GENERALIZED FIBONACCI POLYNOMIALS
V. E. HOGGATT, JR.# and MARJORIE BICKNELL
San Jose State University, San Jose, California
The Fibonacci polynomials and their relationship to diagonals of P a s c a l ' s triangle a r e
generalized in this paper.
The generalized Q-matrix investigated by Ivie [1] occurs a s a
special case.
1. THE FIBONACCI POLYNOMIALS
The Fibonacci polynomials, defined by
(1.1)
F0(x) = 0,
F t (x) = 1,
F2(x) = x,
a r e well known to r e a d e r s of this journal.
F
n+2(
x
) =
x F
n +
i^
+
F
n(
x
)'
That the Fibonacci polynomials a r e generated by
a matrix Q 2 ,
(1.2)
Q2=|t
nl-
Q?='Vl(X)
F (x)
n
( ; ! ) •
Fn(X>
F
Ax)
n-1
can be verified quite easily by mathematical induction. Also, it is apparent that, when x = 1 ,
F (1) = F , the n
Fibonacci number, and when x = 2, F (2) = P , the n Pell number.
n
n
n
n
Further, when P a s c a l ' s triangle is written in left-justified form, the sums of the e l e ments along the rising diagonals give r i s e to the Fibonacci numbers, and, in fact, those e l e ments a r e the coefficients of the Fibonacci polynomials.
That i s ,
[(n-l)/2]
(1.3)
where
F n (x) =
5(-;-)^.
[x] is the greatest integer contained in x,
is a binomial coefficient
and
(")
The first few Fibonacci polynomials a r e displayed below as well a s the a r r a y of their
coefficients.
457
458
[Dec.
GENERALIZED FIBONACCI POLYNOMIALS
Fibonacci Polynomials
Coefficient A r r a y
Ft(x) = 1
F2(x) = x
F3(x) = x 2 + 1
F4(x) = x 3 + 2x
F5(x) = x 4 + 3x2 + 1
F6(x) = x 5 X ^ x 3 + 3x
F7(x) = x 6 + 5x^k,6x 2 + 1
~r
F8(x) = x 7 + 6x 5 + 10x3 + 4x
If one observes that, by rule of formation of the Fibonacci polynomials, if one writes
the polynomials in descending o r d e r , to form the coefficient of the k
adds the coefficients of the k
t e r m of F
n (x)
and the
(k - 1)
t e r m of F (x),
t e r m of
n—x
F
Q (x),
one
the
n—u
a r r a y of coefficients formed has the same rule of formation as Pascal 1 s triangle when it is
written in left-justified form, except that each column is moved one line lower, so that the
coefficients formed a r e those elements that appear along the diagonals formed by beginning
in the left-most column and preceding up one and right one throughout the left-justified Pascal
triangle.
Throughout this paper, this diagonal will be called the rising diagonal of such an
array.
2.
THE TRIBONACCI POLYNOMIALS
Define the Tribonacci polynomials by
T_i(x) = T0(x) = 0,
Wx)
(2.1)
„ = T
2
+T
T2(x)
x T n + 2 (x) + xT n + 1 (x) + T n (x).
When x = 1, T (1) = T , the n
T
Tt(x) = 1,
2
th
Tribonacci number 1, 1, 2, 4, 7, 13, 24, 44, 81,
- + T . The first few Tribonacci polynomials follow.
Tribonacci Polynomials
TjW
= 1
T2(x) = x 2
T3(x) = x 4 + x
T4(x) ^ \ x 6 + 2 ^ + 1
T5(x)
= x 8 HN3X5 +
3x2
T6(x) = x 1 0 '+ 4 x N ^ 6 x 4 + 2x
TT(x) = x12 + 5x8 + 10x6 + 7x3 + 1
T8(x) = x 14 + 6X11 + 15x8 + 16x5 + 6x2
1973]
GENERALIZED FIBONACCI POLYNOMIALS
459
Tribonacci Coefficient A r r a y
1
1
1
1
1
1
1
1
Left-Justified Trinomial Coefficient A r r a y
2
(1 + X + x
A
n = 0, 1,
1
1
1
1
1
2
3
2
1
1
3
6
7
6
1
4
10
16
ft
16
10
4
1
1
5
15
30
45
51
45
30
15
3
1
The Tribonacci coefficient a r r a y has the same rule of formation as the trinomial coefficient
a r r a y , except that each column is placed one line lower.
