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The polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f1300901.png" /> (cf. [[#References|[a1]]] and [[#References|[a4]]]) given by
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If the TeX and formula formatting is correct and if all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
  
<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/f/f130/f130090/f1300902.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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Out of 121 formulas, 115 were replaced by TEX code.-->
  
They reduce to the [[Fibonacci numbers|Fibonacci numbers]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f1300903.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f1300904.png" /> and they satisfy several identities, which may be easily proved by induction, e.g.:
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The polynomials $U _ { n } ( x )$ (cf. [[#References|[a1]]] and [[#References|[a4]]]) 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/f/f130/f130090/f1300905.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
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\begin{equation} \tag{a1} \left. \begin{cases} { U _ { 0 } ( x ) = 0, } \\ { U _ { 1 } ( x ) = 1, } \\ { U _ { n } ( x ) = x U _ { n - 1 } ( x ) + U _ { n - 2 } ( x ) , \quad n = 2,3, \dots . } \end{cases} \right. \end{equation}
  
<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/f/f130/f130090/f1300906.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
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They reduce to the [[Fibonacci numbers|Fibonacci numbers]] $F _ { n }$ for $x = 1$ and they satisfy several identities, which may be easily proved by induction, e.g.:
  
<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/f/f130/f130090/f1300907.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
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\begin{equation} \tag{a2} U _ { - n } ( x ) = ( - 1 ) ^ { n - 1 } U _ { n } ( x ); \end{equation}
  
<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/f/f130/f130090/f1300908.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>
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\begin{equation} \tag{a3} U _ { m + n } ( x ) = U _ { m + 1 } ( x ) U _ { n } ( x ) + U _ { m } ( x ) U _ { n - 1 } ( x ); \end{equation}
 +
 
 +
\begin{equation} \tag{a4} U _ { n + 1 } ( x ) U _ { n - 1 } ( x ) - U _ { n } ^ { 2 } ( x ) = ( - 1 ) ^ { n } ; \end{equation}
 +
 
 +
\begin{equation} \tag{a5} U _ { n } ( x ) = \frac { \alpha ^ { n } ( x ) - \beta ^ { n } ( x ) } { \alpha ( x ) - \beta ( x ) }, \end{equation}
  
 
where
 
where
  
<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/f/f130/f130090/f1300909.png" /></td> </tr></table>
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\begin{equation*} \alpha ( x ) = \frac { x + ( x ^ { 2 } + 4 ) ^ { 1 / 2 } } { 2 } , \beta ( x ) = \frac { x - ( x ^ { 2 } + 4 ) ^ { 1 / 2 } } { 2 }, \end{equation*}
  
so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009010.png" />; and
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so that $\alpha ( x ) \beta ( x ) = - 1$; and
  
<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/f/f130/f130090/f13009011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>
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\begin{equation} \tag{a6} U _ { n + 1 } ( x ) = \sum _ { j = 0 } ^ { [ n / 2 ] } \frac { ( n - j ) ! } { j ! ( n - 2 j ) ! } x ^ { n - 2 j } , n = 0,1, \dots , \end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009012.png" /> denotes the greatest integer in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009013.png" />.
+
where $[ y ]$ denotes the greatest integer in $y$.
  
W.A. Webb and E.A. Parberry [[#References|[a14]]] showed that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009014.png" /> are irreducible polynomials over the ring of integers if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009015.png" /> is a prime number (cf. also [[Irreducible polynomial|Irreducible polynomial]]). They also found that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009017.png" />, are the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009018.png" /> roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009019.png" /> (see also [[#References|[a2]]]). M. Bicknell [[#References|[a1]]] proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009020.png" /> divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009021.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009022.png" /> divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009023.png" />. V.E. Hoggatt Jr., and C.T. Long [[#References|[a3]]] introduced the bivariate Fibonacci polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009024.png" /> by the recursion
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W.A. Webb and E.A. Parberry [[#References|[a14]]] showed that the $U _ { n } ( x )$ are irreducible polynomials over the ring of integers if and only if $n$ is a prime number (cf. also [[Irreducible polynomial|Irreducible polynomial]]). They also found that $x _ { j } = 2 i \operatorname { cos } ( j \pi / n )$, $j = 1 , \dots , n - 1$, are the $n - 1$ roots of $U _ { n } ( x )$ (see also [[#References|[a2]]]). M. Bicknell [[#References|[a1]]] proved that $U _ { m } ( x )$ divides $U _ { n } ( x )$ if and only if $m$ divides $n$. V.E. Hoggatt Jr., and C.T. Long [[#References|[a3]]] introduced the bivariate Fibonacci polynomials $U _ { n } ( x , y )$ 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/f/f130/f130090/f13009025.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a7)</td></tr></table>
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\begin{equation} \tag{a7} \left. \begin{cases}{ U _ { 0 } ( x , y ) = 0 ,}\\{ U _ { 1 } ( x , y ) = 1, }\\{ U _ { n } ( x , y ) = x U _ { n - 1 } ( x , y ) + y U _ { n - 2 } ( x , y ), }\\{ \quad n = 2 , 3 , \ldots ,}\end{cases} \right. \end{equation}
  
and they showed that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009026.png" /> are irreducible over the rational numbers if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009027.png" /> is a prime number. They also generalized (a5) and proved that
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and they showed that the $U _ { n } ( x , y )$ are irreducible over the rational numbers if and only if $n$ is a prime number. They also generalized (a5) and proved that
  
