Research On (p,q)-Fibonacci Numbers And Its Related Problems

Fibonacci numbers is an interesting mathematical problem,which has been widely studied all the time.It was originally a sequence of second-order linear recurrence relation obtained from rabbit reproduction problem,and later it was extended to Lucas number,Pell number,Jacobsthal number,etc.Many scholars studied their Binet formula,Generating functions and identity properties.Later,many scholars extended Fibonacci numbers to polynomial sequences and studied various properties of polynomials.These sequences and their extension played an important role in combinatorial mathematics,number theory,numerical analysis and many other fields.In addition,many scholars combined Fibonacci numbers with quaternions to study Fibonacci quaternions and various generalized Fibonacci quaternions,which enriched the properties of quaternions.Based on the previous research results,this paper discusses the(p,q)-Fibonacci numbers,studies its identity,polynomial properties and related conclusions extended to quaternion field.The main research results are as follows:In the first chapter,we introduce the extended forms of Fibonacci numbers,that is,(p,q)-Fibonacci numbers and(p,q)-Lucas numbers,give their Binet formulas,and study some identities of power sum of(p,q)-Fibonacci numbers.In the second chapter,firstly,we defined(p,q)-Fibonacci polynomial and its matrix representation.Its 9)th power,determinant and some properties are obtained by matrix calculation.Secondly,Hadamard product of matrix of(p,q)-Fibonacci polynomial and its inverse matrix is studied,and finally its determinant and trace are calculated.In the third chapter,we study the Horadam quaternion,which is a special case of(p,q)-Fibonaccci quaternion,calculate the exponential generating function of the quaternion,and derive some identities of Horadam quaternion about binomial sum according to Binet formula,which generalizes the properties of quaternion.