# Explicit coefficients of Lucas Polynomials¶

Yasuaki dot Honda at gmail dot com

Lucas Polynomials are defined using a recurrence relation and initial values. Then, the explicit coefficients of the Lucas Polynomials are given a pri ori and we prove polynomials with such coefficients satisfy the original recurrence relation.

In [1]:
LP[0](x):=2;

Out[1]:
$\tag{{\it \%o}_{1}}{\it LP}_{0}(x):=2$
In [2]:
LP[1](x):=x;

Out[2]:
$\tag{{\it \%o}_{2}}{\it LP}_{1}(x):=x$
In [3]:
LP[n](x):=x*LP[n-1](x)+LP[n-2](x);

Out[3]:
$\tag{{\it \%o}_{3}}{\it LP}_{n}(x):=x\,{\it LP}_{n-1}(x)+{\it LP}_{n-2}(x)$
In [4]:
for n:1 thru 12 do print(expand('LP[n](x)=LP[n](x)));

${\it LP}_{1}(x)=x$
${\it LP}_{2}(x)=x^2+2$
${\it LP}_{3}(x)=x^3+3\,x$
${\it LP}_{4}(x)=x^4+4\,x^2+2$
${\it LP}_{5}(x)=x^5+5\,x^3+5\,x$
${\it LP}_{6}(x)=x^6+6\,x^4+9\,x^2+2$
${\it LP}_{7}(x)=x^7+7\,x^5+14\,x^3+7\,x$
${\it LP}_{8}(x)=x^8+8\,x^6+20\,x^4+16\,x^2+2$
${\it LP}_{9}(x)=x^9+9\,x^7+27\,x^5+30\,x^3+9\,x$
${\it LP}_{10}(x)=x^{10}+10\,x^8+35\,x^6+50\,x^4+25\,x^2+2$
${\it LP}_{11}(x)=x^{11}+11\,x^9+44\,x^7+77\,x^5+55\,x^3+11\,x$
${\it LP}_{12}(x)=x^{12}+12\,x^{10}+54\,x^8+112\,x^6+105\,x^4+36\,x^2+2$
Out[4]:
$\tag{{\it \%o}_{4}}\mathbf{done}$
In [145]:
GLP[n](x):=intosum(sum(n/(n-k)*binomial(n-k,k)*x^(n-2*k),k,0,floor(n/2)));

Out[145]:
$\tag{{\it \%o}_{142}}{\it GLP}_{n}(x):={\it intosum}\left({\it sum}\left(\frac{n}{n-k}\,{{n-k}\choose{k}}\,x^{n-2\,k} , k , 0 , \left \lfloor \frac{n}{2} \right \rfloor\right)\right)$
In [146]:
'GLP[n](x)=GLP[n](x);

Out[146]:
$\tag{{\it \%o}_{143}}{\it GLP}_{n}(x)=n\,\sum_{k=0}^{\left \lfloor \frac{n}{2} \right \rfloor}{\frac{{{n-k}\choose{k}}\,x^{n-2\,k}}{n-k}}$
In [148]:
for i:1 thru 12 do print(ev(GLP[i](x),nouns,expand));

$x$
$x^2+2$
$x^3+3\,x$
$x^4+4\,x^2+2$
$x^5+5\,x^3+5\,x$
$x^6+6\,x^4+9\,x^2+2$
$x^7+7\,x^5+14\,x^3+7\,x$
$x^8+8\,x^6+20\,x^4+16\,x^2+2$
$x^9+9\,x^7+27\,x^5+30\,x^3+9\,x$
$x^{10}+10\,x^8+35\,x^6+50\,x^4+25\,x^2+2$
$x^{11}+11\,x^9+44\,x^7+77\,x^5+55\,x^3+11\,x$
$x^{12}+12\,x^{10}+54\,x^8+112\,x^6+105\,x^4+36\,x^2+2$
Out[148]:
$\tag{{\it \%o}_{145}}\mathbf{done}$
In [72]:
C:n/(n-k)*binomial(n-k,k);

Out[72]:
$\tag{{\it \%o}_{71}}\frac{n\,{{n-k}\choose{k}}}{n-k}$
In [75]:
subst(n-1,n,C)+subst(k-1,k,subst(n-2,n,C));

Out[75]:
$\tag{{\it \%o}_{74}}\frac{\left(n-1\right)\,{{n-k-1}\choose{k}}}{n-k-1}+\frac{\left(n-2\right)\,{{n-k-1}\choose{k-1}}}{n-k-1}$
In [79]:
rec:C-(subst(n-1,n,C)+subst(k-1,k,subst(n-2,n,C)));

Out[79]:
$\tag{{\it \%o}_{78}}\frac{n\,{{n-k}\choose{k}}}{n-k}-\frac{\left(n-1\right)\,{{n-k-1}\choose{k}}}{n-k-1}-\frac{\left(n-2\right)\,{{n-k-1}\choose{k-1}}}{n-k-1}$
In [80]:
makefact(rec),factorial_expand:true,factor;

Out[80]:
$\tag{{\it \%o}_{79}}0$