On some Fibonacci-type polynomials (Q2908294)

The generalized Fibonacci-type (GFT) polynomials being discussed in this paper are polynomials \(\{G^{(k)}_n (x)\}\) in a single variable defined by the recursion NEWLINE\[NEWLINEG_n^{(k)} (x) = x^k G^{(k)}_{(n-1)} (x) + G_{(n-2)}^{(k)}(x),NEWLINE\]NEWLINE with \(G_0^{(k)}(x)=a\); \(G_1^{(k)}(x)=x+b\); \(a,b\in\mathbb Z; a,b>0\).NEWLINENEWLINEThe general class of GFT polynomials is canvassed by letting the initial conditions vary.NEWLINENEWLINEThese polynomials are generalizations of ``polynomials of Fibonacci-type'', in which \(k = 1\), again, letting the initial conditions vary [\textit{T. Amdeberhan}, Integers 10, No. 1, 13--18 (2010; Zbl 1209.11021)]. The GFT polynomials are, in turn generalizations of the classical Fibonacci polynomials: NEWLINE\[NEWLINEF_n(x)=xF_{n-1}(x)+F_{n-2}(x),\quad\text{with }F_1(x)=1,\;F_2(x)=x,\;n>0,NEWLINE\]NEWLINE which, when \(x = 1\), define the Fibonacci numbers [\textit{T. Koshy}, Fibonacci and Lucas numbers with applications. New York, NY: Wiley (2001; Zbl 0984.11010)].NEWLINENEWLINENEWLINEThe standard questions asked about GFT polynomials include questions about the nature and properties of the zeros of the functions \(G_n^{(k)}\), \(n\leq 2, k > 0\). [\textit{H. Yu, Y. Wang} and \textit{M. He}, Fibonacci Q. 34, No. 4, 320--322 (1996; Zbl 0860.11008); \textit{F. Mátyás}, Acta Acad. Paedagog. Agriensis, Sect. Mat. (N.S.) 25, 15--20 (1998; Zbl 0923.11034); \textit{P. E. Ricci}, Riv. Mat. Univ. Parma, V. Ser. 4, 137--146 (1995; Zbl 0866.11008)]NEWLINENEWLINENEWLINEThe paper contains four theorems and a corollary:NEWLINENEWLINENEWLINETheorem 1 consists of a Binet type formula [Koshy, loc. cit.] in terms of the roots of the characteristic equation of the sequence.NEWLINENEWLINENEWLINETheorem 2 gives a generating function for the sequence \(\{G_n^k x\}\).NEWLINENEWLINENEWLINETheorem 3 presents a sequence of tri-diagonal Jacobi matrices (equivalently, doubly Hessenberg matrices) whose determinants are alleged to be the sequence of polynomials \(\{G_n^{(k)} x\}\).NEWLINENEWLINENEWLINETheorem 4. Letting \(\{g_n^{(k)}\}\) be the maximal real roots of \(\{G_n^{(k)} x\}\), asserts that \(\{g_{2n}^{(k)}\}\) is a decreasing sequence, and that \(\{g_{2n-1}^{(k)}\}\) is an increasing sequence.NEWLINENEWLINENEWLINECorollary 5 asserts that the zeros of \(G_n^{(k)}(x)\) lie in the open interval \((0,\infty)\).NEWLINENEWLINETheorem 3 is obviously incorrect. The matrix in question contains no symbol \(a\). But, \(G_0^k=a\). However, \(a\) is used in the proof of the theorem. Moreover, its existence in the construction of the matrix is implied in the use of Gershgorin's circle theorem at the end of the proof of Corollary 5. So one suspects that the matrix in Theorem 3 is the result of poor proof-reading and a typographical error. There is also a sign problem down one of the diagonals.NEWLINENEWLINEThe statement of Corollary 5 is ambiguous, as it stands, though a reading of the proof reveals its intent.NEWLINENEWLINENEWLINEThe poor use of English makes the paper difficult to read, and in some instances, to understand. It should have had the advice of someone competent in the English language. Otherwise, the results are interesting.