[ Generating functions Special polynomials (p) Find the generating functions of the Fibonacci polynomial sequence $F(x, z)=F_{0}(x)+F_{1}(x) z+F_{2}(x) z^{2}$ $+\ldots+F_{n}(x) z^{n}+\ldots$

and the Lucas polynomial sequence $L(x, z)=L_{0}(x)+L_{1}(x) z+L_{2}(x) z^{2}+\ldots+L_{n}(x) z^{n}+\ldots$

Definitions of Fibonacci and Lucas polynomials can be found in the reference manual.

#
From the equalities $F_{0}(x)=0, F_{1}(x)=1, F_{n}(x)=x F_{n-1}(x)+F_{n-2}(x)$ we get $F(x, z)=z+\left(x z+z^{2}\right) F(x, z)$.

The generating function of the sequence of Lucas polynomials can be obtained similarly or

## Answer

$F(x, z)=z\left(1-x z-z^{2}\right)^{-1}, L(x, z)=(2-x z)\left(1-x z-z^{2}\right)^{-1}$.