[seqfan] Fibonacci derivatives
Richard Mathar
mathar at strw.leidenuniv.nl
Fri Jul 6 20:12:52 CEST 2012
Here is another triangle s.o. might submit (rows n>=0, columns k>=0):
0
1,0
1,1,0
2,2,2,0
3,5,6,6,0
5,10,18,24,24,0
8,20,44,84,120,120,0
13,38,102,240,480,720,720,0
Defined starting with the Fibonacci polynomials
U_n(x) = x*U_{n-1} + U_{n-2) with U_0=0, U_1=1,
then taking the k'th derivative, for example
U_n^(1) = (d/dx) U_n(x) = (n*Lucas(n)-fib(n))/5
U_n^(2)= (d/dx) U_n^(1)(x)
and finally setting the argument x in these higher derivatives to 1.
P. Filipponi, A. F. Horadam, Second derivative sequences of Fibonacci
and Lucas Polynomials, Fib Quart 31 (1993) 194-204 http://www.ams.org/mathscinet-getitem?mr=1228557
I am too lazy to do this myself.
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