View source for Lucas polynomials


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The polynomials $V_n(x)$ (cf.
[[#References|[a1]]] and
[[#References|[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|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|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
[[#References|[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
[[#References|[a4]]] found the roots of $V_n(x)$. These are $x_j = 2i\cos((2j+1)\pi/2n)$,
$j=1,\dots,n-1$. Bicknell
[[#References|[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
[[#References|[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|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|generating function]] of $V_n(x,y)$, generalizes (a8).

Ch.A. Charalambides
[[#References|[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.
[[#References|[a6]]] and
[[#References|[a8]]] and
[[Fibonacci polynomials|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.
[[#References|[a7]]]). Charalambides
[[#References|[a3]]] showed that the reliability of a circular
[[Consecutive k out of n-system|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====
<table><TR><TD valign="top">[a1]</TD>
 <TD valign="top"> G.E. Bergum, V.E. Hoggatt, Jr., "Irreducibility of Lucas and generalized Lucas polynomials" ''Fibonacci Quart.'' , '''12''' (1974) pp. 95–100</TD>
</TR><TR><TD valign="top">[a2]</TD>
 <TD valign="top"> M. Bicknell, "A primer for the Fibonacci numbers VII" ''Fibonacci Quart.'' , '''8''' (1970) pp. 407–420</TD>
</TR><TR><TD valign="top">[a3]</TD>
 <TD valign="top"> 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</TD>
</TR><TR><TD valign="top">[a4]</TD>
 <TD valign="top"> V.E. Hoggatt Jr., M. Bicknell, "Roots of Fibonacci polynomials" ''Fibonacci Quart.'' , '''11''' (1973) pp. 271–274</TD>
</TR><TR><TD valign="top">[a5]</TD>
 <TD valign="top"> E. Lucas, "Theorie de fonctions numeriques simplement periodiques" ''Amer. J. Math.'' , '''1''' (1878) pp. 184–240; 289–321</TD>
</TR><TR><TD valign="top">[a6]</TD>
 <TD valign="top"> 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</TD>
</TR><TR><TD valign="top">[a7]</TD>
 <TD valign="top"> 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</TD>
</TR><TR><TD valign="top">[a8]</TD>
 <TD valign="top"> A.N. Philippou, C. Georghiou, G.N. Philippou, "Fibonacci-type polynomials of order $k$ with probability applications" ''Fibonacci Quart.'' , '''23''' (1985) pp. 100–105</TD>
</TR><TR><TD valign="top">[a9]</TD>
 <TD valign="top"> J. Riordan, "Combinatorial Identities" , Wiley (1968)</TD>
</TR></table>