Lucas polynomials

The polynomials $V_n(x)$ (cf. [a1] and [a5]) given by $$\left.\begin{align}V_0(x) &= 2,\\ V_1(x) &= x,\\ V_n(x) &= x V_{n-1}(x)+V_{n-2}(x),\quad n = 2,3,\dots \end{align}\quad\right\}\tag{a1}$$ They reduce to the Lucas numbers $L_n$ for $x=1$, and they satisfy several identities, which may be easily proved by induction, e.g.: \begin{alignat}{1} &V_{-n}(x) &=\ & (-1)^nV_n(x);\tag{a2}\\ &V_{m+n}(x) &=& V_m(x)V_n(x) - (-1)^nV_{m-n}(x);\tag{a3}\\ &V_{2n}(x) &=& V_n^{\;2}(x)-2(-1)^n;\tag{a4}\\ &V_{2n+1}(x)\ &=& V_{n+1}(x)V_n(x)-(-1)^nx;\tag{a5}\\ &U_{2n}(x) &=& U_n(x)V_n(x),\tag{a6} \end{alignat} where $U_m(x)$ denote the Fibonacci polynomials; $$V_n(x) = \alpha^n(x)+\beta^n(x),\tag{a7}$$ where $$\alpha(x) = \frac{x+(x^2+4)^{1/2}}{2},\quad \beta(x) = \frac{x-(x^2+4)^{1/2}}{2},$$ so that $\alpha(x)\beta(x) = -1$; and $$V_n(x) = \sum_{j=0}^{[n/2]} \frac{n}{n-j}\; \frac{(n-j)!}{j!(n-2j)!}\; x^{n-2j},\quad n=1,2,\dots,\tag{a8} $$ where $[y]$ denotes the greatest integer in $y$.

The Lucas polynomials are related to the Chebyshev polynomials $T_n(x) = \cos(n\theta)$, $\cos(\theta) = x$, by $$V_n(x) = 2i^{-n} T_n\Big(\frac{ix}{2}\Big),\; i = (-1)^{1/2}.\tag{a9}$$ J. Riordan [a9] considered the polynomials $h_n(x) = i^{-n}V_n(ix)$ and the Lucas-type polynomials $$L_n(x) = \sum_{j=0}^{[n/2]} \frac{n}{n-j}\; \frac{(n-j)!}{j!(n-2j)!}\; x^{n-j} = x^{n/2}V_n(x^{1/2}),\; n = 1,2,\dots,\tag{a10}$$ in a derivation of Chebyshev-type pairs of inverse relations. V.E. Hoggatt Jr. and M. Bicknell [a4] found the roots of $V_n(x)$. These are $x_j = 2i\cos((2j+1)\pi/2n)$, $j=1,\dots,n-1$. Bicknell [a2] showed that $V_m(x)$ divides $V_n(x)$ if and only if $n$ is an odd multiple of $m$. G.E. Bergum and Hoggatt Jr. introduced in [a1] the bivariate Lucas polynomials $V_n(x,y)$ by the recursion $$\begin{equation} \left.\begin{align}V_0(x,y) &= 2,\\ V_1(x,y) &= x,\\ V_n(x,y) &= x V_{n-1}(x,y)+ y V_{n-2}(x,y),\quad n = 2,3,\dots \end{align}\right\}\end{equation}\tag{a11}$$ generalized (a7) for $V_n(x,y)$, and showed that the $V_n(x,y)$ are irreducible polynomials over the rational numbers if and only if $n=2^k$ for some positive integer (cf. also Irreducible polynomial). The formula $$V_n(x,y) = \sum_{j=0}^{[n/2]} \frac{n}{n-j}\; \frac{(n-j)!}{j!(n-2j)!} \; x^{n-2j}y^j,\; n = 1,2,\dots,\tag{a12}$$ which may be derived by induction on $n$ or by expanding the generating function of $V_n(x,y)$, generalizes (a8).

