What is a name for given below polynomials?

Question

Physicists has a lot work with special polynomials, so I want to ask:

Is there anybody who knows what are called this polynomials f(q) ( q as variable)?

$1$

$q$

$q^2+1$

$q^3+2q$

$q^4+3q^2+1$

$q^5+4q^3+3q$

$q^6+5q^4+6q^2+1$

$q^7+6q^5+10q^3+4q$

$q^8+7q^6+15q^4+10q^2+1$

$q^9+8q^7+21q^5+20q^3+5q$

$q^{10}+9q^8+28q^6+35q^4+15q^2+1$

and so on?

They are generated by simple generating function $(1+qT)^k$ assuming that there is a relation $T^2 = 1+qT$

For every f(q) $f(1) =F_q$ where $F_q$ - q'th Fibonacci number.

It looks like for every of them, it is possible to find ( complex) roots as finite expression, and f(q)=0 is solvable ( by Sage) however roots are rather complicated numbers.

Are they known? Analyzed?

Answer 1

A quick search on The On-Line Encyclopedia of Integer Sequences for some of the coefficients (5,20,21,8,1,1,15,35,28,9,1) gives an entry that mentions the Fibonacci polynomials in its comments.