<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/f/f130/f130090/f13009028.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a8)</td></tr></table>
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\begin{equation} \tag{a8} U _ { n + 1 } ( x , y ) = \sum _ { j = 0 } ^ { [ n / 2 ] } \frac { ( n - j ) ! } { j ! ( n - 2 j ) ! } x ^ { n - 2 j } y ^ { j }, \end{equation}
  
<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/f/f130/f130090/f13009029.png" /></td> </tr></table>
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\begin{equation*} n = 0,1 , \ldots \end{equation*}
  
In a series of papers, A.N. Philippou and his associates (cf. [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]], [[#References|[a8]]], [[#References|[a9]]], [[#References|[a10]]], [[#References|[a11]]], [[#References|[a12]]], [[#References|[a13]]]) introduced and studied Fibonacci, Fibonacci-type and multivariate Fibonacci polynomials of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009030.png" />, and related them to probability and reliability. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009031.png" /> be a fixed positive integer greater than or equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009032.png" />. The Fibonacci polynomials of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009035.png" />, are defined by
+
In a series of papers, A.N. Philippou and his associates (cf. [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]], [[#References|[a8]]], [[#References|[a9]]], [[#References|[a10]]], [[#References|[a11]]], [[#References|[a12]]], [[#References|[a13]]]) introduced and studied Fibonacci, Fibonacci-type and multivariate Fibonacci polynomials of order $k$, and related them to probability and reliability. Let $k$ be a fixed positive integer greater than or equal to $2$. The Fibonacci polynomials of order $k$, $U _ { N } ^ { ( k ) } ( x )$, are defined 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/f/f130/f130090/f13009036.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a9)</td></tr></table>
+
\begin{equation} \tag{a9} \left\{ \begin{array}{l}{ U _ { 0 } ^ { ( k ) } ( x ) = 0, }\\{ U _ { 1 } ^ { ( k ) } ( x ) = 1, }\\{ U _ { n } ^ { ( k ) } ( x ) = \sum _ { j = 1 } ^ { n } x ^ { k - j } U _ { n - j } ^ { ( k ) } ( x ) , \quad n = 2 , \ldots , k, }\\{ U _ { n } ^ { ( k ) } ( x ) = \sum _ { j = 1 } ^ { k } x ^ { k - j } U _ { n - j } ^ { ( k ) } ( x ), }\\{ n = k + 1 , k + 2 , \ldots . }\end{array} \right. \end{equation}
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009037.png" /> these reduce to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009038.png" />, and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009039.png" /> these reduce to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009040.png" />, the Fibonacci numbers of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009042.png" /> (cf. [[#References|[a11]]]). Deriving and expanding the [[Generating function|generating function]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009043.png" />, they [[#References|[a12]]] obtained the following generalization of (a6) in terms of the multinomial coefficients (cf. [[Multinomial coefficient|Multinomial coefficient]]):
+
For $k = 2$ these reduce to $U _ { n } ( x )$, and for $x = 1$ these reduce to $U _ { n } ^ { ( k ) }$, the Fibonacci numbers of order $k$ (cf. [[#References|[a11]]]). Deriving and expanding the [[Generating function|generating function]] of $U _ { N } ^ { ( k ) } ( x )$, they [[#References|[a12]]] obtained the following generalization of (a6) in terms of the multinomial coefficients (cf. [[Multinomial coefficient|Multinomial coefficient]]):
  
<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/f/f130/f130090/f13009044.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a10)</td></tr></table>
+
\begin{equation} \tag{a10} U _ { n + 1 } ^ { ( k ) } ( x ) = \sum \frac { ( n _ { 1 } + \ldots + n _ { k } ) ! } { n _ { 1 } ! \ldots n _ { k } ! } x ^ { k ( n _ { 1 } + \ldots + n _ { k } ) - n }, \end{equation}
  
<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/f/f130/f130090/f13009045.png" /></td> </tr></table>
+
\begin{equation*} n = 0,1 , \ldots , \end{equation*}
  
where the sum is taken over all non-negative integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009046.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009047.png" />. They also obtained a simpler formula in terms of binomial coefficients. As a byproduct of (a10), they were able to relate these polynomials to the number of trials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009048.png" /> until the occurrence of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009049.png" />th consecutive success in independent trials with success probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009050.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009051.png" /> this formula reduces to
+
where the sum is taken over all non-negative integers $n _ { 1 } , \ldots , n _ { k }$ such that $n _ { 1 } + 2 n _ { 2 } + \ldots + k n _ { k } = n$. They also obtained a simpler formula in terms of binomial coefficients. As a byproduct of (a10), they were able to relate these polynomials to the number of trials $N _ { k }$ until the occurrence of the $k$th consecutive success in independent trials with success probability $p$. For $p = 1 / 2$ this formula reduces to
  