Thus, the sums of the rows are the
same as the sums of the rising diagonals of the trinomial coefficient a r r a y , both sums yielding the Tribonacci numbers 1, 1, 2, 4, 7, 13, 24, • • • , and the coefficients of the Tribonacci
polynomials a r e the trinomial coefficients found on those same rising diagonals.
That i s ,
. 2n-3j-2
(2.2)
j=0
X
**
where
("),
is the trinomial coefficient in the n
column is the zero
row and j
column where, as is usual, the left-most
column and the top row the zero
0), = °
row, and
B)>
-
The Tribonacci polynomials a r e generated by the matrix Q 3 ,
460
[Dec.
GENEEALIZED FIBONACCI POLYNOMIAL
Q3
so that
Wx)
(2.3)
Q3
T n (x)
xT (x) + T ,(x)
n
n-1
xT
T
Ax) + T 0 (x)
n-1
n-2
T n (x)
T
xT
n-2
n-l(x)
A proof could be made by mathematical induction.
n-1«
0 (x)
T
+ T
Q(x)
n-3
n-2(x)
That Q3 has the given form for n = 1 is
apparent by inspection of Q 3 , element-by-element. Expansion of the matrix product Q3
Q3Q3 gives the elements of Q3
=
in the required form, making use of the recursion (2.1).
Notice that det Q3 = l n = 1, analogous to the Fibonacci case.
In fact, we can write
an interesting determinant identity. Again using (2.1), we multiply row one of Q3 by x2 and
add to row 2. Then we exchange rows 1 and 2 to write
(-1) =
(2.4)
T
n+2(x)
T
n+lW
T
T n (x)
T
T n (x)
n+l(x)
T n (x)
T
n-l(x>
n-lW
T
n-2(x)
which becomes an identity for Tribonacci numbers when x = 1.
3.
THE QUADRANACCI POLYNOMIALS
The Quadranacci polynomials are defined by T* (x) = T* (x) = T*(x) = 0, T*(x) = 1,
(3.1)
T* +4 (x) = x 3 T* + 3 (x) + x 2 T* + 2 (x) + xT* + 1 (x) + T*(x) .
The first few values a r e
Ti(x) = 1
T2(x) = x 3
T3(x)
T4(x) = x 9 4 ^ x 5 +
T5(x) = x 12 + S x
8
x
^ ^
+ 1
T6(x) = x 15 + 4X11 + 6 x 7 > ^ x 3
T7(x) = x 18 + 5x14 + 10x10 + 10x6 + 3x2
1973]
GENERALIZED FIBONACCI POLYNOMIALS
Notice that the coefficient of the j
term of T*(x) i s the sum of the coefficients of
t e r m of T J . J W , (j - l )
term of T* 9 (x), (j - 2 ) n d term of T* Q (x), and
n
rd
*
~*
(j - 3)
term of T ,(x) when the polynomials a r e arranged in descending o r d e r . Then,
the a r r a y of coefficients, if each row were moved up one line, would have the same rule of
formation a s the left-justified a r r a y of quadranomial coefficients, arising from expansions of
the
t h
461
j
St
(1 +.x + x 2 + x 3 ) 11 , n = 0 , 1 , 2 , • • • . Thus, the coefficients of T*(x) a r e those found on the
n ^ rising diagonal of the quadranomial triangle. Also, T*(l) = T*,
number 1, 1, 2, 4 , 8, 15, 29, 56, 108, • • • , T * + 4 = T*
+ T*
the n
+ T*
Quadranacci
+ T*.
The Quadranacci polynomials are generated by the matrix Q 4 ,
x3
1
2
x
Q4
so that
xT
n-J
0
0 \
0
1
0
x
0
0
1
1
0
0
0 /
.th
Q? = (a..) has i t s j
column given by a ^ = T * + 2 _ / x > >
(x)+T
n-l-3
(x)
a
'
3j
=
^U-j
W+T
5-j
W
'
and
%
a
2
j
= T* + 1 _.(x),
=
x2T
n+l-j(x)
+
j = 1, 2, 3,4..
That Q £ has the form claimed above can be established by mathematical induction. That Q4
Let Q^ + = (b ) and Q4 = (q.^).
has the stated form follows by inspection.
pand Q f
+1
b
= Q4Q£.
