Rocky Mountain Journal of Mathematics

Determinantal and permanental representations of Fibonacci type numbers and polynomials

Kenan Kaygisiz and Adem Şahin

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In this paper, we compute terms of the matrix $A_{(k)}^{\infty }$, which contains Fibonacci type numbers and polynomials, with the help of determinants and permanents of various Hessenberg matrices. In addition, we show that determinants of these Hessenberg matrices can be obtained by using combinations. The results that we obtain are important, since the matrix $A_{(k)}^{\infty } $ is a general form of Fibonacci type numbers and polynomials, such as $k$ sequences of the generalized order-$k$ Fibonacci and Pell numbers, generalized bivariate Fibonacci $p$-polynomials, bivariate Fibonacci and Pell $p$-polynomials, second kind Chebyshev polynomials and bivariate Jacobsthal polynomials, etc.

Article information

Rocky Mountain J. Math., Volume 46, Number 1 (2016), 227-242.

First available in Project Euclid: 23 May 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11B37: Recurrences {For applications to special functions, see 33-XX}
Secondary: 15A15: Determinants, permanents, other special matrix functions [See also 19B10, 19B14] 15A51

Matrix $A_(k)^\infty $ generalized Fibonacci polynomials Hessenberg matrix


Kaygisiz, Kenan; Şahin, Adem. Determinantal and permanental representations of Fibonacci type numbers and polynomials. Rocky Mountain J. Math. 46 (2016), no. 1, 227--242. doi:10.1216/RMJ-2016-46-1-227.

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