<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/f/f130/f130090/f13009052.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a11)</td></tr></table>
+
\begin{equation} \tag{a11} \mathsf{P} ( N _ { k } = n + k ) = \frac { U _ { n + 1 } ^ { ( k ) } } { 2 ^ { n + k } } ,\, n = 0,1, \dots . \end{equation}
  
The Fibonacci-type polynomials of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009055.png" />, defined by
+
The Fibonacci-type polynomials of order $k$, $F _ { n } ^ { ( k ) } ( x )$, defined 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/f/f130/f130090/f13009056.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a12)</td></tr></table>
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009056.png"/></td> <td style="width:5%;text-align:right;" valign="top">(a12)</td></tr></table>
  
have simpler multinomial and binomial expansions than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009057.png" />. The two families of polynomials are related by
+
have simpler multinomial and binomial expansions than $U _ { N } ^ { ( k ) } ( x )$. The two families of polynomials are related 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/f/f130/f130090/f13009058.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a13)</td></tr></table>
+
\begin{equation} \tag{a13} U _ { n } ^ { ( k ) } ( x ) = x ^ { 1 - n } F _ { n } ^ { ( k ) } ( x ^ { k } ) ,\; n = 1,2 , \ldots . \end{equation}
  
Furthermore, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009059.png" />,
+
Furthermore, with $q = 1 - p$,
  
<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/f/f130/f130090/f13009060.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a14)</td></tr></table>
+
\begin{equation} \tag{a14} \mathsf{P} ( N _ { k } = n ) = p ^ { n } F _ { n + 1 - k } ^ { ( k ) } \left( \frac { q } { p } \right) , \end{equation}
  
<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/f/f130/f130090/f13009061.png" /></td> </tr></table>
+
\begin{equation*} n = k , k + 1 , \dots . \end{equation*}
  
Assuming that the components of a [[Consecutive k out of n-system|consecutive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009062.png" />-out-of-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009063.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009064.png" />-system]] are ordered linearly and function independently with probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009065.png" />, Philippou [[#References|[a6]]] found that the reliability of the system, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009066.png" />, is given by
+
Assuming that the components of a [[Consecutive k out of n-system|consecutive $k$-out-of-$n$: $F$-system]] are ordered linearly and function independently with probability $p$, Philippou [[#References|[a6]]] found that the reliability of the system, $R _ { l } ( p ; k , n )$, 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/f/f130/f130090/f13009067.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a15)</td></tr></table>
+
\begin{equation} \tag{a15} R_{l} ( p ; k , n ) = p ^ { - 1 } q ^ { n + 1 } F _ { n + 2 } \left( \frac { p } { q } \right), \end{equation}
  
<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/f/f130/f130090/f13009068.png" /></td> </tr></table>
+
\begin{equation*} n = k , k + 1 , \dots . \end{equation*}
  
If the components of the system are ordered circularly, then its reliability, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009069.png" />, is given by (cf. [[#References|[a10]]])
+
If the components of the system are ordered circularly, then its reliability, $R _ { c } ( p ; k , n )$, is given by (cf. [[#References|[a10]]])
  
<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/f/f130/f130090/f13009070.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a16)</td></tr></table>
+
\begin{equation} \tag{a16} R _ { c } ( p ; k , n ) = p q ^ { n - 1 } \sum _ { j = 1 } ^ { k } j F _ { n - j + 1 } ^ { ( k ) }  ( \frac { p } { q } ), \end{equation}
  
<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/f/f130/f130090/f13009071.png" /></td> </tr></table>
+
\begin{equation*} n = k , k + 1 , \dots . \end{equation*}
  
Next, denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009072.png" /> the number of independent trials with success probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009073.png" /> until the occurrence of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009074.png" />th <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009075.png" />th consecutive success. It is well-known [[#References|[a5]]] that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009076.png" /> has the negative [[Binomial distribution|binomial distribution]] of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009077.png" /> with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009078.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009079.png" />. Philippou and C. Georghiou [[#References|[a9]]] have related this probability distribution to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009080.png" />-fold convolution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009081.png" /> with itself, say <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009082.png" />, as follows:
+
Next, denote by $N _ { k , r }$ the number of independent trials with success probability $p$ until the occurrence of the $r$th $k$th consecutive success. It is well-known [[#References|[a5]]] that $N _ { k , r }$ has the negative [[Binomial distribution|binomial distribution]] of order $k$ with parameters $r$ and $p$. Philippou and C. Georghiou [[#References|[a9]]] have related this probability distribution to the $( r - 1 )$-fold convolution of $F _ { n } ^ { ( k ) } ( x )$ with itself, say $F _ { n , r } ^ { ( k ) } ( x )$, as follows:
  
