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The polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l1301201.png" /> (cf. [[#References|[a1]]] and [[#References|[a5]]]) given by
+
{{MSC|11B39,11K99,60C05}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l1301202.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
The polynomials $V_n(x)$ (cf.
 +
{{Cite|BeHo}} and
 +
{{Cite|Lu}}) 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 $\alpha(x) + \beta(x) = x$ and $\alpha(x)\beta(x) = -1$; 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 $y$.
  
They reduce to the Lucas numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l1301203.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l1301204.png" />, and they satisfy several identities, which may be easily proved by induction, e.g.:
+
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
 +
{{Cite|Ri}} 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
 +
{{Cite|HoBi}} found the roots of $V_n(x)$. These are $x_j = 2i\cos((2j+1)\pi/2n)$,
 +
$j=1,\dots,n-1$. Bicknell
 +
{{Cite|Bi}} 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
 +
{{Cite|BeHo}} the bivariate Lucas polynomials $V_n(x,y)$ by the
 +
recursion
 +
$$\left.\begin{align*}
 +
V_0(x,y) &= 2,\\
 +
V_1(x,y) &= x,\\
 +
V_n(x,y) &= x V_{n-1}(x,y)+ y V_{n-2}(x,y),\quad n = 2,3,\ldots
 +
\end{align*}\right\rbrace\tag{a11}$$
 +
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).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l1301205.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
Ch.A. Charalambides
 +
{{Cite|Ch}} 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.
 +
{{Cite|Ph}} and
 +
{{Cite|PhGePh2}} 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.
 +
{{Cite|PhGePh}}). Charalambides
 +
{{Cite|Ch}} 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
 +
$$\begin{align*}R_c(p; k,n) &= q^n L_n^{\;(k)}\Big(\frac{p}{q}\Big)\\
 +
&= -q^n + \sum_{j=0}^{[n/(k+1)]}(-1)^j\frac{n}{n-jk}\;
 +
\frac{(n-jk)!}{j!(n-jk-j)!}p^jq^{jk}.\end{align*}\tag{a17}$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l1301206.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
+
====References====
 
+
{|
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l1301207.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
+
|-
 
+
|valign="top"|{{Ref|BeHo}}||valign="top"| G.E. Bergum, V.E. Hoggatt, Jr., "Irreducibility of Lucas and generalized Lucas polynomials" ''Fibonacci Quart.'', '''12''' (1974) pp. 95–100  {{MR|0349581}}  {{ZBL|0277.12002}}   
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l1301208.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>
+
|-
 
+
|valign="top"|{{Ref|Bi}}||valign="top"| M. Bicknell, "A primer for the Fibonacci numbers. VII" ''Fibonacci Quart.'', '''8''' (1970) pp. 407–420     
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l1301209.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>
+
|-
 
+
|valign="top"|{{Ref|Ch}}||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  {{MR|1131401}}  {{ZBL|0745.11014}}   
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012010.png" /> denote the [[Fibonacci polynomials|Fibonacci polynomials]];
+
|-
 
+
|valign="top"|{{Ref|HoBi}}||valign="top"| V.E. Hoggatt Jr., M. Bicknell, "Roots of Fibonacci polynomials" ''Fibonacci Quart.'', '''11''' (1973) pp. 271–274  {{MR|0323700}}  {{ZBL|0272.33004}}   
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a7)</td></tr></table>
+
|-
 
+
|valign="top"|{{Ref|Lu}}||valign="top"| E. Lucas, "Theorie de fonctions numeriques simplement periodiques" ''Amer. J. Math.'', '''1''' (1878) pp. 184–240; 289–321  {{MR|1505176}} {{MR|1505164}} {{MR|1505161}}   
where
+
|-
 
+
|valign="top"|{{Ref|Ph}}||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  {{MR|0857826}}  {{ZBL|0602.60023}}   
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012012.png" /></td> </tr></table>
+
|-
 
+
|valign="top"|{{Ref|PhGePh}}||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  {{MR|0712573}}  {{ZBL|0524.10008}}   
so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012013.png" />; and
+
|-
 
+
|valign="top"|{{Ref|PhGePh2}}||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  {{MR|0797126}}  {{ZBL|0563.10014}}   
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a8)</td></tr></table>
+
|-
 
+
|valign="top"|{{Ref|Ri}}||valign="top"| J. Riordan, "Combinatorial Identities", Wiley (1968) {{MR|0231725}}  {{ZBL|0194.00502}}   
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012015.png" /></td> </tr></table>
+
|-
 
+
|}
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012016.png" /> denotes the greatest integer in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012017.png" />.
 
