Some Properties of Convolved k-Fibonacci Numbers

are the terms of the sequence 0, 1, 1, 2, 3, 5, . . . , wherein each term is the sum of the two previous terms, beginning with the values F 0 = 0 and F 1 = 1. Besides the usual Fibonacci numbers many kinds of generalizations of these numbers have been presented in the literature. In particular, a generalization is the k-Fibonacci numbers. For any positive real number k, the k-Fibonacci sequence, say {F k,n } n∈N, is defined recurrently by


Introduction
Fibonacci numbers and their generalizations have many interesting properties and applications to almost every field of science and art (e.g., see [1]).The Fibonacci numbers   are the terms of the sequence 0, 1, 1, 2, 3, 5, . . ., wherein each term is the sum of the two previous terms, beginning with the values  0 = 0 and  1 = 1.
Besides the usual Fibonacci numbers many kinds of generalizations of these numbers have been presented in the literature.In particular, a generalization is the -Fibonacci numbers.
(i) Binet formula: Definition 1.For any positive real number , the -Lucas sequence, say { , } ∈N , is defined recurrently by If  = 1 we have the classical Lucas numbers.Some properties that the -Lucas numbers verify are summarized below (see [13] for the proofs).

Convolved 𝑘-Fibonacci Numbers
Definition 2. The convolved -Fibonacci numbers  () , are defined by Note that Moreover, from Humbert polynomials (with  = ,  = 2,  = /2,  = −1,  = −, and  = 1), we have If  = 1 we obtain the combinatorial formula of -Fibonacci numbers.In Tables 1, 2, and 3 some values of convolved -Fibonacci numbers are provided.The purpose of this paper is to investigate the properties of these numbers.

Hessenberg Matrices and Convolved 𝑘-Fibonacci Numbers
An upper Hessenberg matrix,   , is an  ×  matrix, where  , = 0 whenever  >  + 1 and  +1, ̸ = 0 for some .That is, all entries below the superdiagonal are 0 but the matrix is not upper triangular: We consider a type of upper Hessenberg matrix whose determinants are -Fibonacci numbers.Some results about Fibonacci numbers and Hessenberg can be found in [14].The following known result about upper Hessenberg matrices will be used.) ) ) , then In particular, if then from Theorem 5 we have that det  ()  =  ,+1 , ( = 1, 2, . . . ) .
It is clear that the principal minor  () () of  ()  is equal to  ,  ,−+1 .It follows that the principal minor  () ( 1 ,  2 , . . .,   ) of the matrix  ()   is obtained by deleting rows and columns with indices 1 ≤  1 <  2 < ⋅ ⋅ ⋅ <   ≤ : Then we have the following theorem.Theorem 6.Let  ()  − , ( = 0, 1, 2, . . .,  − 1) be the sum of all principal minors of  ()   or order  − .Then Since the coefficients of the characteristic polynomial of a matrix are, up to the sign, sums of principal minors of the matrix, then we have the following.(31) Therefore the corollary is obtained.