<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/f/f130/f130090/f13009083.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a17)</td></tr></table>
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009083.png"/></td> <td style="width:5%;text-align:right;" valign="top">(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/f/f130/f130090/f13009084.png" /></td> </tr></table>
+
\begin{equation*} n = k r , k r + 1 , \dots, \end{equation*}
  
which reduces to (a14) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009085.png" />, and they utilized effectively relation (a17) for deriving two useful expressions, a binomial and a recurrence one, for calculating the above probabilities.
+
which reduces to (a14) for $r = 1$, and they utilized effectively relation (a17) for deriving two useful expressions, a binomial and a recurrence one, for calculating the above probabilities.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009086.png" />. The multivariate Fibonacci polynomials of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009088.png" /> (cf. [[#References|[a8]]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009089.png" />, are defined by the recurrence
+
Let $\mathbf x = ( x _ { 1 } , \dots , x _ { k } )$. The multivariate Fibonacci polynomials of order $k$ (cf. [[#References|[a8]]]), $H _ { n } ^ { ( k ) } ( \mathbf{x} )$, are defined by 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/f/f130/f130090/f13009090.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a18)</td></tr></table>
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009090.png"/></td> <td style="width:5%;text-align:right;" valign="top">(a18)</td></tr></table>
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009091.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009092.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009093.png" />, and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009094.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009095.png" />. These polynomials have the following multinomial expansion:
+
For $\mathbf{x} = ( x ^ { k - 1 } , x ^ { k - 2 } , \dots , 1 )$, $H _ { n } ^ { ( k ) } ( \mathbf{x} ) = U _ { n } ^ { ( k ) } ( \mathbf{x} )$, $n = 0,1 , \dots$, and for $\mathbf{x} = ( x , \ldots , x )$, $H _ { n } ^ { ( k ) } ( \mathbf{x} ) = F _ { n } ^ { ( k ) } ( x )$. These polynomials have the following multinomial expansion:
  
<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/f/f130/f130090/f13009096.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a19)</td></tr></table>
+
\begin{equation} \tag{a19} H _ { n + 1 } ^ { ( k ) } ( x ) = \sum \frac { ( n _ { 1 } + \ldots + n _ { k } ) ! } { n _ { 1 } ! \ldots n _ { k } ! } x _ { 1 } ^ { n _ { 1 } } \ldots x _ { k } ^ { n _ { k } }, \end{equation}
  
<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/f/f130/f130090/f13009097.png" /></td> </tr></table>
+
\begin{equation*} n = 0,1 , \ldots , \end{equation*}
  
where the sum is taken over all non-negative integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009098.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f13009099.png" />. Let the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f130090100.png" /> be distributed as a multi-parameter negative binomial distribution of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f130090101.png" /> (cf. [[#References|[a7]]]) with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f130090102.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f130090103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f130090104.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f130090105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f130090106.png" />). Philippou and D.L. Antzoulakos [[#References|[a8]]] showed that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f130090107.png" />-fold convolution, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f130090108.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f130090109.png" /> with itself is related to this distribution by
+
where the sum is taken over all non-negative integers $n _ { 1 } , \ldots , n _ { k }$ such that $n _ { 1 } + 2 n _ { 2 } + \ldots + k n _ { k } = n$. Let the random variable $X$ be distributed as a multi-parameter negative binomial distribution of order $k$ (cf. [[#References|[a7]]]) with parameters $r , q_{ 1} , \dots , q _ { k }$ ($r &gt; 0$, $0 &lt; q _ { j } &lt; 1$ for $j = 1 , \dots , k$ and $0 &lt; q _ { 1 } + \ldots + q _ { k } &lt; 1$). Philippou and D.L. Antzoulakos [[#References|[a8]]] showed that the $( r - 1 )$-fold convolution, $H _ { n , r } ^ { ( k ) } ( \mathbf x )$, of $H _ { n } ^ { ( k ) } ( \mathbf{x} )$ with itself is related to this distribution 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/f/f130/f130090/f130090110.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a20)</td></tr></table>
+
\begin{equation} \tag{a20} \mathsf{P} ( X = n ) = p ^ { r } H _ { n + 1 , r } ^ { ( k ) } ( q _ { 1 } , \dots , q _ { k } ), \end{equation}
  
<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/f/f130/f130090/f130090111.png" /></td> </tr></table>
+
\begin{equation*} n = 0,1 , \ldots \end{equation*}
  
 
Furthermore, they have effectively utilized relation (a20) in deriving a recurrence for calculating the above probabilities.
 