 
 
The Lucas polynomials are related to the [[Chebyshev polynomials|Chebyshev polynomials]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012019.png" />, by
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a9)</td></tr></table>
 
 
 
J. Riordan [[#References|[a9]]] considered the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012021.png" /> and the Lucas-type polynomials
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a10)</td></tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012023.png" /></td> </tr></table>
 
 
 
in a derivation of Chebyshev-type pairs of inverse relations. V.E. Hoggatt Jr. and M. Bicknell [[#References|[a4]]] found the roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012024.png" />. These are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012026.png" />. Bicknell [[#References|[a2]]] showed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012027.png" /> divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012028.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012029.png" /> is an odd multiple of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012030.png" />. G.E. Bergum and Hoggatt Jr. introduced in [[#References|[a1]]] the bivariate Lucas polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012031.png" /> by the recursion
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012032.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a11)</td></tr></table>
 
 
 
generalized (a7) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012033.png" />, and showed that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012034.png" /> are irreducible polynomials over the rational numbers if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012035.png" /> for some positive integer (cf. also [[Irreducible polynomial|Irreducible polynomial]]). The formula
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012036.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a12)</td></tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012037.png" /></td> </tr></table>
 
 
 
which may be derived by induction on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012038.png" /> or by expanding the [[Generating function|generating function]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012039.png" />, generalizes (a8).
 
 
 
Ch.A. Charalambides [[#References|[a3]]] introduced and studied the Lucas and Lucas-type polynomials of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012042.png" />. The Lucas-type polynomials of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012044.png" /> satisfy the recurrence
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012045.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a13)</td></tr></table>
 
  
These polynomials have the binomial and multinomial expansions
+
====Comments====
 
+
The Lucas polynomials $V_n(x)$ are a special case of the [[Dickson polynomial]]s $D_n(x,a)$ by taking $a$ to be $-1$.
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012046.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a14)</td></tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012047.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012048.png" /></td> </tr></table>
 
 
 
where the second summation is taken over all non-negative integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012049.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012050.png" />, and they are related to the Fibonacci-type polynomials of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012051.png" /> (cf. [[#References|[a6]]] and [[#References|[a8]]] and [[Fibonacci polynomials|Fibonacci polynomials]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012052.png" />, by
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012053.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a15)</td></tr></table>
 
 
 
Furthermore,
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012054.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a16)</td></tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012055.png" /></td> </tr></table>
 
 
 
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012056.png" /> are the Fibonacci polynomials of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012057.png" /> (cf. [[#References|[a7]]]). Charalambides [[#References|[a3]]] showed that the reliability of a circular [[Consecutive k out of n-system|consecutive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012058.png" />-out-of-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012059.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012060.png" />-system]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012061.png" />, whose components function independently with probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012062.png" /> is given by
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012063.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a17)</td></tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012064.png" /></td> </tr></table>
 
 
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012065.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012066.png" />, longest runs, and reliability of consecutive-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012067.png" />-out-of-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012068.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012069.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012070.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130120/l13012071.png" /> 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>
 

Latest revision as of 16:05, 22 January 2021

2020 Mathematics Subject Classification: Primary: 11B39,11K99,60C05 [MSN][ZBL]

The polynomials $V_n(x)$ (cf. [BeHo] and [Lu]) 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 $\alpha(x) + \beta(x) = x$ and $\alpha(x)\beta(x) = -1$; 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 [Ri] 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 [HoBi] found the roots of $V_n(x)$. These are $x_j = 2i\cos((2j+1)\pi/2n)$, $j=1,\dots,n-1$. Bicknell [Bi] 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 [BeHo] the bivariate Lucas polynomials $V_n(x,y)$ by the recursion $$\left.\begin{align*} V_0(x,y) &= 2,\\ V_1(x,y) &= x,\\ V_n(x,y) &= x V_{n-1}(x,y)+ y V_{n-2}(x,y),\quad n = 2,3,\ldots \end{align*}\right\rbrace\tag{a11}$$ 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 [Ch] 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. [Ph] and [PhGePh2] 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. [PhGePh]). Charalambides [Ch] 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

[BeHo] G.E. Bergum, V.E. Hoggatt, Jr., "Irreducibility of Lucas and generalized Lucas polynomials" Fibonacci Quart., 12 (1974) pp. 95–100 MR0349581 Zbl 0277.12002
[Bi] M. Bicknell, "A primer for the Fibonacci numbers. VII" Fibonacci Quart., 8 (1970) pp. 407–420
[Ch] 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 MR1131401 Zbl 0745.11014
[HoBi] V.E. Hoggatt Jr., M. Bicknell, "Roots of Fibonacci polynomials" Fibonacci Quart., 11 (1973) pp. 271–274 MR0323700 Zbl 0272.33004
[Lu] E. Lucas, "Theorie de fonctions numeriques simplement periodiques" Amer. J. Math., 1 (1878) pp. 184–240; 289–321 MR1505176 MR1505164 MR1505161
[Ph] 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 MR0857826 Zbl 0602.60023
[PhGePh] 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 MR0712573 Zbl 0524.10008
[PhGePh2] A.N. Philippou, C. Georghiou, G.N. Philippou, "Fibonacci-type polynomials of order $k$ with probability applications" Fibonacci Quart., 23 (1985) pp. 100–105 MR0797126 Zbl 0563.10014
[Ri] J. Riordan, "Combinatorial Identities", Wiley (1968) MR0231725 Zbl 0194.00502

Comments

The Lucas polynomials $V_n(x)$ are a special case of the Dickson polynomials $D_n(x,a)$ by taking $a$ to be $-1$.

How to Cite This Entry:
Lucas polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lucas_polynomials&oldid=17297
This article was adapted from an original article by Andreas N. Philippou (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article