinates are (rnl9 m2S »*»s mn) and vector M5 whose components are the same as the coordinates of point M9 will be used interchangeably for convenience. The multivariable function B is said to satisfy a partial difference equation when its value at point M9 5(M), is linearly related to its values at shifted arguments such as M Afe3 i.e.,