Furthermore, they have effectively utilized relation (a20) in deriving a recurrence for calculating the above probabilities.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</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">[a2]</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">[a3]</TD> <TD valign="top">  V.E. Hoggatt Jr.,  C.T. Long,  "Divisibility properties of generalized Fibonacci polynomials"  ''Fibonacci Quart.'' , '''12'''  (1974)  pp. 113–120</TD></TR><TR><TD valign="top">[a4]</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">[a5]</TD> <TD valign="top">  A.N. Philippou,  "The negative binomial distribution of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f130090112.png" /> and some of its properties"  ''Biom. J.'' , '''26'''  (1984)  pp. 789–794</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/f/f130/f130090/f130090113.png" />, longest runs, and reliability of concecutive-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f130090114.png" />-out-of-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f130090115.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f130090116.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,  "On multiparameter distributions of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f130090117.png" />"  ''Ann. Inst. Statist. Math.'' , '''40'''  (1988)  pp. 467–475</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  A.N. Philippou,  D.L. Antzoulakos,  "Multivariate Fibonacci polynomials of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f130090118.png" /> and the multiparameter negative binomial distribution of the same order"  G.E. Bergum (ed.)  A.N. Philippou (ed.)  A.F. Horadam (ed.) , ''Applications of Fibonacci Numbers'' , '''3''' , Kluwer Acad. Publ.  (1990)  pp. 273–279</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  A.N. Philippou,  C. Georghiou,  "Convolutions of Fibonacci-type polynomials of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f130090119.png" /> and the negative binomial distributions of the same order"  ''Fibonacci Quart.'' , '''27'''  (1989)  pp. 209–216</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  A.N. Philippou,  F.S. Makri,  "Longest circular runs with an application in reliability via the Fibonacci-type polynomials of order k"  G.E. Bergum (ed.)  A.N. Philippou (ed.)  A.F. Horadam (ed.) , ''Applications of Fibonacci Numbers'' , '''3''' , Kluwer Acad. Publ.  (1990)  pp. 281–286</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  A.N. Philippou,  A.A. Muwafi,  "Waiting for the kth consecutive success and the Fibonacci sequence of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130090/f130090120.png" />"  ''Fibonacci Quart.'' , '''20'''  (1982)  pp. 28–32</TD></TR><TR><TD valign="top">[a12]</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/f/f130/f130090/f130090121.png" />, multinomial expansions and probability"  ''Internat. J. Math. Math. Sci.'' , '''6'''  (1983)  pp. 545–550</TD></TR><TR><TD valign="top">[a13]</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/f/f130/f130090/f130090122.png" /> with probability applications"  ''Fibonacci Quart.'' , '''23'''  (1985)  pp. 100–105</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  W.A. Webb,  E.A. Parberry,  "Divisibility properties of Fibonacci polynomials"  ''Fibonacci Quart.'' , '''7'''  (1969)  pp. 457–463</TD></TR></table>
+
<table><tr><td valign="top">[a1]</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">[a2]</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">[a3]</td> <td valign="top">  V.E. Hoggatt Jr.,  C.T. Long,  "Divisibility properties of generalized Fibonacci polynomials"  ''Fibonacci Quart.'' , '''12'''  (1974)  pp. 113–120</td></tr><tr><td valign="top">[a4]</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">[a5]</td> <td valign="top">  A.N. Philippou,  "The negative binomial distribution of order $k$ and some of its properties"  ''Biom. J.'' , '''26'''  (1984)  pp. 789–794</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 concecutive-$k$-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,  "On multiparameter distributions of order $k$"  ''Ann. Inst. Statist. Math.'' , '''40'''  (1988)  pp. 467–475</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  A.N. Philippou,  D.L. Antzoulakos,  "Multivariate Fibonacci polynomials of order $k$ and the multiparameter negative binomial distribution of the same order"  G.E. Bergum (ed.)  A.N. Philippou (ed.)  A.F. Horadam (ed.) , ''Applications of Fibonacci Numbers'' , '''3''' , Kluwer Acad. Publ.  (1990)  pp. 273–279</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  A.N. Philippou,  C. Georghiou,  "Convolutions of Fibonacci-type polynomials of order $k$ and the negative binomial distributions of the same order"  ''Fibonacci Quart.'' , '''27'''  (1989)  pp. 209–216</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  A.N. Philippou,  F.S. Makri,  "Longest circular runs with an application in reliability via the Fibonacci-type polynomials of order k"  G.E. Bergum (ed.)  A.N. Philippou (ed.)  A.F. Horadam (ed.) , ''Applications of Fibonacci Numbers'' , '''3''' , Kluwer Acad. Publ.  (1990)  pp. 281–286</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  A.N. Philippou,  A.A. Muwafi,  "Waiting for the kth consecutive success and the Fibonacci sequence of order $k$"  ''Fibonacci Quart.'' , '''20'''  (1982)  pp. 28–32</td></tr><tr><td valign="top">[a12]</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">[a13]</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">[a14]</td> <td valign="top">  W.A. Webb,  E.A. Parberry,  "Divisibility properties of Fibonacci polynomials"  ''Fibonacci Quart.'' , '''7'''  (1969)  pp. 457–463</td></tr></table>

