Multivariate Fibonacci Polynomials of Order K and the Multiparameter Negative Binomial Distribution of the Same Order

Abstract

Unless otherwise explicitly stated, in this paper k and r are fixed positive integers, n and n _i (1≤i≤k) are non-negative integers as specified, p and q _i (1≤i≤k) are real numbers in the interval (0,1) which satisfy the relation p+q₁+…+q _k =1, and x and x _i (1≤i≤k) are real numbers in the interval (0,∞). Let {F ^(k) _n (x)} ^∞ _n be the sequence of Fibonacci-type polynomials of order k, i.e. F ^(k)₀ (x)=0, F ^(k)₁ (x)=1, and

$$F_n^{\left( k \right)}\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {x\sum\limits_{i = 1}^n {F_{n - i}^{\left( k \right)}\left( x \right)} if 2 \leqslant n \leqslant k + 1,} \\ {x\sum\limits_{i = 1}^k {F_{n - i}^{\left( k \right)}\left( x \right)} if n \geqslant k + 2.} \end{array}} \right.$$

((1.1))