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.
LP[0](x):=2;
LP[1](x):=x;
LP[n](x):=x*LP[n-1](x)+LP[n-2](x);
for n:1 thru 12 do print(expand('LP[n](x)=LP[n](x)));
GLP[n](x):=intosum(sum(n/(n-k)*binomial(n-k,k)*x^(n-2*k),k,0,floor(n/2)));
'GLP[n](x)=GLP[n](x);
for i:1 thru 12 do print(ev(GLP[i](x),nouns,expand));
C:n/(n-k)*binomial(n-k,k);
subst(n-1,n,C)+subst(k-1,k,subst(n-2,n,C));
rec:C-(subst(n-1,n,C)+subst(k-1,k,subst(n-2,n,C)));
makefact(rec),factorial_expand:true,factor;