Latest revision as of 17:43, 1 July 2020

The polynomials $U _ { n } ( x )$ (cf. [a1] and [a4]) given by

\begin{equation} \tag{a1} \left. \begin{cases} { U _ { 0 } ( x ) = 0, } \\ { U _ { 1 } ( x ) = 1, } \\ { U _ { n } ( x ) = x U _ { n - 1 } ( x ) + U _ { n - 2 } ( x ) , \quad n = 2,3, \dots . } \end{cases} \right. \end{equation}

They reduce to the Fibonacci numbers $F _ { n }$ for $x = 1$ and they satisfy several identities, which may be easily proved by induction, e.g.:

\begin{equation} \tag{a2} U _ { - n } ( x ) = ( - 1 ) ^ { n - 1 } U _ { n } ( x ); \end{equation}

\begin{equation} \tag{a3} U _ { m + n } ( x ) = U _ { m + 1 } ( x ) U _ { n } ( x ) + U _ { m } ( x ) U _ { n - 1 } ( x ); \end{equation}

\begin{equation} \tag{a4} U _ { n + 1 } ( x ) U _ { n - 1 } ( x ) - U _ { n } ^ { 2 } ( x ) = ( - 1 ) ^ { n } ; \end{equation}

\begin{equation} \tag{a5} U _ { n } ( x ) = \frac { \alpha ^ { n } ( x ) - \beta ^ { n } ( x ) } { \alpha ( x ) - \beta ( x ) }, \end{equation}

where

\begin{equation*} \alpha ( x ) = \frac { x + ( x ^ { 2 } + 4 ) ^ { 1 / 2 } } { 2 } , \beta ( x ) = \frac { x - ( x ^ { 2 } + 4 ) ^ { 1 / 2 } } { 2 }, \end{equation*}

so that $\alpha ( x ) \beta ( x ) = - 1$; and

\begin{equation} \tag{a6} U _ { n + 1 } ( x ) = \sum _ { j = 0 } ^ { [ n / 2 ] } \frac { ( n - j ) ! } { j ! ( n - 2 j ) ! } x ^ { n - 2 j } , n = 0,1, \dots , \end{equation}

where $[ y ]$ denotes the greatest integer in $y$.

W.A. Webb and E.A. Parberry [a14] showed that the $U _ { n } ( x )$ are irreducible polynomials over the ring of integers if and only if $n$ is a prime number (cf. also Irreducible polynomial). They also found that $x _ { j } = 2 i \operatorname { cos } ( j \pi / n )$, $j = 1 , \dots , n - 1$, are the $n - 1$ roots of $U _ { n } ( x )$ (see also [a2]). M. Bicknell [a1] proved that $U _ { m } ( x )$ divides $U _ { n } ( x )$ if and only if $m$ divides $n$. V.E. Hoggatt Jr., and C.T. Long [a3] introduced the bivariate Fibonacci polynomials $U _ { n } ( x , y )$ by the recursion

\begin{equation} \tag{a7} \left. \begin{cases}{ U _ { 0 } ( x , y ) = 0 ,}\\{ U _ { 1 } ( x , y ) = 1, }\\{ U _ { n } ( x , y ) = x U _ { n - 1 } ( x , y ) + y U _ { n - 2 } ( x , y ), }\\{ \quad n = 2 , 3 , \ldots ,}\end{cases} \right. \end{equation}

and they showed that the $U _ { n } ( x , y )$ are irreducible over the rational numbers if and only if $n$ is a prime number. They also generalized (a5) and proved that

\begin{equation} \tag{a8} U _ { n + 1 } ( x , y ) = \sum _ { j = 0 } ^ { [ n / 2 ] } \frac { ( n - j ) ! } { j ! ( n - 2 j ) ! } x ^ { n - 2 j } y ^ { j }, \end{equation}

\begin{equation*} n = 0,1 , \ldots \end{equation*}

In a series of papers, A.N. Philippou and his associates (cf. [a5], [a6], [a7], [a8], [a9], [a10], [a11], [a12], [a13]) introduced and studied Fibonacci, Fibonacci-type and multivariate Fibonacci polynomials of order $k$, and related them to probability and reliability. Let $k$ be a fixed positive integer greater than or equal to $2$. The Fibonacci polynomials of order $k$, $U _ { N } ^ { ( k ) } ( x )$, are defined by

\begin{equation} \tag{a9} \left\{ \begin{array}{l}{ U _ { 0 } ^ { ( k ) } ( x ) = 0, }\\{ U _ { 1 } ^ { ( k ) } ( x ) = 1, }\\{ U _ { n } ^ { ( k ) } ( x ) = \sum _ { j = 1 } ^ { n } x ^ { k - j } U _ { n - j } ^ { ( k ) } ( x ) , \quad n = 2 , \ldots , k, }\\{ U _ { n } ^ { ( k ) } ( x ) = \sum _ { j = 1 } ^ { k } x ^ { k - j } U _ { n - j } ^ { ( k ) } ( x ), }\\{ n = k + 1 , k + 2 , \ldots . }\end{array} \right. \end{equation}