Ch.A. Charalambides [a3] introduced and studied the Lucas and Lucas-type polynomials of order $k$, $V_n^{\;(k)}(x)$ and $L_n^{\;(k)}(x)$. The Lucas-type polynomials of order $k$ satisfy the recurrence $$\left.\begin{alignat}{1}L_1^{\;(k)}(x) &= x,\\ L_n^{\;(k)}(x) &= x\Big(n+\sum_{j=1}^{n-1}L_{n-j}^{\;(k)}(x)\Big),\; & n = 2,\dots,k, \\ L_n^{\;(k)}(x) &= x\sum_{j=1}^k L_{n-j}^{\;(k)}(x),\; & n = k+1,k+2,\dots \end{alignat}\right\}\tag{a13}$$ These polynomials have the binomial and multinomial expansions $$\begin{alignat}{1}L_n^{\;(k)}(x) &= -1 + \sum_{j=0}^{[n/(k+1)]}(-1)^j\frac{n}{n-jk}\;\frac{(n-jk)!}{j!(n-jk-j)!}x^j(1+x)^{n-jk-j}\\ &=\sum \frac{n_1+2n_2+\cdots+kn_k}{n_1+\cdots+n_k}\;\frac{(n_1+\cdots+n_k)!}{n_1!\cdots n_k!} x^{n_1+\cdots+n_k},\end{alignat}\tag{a14} $$ where the second summation is taken over all non-negative integers $n_1,\dots,n_k$ such that $n_1+2n_2+\cdots +kn_k = n$, and they are related to the Fibonacci-type polynomials of order $k$ (cf. [a6] and [a8] and Fibonacci polynomials), $F_n^{\;(k)}(x)$, by $$L_n^{\;(k)}(x) = x \sum_{j=1}^{\min\{n,k\}} jF_{n-j+1}^{\;(k)}(x).\tag{a15}$$ Furthermore, $$\begin{align}V_n^{\;(k)}(x) &= x^{-n} L_n^{\;(k)}(x^k)\\ &= \sum_{j=1}^{\min\{n,k\}} jx^{k-j+1}U_{n-j+1}^{\;(k)}(x),\; n=1,2,\dots,\; k=2,3,\dots,\end{align}\tag{a16}$$ where the $U_n^{\;(k)}(x)$ are the Fibonacci polynomials of order $k$ (cf. [a7]). Charalambides [a3] showed that the reliability of a circular consecutive $k$-out-of-$n$: $F$-system, $R_c(p; k,n)$, whose components function independently with probability $p$ (and $q = 1-p$) is given by $$\begin{align}R_c(p; k,n) &= q^n L_n^{\;(k)}\Big(\frac{p}{q}\Big)\\ &= -q^n + \sum_{j=0}^{[n/(k+1)]}(-1)^j\frac{n}{n-jk}\; \frac{(n-jk)!}{j!(n-jk-j)!}p^jq^{jk}.\end{align}\tag{a17}$$

References

[a1]	G.E. Bergum, V.E. Hoggatt, Jr., "Irreducibility of Lucas and generalized Lucas polynomials" Fibonacci Quart. , 12 (1974) pp. 95–100
[a2]	M. Bicknell, "A primer for the Fibonacci numbers VII" Fibonacci Quart. , 8 (1970) pp. 407–420
[a3]	Ch.A. Charalambides, "Lucas numbers and polynomials of order $k$ and the length of the longest circular success run" Fibonacci Quart. , 29 (1991) pp. 290–297
[a4]	V.E. Hoggatt Jr., M. Bicknell, "Roots of Fibonacci polynomials" Fibonacci Quart. , 11 (1973) pp. 271–274
[a5]	E. Lucas, "Theorie de fonctions numeriques simplement periodiques" Amer. J. Math. , 1 (1878) pp. 184–240; 289–321
[a6]	A.N. Philippou, "Distributions and Fibonacci polynomials of order $k$, longest runs, and reliability of consecutive-$k$-out-of-$n$: $F$ systems" A.N. Philippou (ed.) G.E. Bergum (ed.) A.F. Horadam (ed.) , Fibonacci Numbers and Their Applications , Reidel (1986) pp. 203–227
[a7]	A.N. Philippou, C. Georghiou, G.N. Philippou, "Fibonacci polynomials of order $k$, multinomial expansions and probability" Internat. J. Math. Math. Sci. , 6 (1983) pp. 545–550
[a8]	A.N. Philippou, C. Georghiou, G.N. Philippou, "Fibonacci-type polynomials of order $k$ with probability applications" Fibonacci Quart. , 23 (1985) pp. 100–105
[a9]	J. Riordan, "Combinatorial Identities" , Wiley (1968)