Fibonacci polynomials

Abstract

The Fibonacci polynomials ${\lbrace F_n (x) \rbrace}_{n \geq 0}$ have been studied in multiple ways, [$\href{https://www.imsc.res.in/~viennot/wa_files/viennotop1983-ocr.pdf}{1}$,$\href{https://www.fq.math.ca/Scanned/11-3/hoggatt1.pdf}{6}$,$\href{https://www.fq.math.ca/Scanned/12-2/hoggatt1.pdf}{7}$,$\href{https://www.rivmat.unipr.it/fulltext/1995-4/1995-4-15.pdf}{9}$].In this paper we study them by means of the theory of heaps of Viennot [11, 12]. In this setting our polynomials form a basis ${\lbrace P_n (x) \rbrace}_{n \geq 0}$ with $P_n (x)$ monic of degree $n$. This given, we are forced to set $P_n (x) = F_{n+1} (x)$. The heaps setting extends the Flajolet view$\href{https://doi.org/10.1016/0012-365X(80)90050-3}{[4]}$ of the classical theory of orthogonal polynomials given by a three term recursion [3, 10]. Thus with heaps most of the identities for the polynomials $P_n (x)$’s can be derived by combinatorial arguments. Using the present setting we derive a variety of new identities. We must mention that the theory of heaps is presented here without restrictions. This is much more than needed to deal with the Fibonacci polynomials. We do this to convey a flavor of the power of heaps. In $\href{https://link.springer.com/book/10.1007/978-3-030-58373-6 }{[5]}$ there is a chapter dedicated to heaps where most of its contents are dedicated to applications of the theory. In this paper we improve upon the developments in $\href{https://link.springer.com/book/10.1007/978-3-030-58373-6 }{[5]}$ by adding details that were omitted there.