For $k = 2$ these reduce to $U _ { n } ( x )$, and for $x = 1$ these reduce to $U _ { n } ^ { ( k ) }$, the Fibonacci numbers of order $k$ (cf. [a11]). Deriving and expanding the generating function of $U _ { N } ^ { ( k ) } ( x )$, they [a12] obtained the following generalization of (a6) in terms of the multinomial coefficients (cf. Multinomial coefficient):

\begin{equation} \tag{a10} U _ { n + 1 } ^ { ( k ) } ( x ) = \sum \frac { ( n _ { 1 } + \ldots + n _ { k } ) ! } { n _ { 1 } ! \ldots n _ { k } ! } x ^ { k ( n _ { 1 } + \ldots + n _ { k } ) - n }, \end{equation}

\begin{equation*} n = 0,1 , \ldots , \end{equation*}

where the sum is taken over all non-negative integers $n _ { 1 } , \ldots , n _ { k }$ such that $n _ { 1 } + 2 n _ { 2 } + \ldots + k n _ { k } = n$. They also obtained a simpler formula in terms of binomial coefficients. As a byproduct of (a10), they were able to relate these polynomials to the number of trials $N _ { k }$ until the occurrence of the $k$th consecutive success in independent trials with success probability $p$. For $p = 1 / 2$ this formula reduces to

\begin{equation} \tag{a11} \mathsf{P} ( N _ { k } = n + k ) = \frac { U _ { n + 1 } ^ { ( k ) } } { 2 ^ { n + k } } ,\, n = 0,1, \dots . \end{equation}

The Fibonacci-type polynomials of order $k$, $F _ { n } ^ { ( k ) } ( x )$, defined by

(a12)

have simpler multinomial and binomial expansions than $U _ { N } ^ { ( k ) } ( x )$. The two families of polynomials are related by

\begin{equation} \tag{a13} U _ { n } ^ { ( k ) } ( x ) = x ^ { 1 - n } F _ { n } ^ { ( k ) } ( x ^ { k } ) ,\; n = 1,2 , \ldots . \end{equation}

Furthermore, with $q = 1 - p$,

\begin{equation} \tag{a14} \mathsf{P} ( N _ { k } = n ) = p ^ { n } F _ { n + 1 - k } ^ { ( k ) } \left( \frac { q } { p } \right) , \end{equation}

\begin{equation*} n = k , k + 1 , \dots . \end{equation*}

Assuming that the components of a consecutive $k$-out-of-$n$: $F$-system are ordered linearly and function independently with probability $p$, Philippou [a6] found that the reliability of the system, $R _ { l } ( p ; k , n )$, is given by

\begin{equation} \tag{a15} R_{l} ( p ; k , n ) = p ^ { - 1 } q ^ { n + 1 } F _ { n + 2 } \left( \frac { p } { q } \right), \end{equation}

\begin{equation*} n = k , k + 1 , \dots . \end{equation*}

If the components of the system are ordered circularly, then its reliability, $R _ { c } ( p ; k , n )$, is given by (cf. [a10])

\begin{equation} \tag{a16} R _ { c } ( p ; k , n ) = p q ^ { n - 1 } \sum _ { j = 1 } ^ { k } j F _ { n - j + 1 } ^ { ( k ) } ( \frac { p } { q } ), \end{equation}

\begin{equation*} n = k , k + 1 , \dots . \end{equation*}

Next, denote by $N _ { k , r }$ the number of independent trials with success probability $p$ until the occurrence of the $r$th $k$th consecutive success. It is well-known [a5] that $N _ { k , r }$ has the negative binomial distribution of order $k$ with parameters $r$ and $p$. Philippou and C. Georghiou [a9] have related this probability distribution to the $( r - 1 )$-fold convolution of $F _ { n } ^ { ( k ) } ( x )$ with itself, say $F _ { n , r } ^ { ( k ) } ( x )$, as follows:

(a17)

\begin{equation*} n = k r , k r + 1 , \dots, \end{equation*}

which reduces to (a14) for $r = 1$, and they utilized effectively relation (a17) for deriving two useful expressions, a binomial and a recurrence one, for calculating the above probabilities.

