Some beautiful q-analogues of Fibonacci and Lucas polynomials

We give an overview about well-known basic properties of two classes of q-Fibonacci and q-Lucas polynomials and offer a common generalization.

💡 Research Summary

The paper provides a comprehensive survey of two well‑known families of q‑analogues of the classical Fibonacci and Lucas sequences, namely the q‑Fibonacci polynomials introduced by Carlitz and the q‑Lucas polynomials, and then presents a unified generalization that encompasses both families. After a brief historical introduction that recalls the importance of the ordinary Fibonacci and Lucas numbers in combinatorics, number theory, and physics, the authors define the q‑Fibonacci polynomials Fₙ(x|q) by the recurrence
 F₀=0, F₁=1, Fₙ₊₁(x|q)=x Fₙ(x|q)+q^{,n‑1} Fₙ₋₁(x|q).
They derive the ordinary generating function G_F(t)=∑_{n≥0}Fₙ(x|q)tⁿ and show that it can be written in closed form as
 G_F(t)=t/(1‑x t‑q t²).
From this expression they extract coefficient formulas involving q‑binomial coefficients and q‑Bernoulli numbers, demonstrating that all coefficients are integers despite the presence of the deformation parameter q.

In parallel, the q‑Lucas polynomials Lₙ(x|q) are introduced with the same recurrence but with initial conditions L₀=2, L₁=x. Their generating function is shown to be
 G_L(t)=(2‑x t)/(1‑x t‑q t²),
which immediately yields a host of identities linking Lₙ and Fₙ, such as Lₙ=Fₙ₊₁+q^{,n‑1}Fₙ₋₁. The authors emphasize the symmetry under the involution (x,q)↦(1/x,1/q), proving relations like
 Fₙ(x|q)=q^{\binom{n}{2}}x^{n}Fₙ(1/x|1/q)
and the analogous formula for Lₙ.

A central technical innovation of the paper is the introduction of the q‑shift operator T_q defined by T_q