Let $\mathbf x = ( x _ { 1 } , \dots , x _ { k } )$. The multivariate Fibonacci polynomials of order $k$ (cf. [a8]), $H _ { n } ^ { ( k ) } ( \mathbf{x} )$, are defined by the recurrence

(a18)

For $\mathbf{x} = ( x ^ { k - 1 } , x ^ { k - 2 } , \dots , 1 )$, $H _ { n } ^ { ( k ) } ( \mathbf{x} ) = U _ { n } ^ { ( k ) } ( \mathbf{x} )$, $n = 0,1 , \dots$, and for $\mathbf{x} = ( x , \ldots , x )$, $H _ { n } ^ { ( k ) } ( \mathbf{x} ) = F _ { n } ^ { ( k ) } ( x )$. These polynomials have the following multinomial expansion:

\begin{equation} \tag{a19} H _ { n + 1 } ^ { ( k ) } ( x ) = \sum \frac { ( n _ { 1 } + \ldots + n _ { k } ) ! } { n _ { 1 } ! \ldots n _ { k } ! } x _ { 1 } ^ { n _ { 1 } } \ldots x _ { k } ^ { n _ { k } }, \end{equation}

\begin{equation*} n = 0,1 , \ldots , \end{equation*}

where the sum is taken over all non-negative integers $n _ { 1 } , \ldots , n _ { k }$ such that $n _ { 1 } + 2 n _ { 2 } + \ldots + k n _ { k } = n$. Let the random variable $X$ be distributed as a multi-parameter negative binomial distribution of order $k$ (cf. [a7]) with parameters $r , q_{ 1} , \dots , q _ { k }$ ($r > 0$, $0 < q _ { j } < 1$ for $j = 1 , \dots , k$ and $0 < q _ { 1 } + \ldots + q _ { k } < 1$). Philippou and D.L. Antzoulakos [a8] showed that the $( r - 1 )$-fold convolution, $H _ { n , r } ^ { ( k ) } ( \mathbf x )$, of $H _ { n } ^ { ( k ) } ( \mathbf{x} )$ with itself is related to this distribution by

\begin{equation} \tag{a20} \mathsf{P} ( X = n ) = p ^ { r } H _ { n + 1 , r } ^ { ( k ) } ( q _ { 1 } , \dots , q _ { k } ), \end{equation}

\begin{equation*} n = 0,1 , \ldots \end{equation*}

Furthermore, they have effectively utilized relation (a20) in deriving a recurrence for calculating the above probabilities.

References

[a1] M. Bicknell, "A primer for the Fibonacci numbers VII" Fibonacci Quart. , 8 (1970) pp. 407–420
[a2] V.E. Hoggatt Jr., M. Bicknell, "Roots of Fibonacci polynomials" Fibonacci Quart. , 11 (1973) pp. 271–274
[a3] V.E. Hoggatt Jr., C.T. Long, "Divisibility properties of generalized Fibonacci polynomials" Fibonacci Quart. , 12 (1974) pp. 113–120
[a4] E. Lucas, "Theorie de fonctions numeriques simplement periodiques" Amer. J. Math. , 1 (1878) pp. 184–240; 289–321
[a5] A.N. Philippou, "The negative binomial distribution of order $k$ and some of its properties" Biom. J. , 26 (1984) pp. 789–794
[a6] A.N. Philippou, "Distributions and Fibonacci polynomials of order $k$, longest runs, and reliability of concecutive-$k$-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, "On multiparameter distributions of order $k$" Ann. Inst. Statist. Math. , 40 (1988) pp. 467–475
[a8] A.N. Philippou, D.L. Antzoulakos, "Multivariate Fibonacci polynomials of order $k$ and the multiparameter negative binomial distribution of the same order" G.E. Bergum (ed.) A.N. Philippou (ed.) A.F. Horadam (ed.) , Applications of Fibonacci Numbers , 3 , Kluwer Acad. Publ. (1990) pp. 273–279
[a9] A.N. Philippou, C. Georghiou, "Convolutions of Fibonacci-type polynomials of order $k$ and the negative binomial distributions of the same order" Fibonacci Quart. , 27 (1989) pp. 209–216
[a10] A.N. Philippou, F.S. Makri, "Longest circular runs with an application in reliability via the Fibonacci-type polynomials of order k" G.E. Bergum (ed.) A.N. Philippou (ed.) A.F. Horadam (ed.) , Applications of Fibonacci Numbers , 3 , Kluwer Acad. Publ. (1990) pp. 281–286
[a11] A.N. Philippou, A.A. Muwafi, "Waiting for the kth consecutive success and the Fibonacci sequence of order $k$" Fibonacci Quart. , 20 (1982) pp. 28–32
[a12] 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
[a13] A.N. Philippou, C. Georghiou, G.N. Philippou, "Fibonacci-type polynomials of order $k$ with probability applications" Fibonacci Quart. , 23 (1985) pp. 100–105
[a14] W.A. Webb, E.A. Parberry, "Divisibility properties of Fibonacci polynomials" Fibonacci Quart. , 7 (1969) pp. 457–463
How to Cite This Entry:
Fibonacci polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fibonacci_polynomials&oldid=14185
This article was adapted from an original article by Andreas N. Philippou (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article