= q
lj
The first row of Qf
liaij
+q
i2a2j
+q
+
Then w e e x -
has the required form, for
i3a3j
+q
i4a4j
= [x 3 T* + 2 _.(x)] + [x 2 T* + 1 _.(x) + xT*_.(x) + T * - ; U j ( x ) ] + 0 + 0
i
(n+l)+2-j
where we make use of (3.1). Computation of b 9 . , b Q . , and hA. i s s i m i l a r , and shows that
3
J
3
n+1
Q4
has the required form, which would complete the proof.
We derive a determinant identity for Quadranacci polynomials from Q4 by forming the
matrix Q 4 n a s follows.
Add x 3 times row 1 to row 2, making a'
times row 1 and x 3 times row 2 to row 3 , producing a' = T* + 4 _.(x).
n
J
J
3. Then matrix Qf h a s
a
ij = Tn+4-j(x)'
a
2j = Tn+3-j(x)'
= T*
Exchange rows l a n d
a
3j = Tn+2-j(x) •
det
Q4*n = ( - D n + 1
and
a
4j
=
T
n+l-j(x)'
j = lj 2
because there was one row exchange.
'
3
' 4>
and
(x). Add x 2
That i s , for example, when x = 1,
462
GENEBALIZED FIBONACCI POLYNOMIALS
T*
IT*
n+3
(3.2)
where T* is the n
T* !
n
1 n+2
T*
n+1
n+2
T*
n+1
T*
n
T*
n
n-1
T* n
n-1
T*
n-2
T*
n
T* ,
n-1
n-2
T*
n-3
IT*
n+1
(-I)'
[Dec.
n+1
Quadranacci number.
4.
THE R-BONACCI POLYNOMIALS
Define the r-bonacci polynomials by
R
-(r-2)(x) =
R
-(r-l)(x)
= R_ 1 (x) = RQ(x) = 0,
=
R^x) = 1,
R 2 (x) = x r - 1
R ^_ (x) = x r " 1 R __, ,(x) + x r " 2 R ^ Q(x) + . . . + R (x)
n+r
n+r-1
n+r-2
n
(4.1)
The r-bonacci polynomials, by their recursive definition will have the coefficients of R (x),
written in descending o r d e r , given by the coefficients on the n
rising diagonal of the left-
justified r-nomial coefficient a r r a y , the coefficients arising from expansions of
r l
2
That i s ,
(1 + x + x + -. . + x
(4.2)
n
n = 0, 1, 2,
) ,
(r-D(n-l)-rj
R n (x)
where
(-I
is the element in the n
row and j
0).
column of the left-justified r-nomial triangle, and
0
when
j > n
The r-bonacci polynomials a r e generated by the r x r matrix Q ,
1
0
0
0
1
0
0
0
0
1
0
X
0
0
0
1
1
0
0
0
r-2
1x
r-3
X
Q.
\
• • o\
1
X
••
o)
1963]
GENERALIZED FIBONACCI POLYNOMIALS
463
which is an identity matrix of o r d e r (r - 1) bordered on the left by a column of descending
powers of x and followed by a bottom row of z e r o s .
The matrix
Q n has R
(x) a s the
element in the upper left and R (x) in the lower left, with general element a., given by
ii
lj
r-1+1
k=l
Proof of (4.3) is byJ mathematical induction.
Let Q
^ r = (b..).
IJ
Then
b i. i- = x ~ ', i = 1,
' 2
' ,
• • • , r; b.. = 1, j = i + 1, i = 1, 2, • • • , r; and b.. = 0 whenever j f 1 and j f i + 1.
Let
~n+l
_, _n
/ v
Q
=QQ = ( c ) .
^r
^r r
lj
Then
r
3
=
ij
b
a
S ik kj
= b
k=l
a
+
i l l k 2 b^ -i ka-'k j
k=2
= x r - 1 R _,, .(x) + a . ^ . + 0
n+l-j
1+1,3
k=l
r-i+1
/
j
x
ri +
, xl - k - i
" '
R ^ ^ u i
A i,(x)
^(n+D+1-j-k
,
k=l
which is the required form for the general element of Q
, completing the proof.
If we operate upon Q as before, we can again make a determinant identity. Repeatr—1
st
r—2
nd
r~f~l—i
edly add x
times the (i - 1)
row, x
times the (i - 2)
row, • • • , x
times
the first row to the i
row, to produce a new i
row with R ._- in its first column, for
i = 2, 3 , • • • , r - 1. Then make (r - l ) / 2 row exchanges to put the elements in the columns
in descending order.
The matrix R formed has its general element given by
r.. = R , ,.. . .(x)
,
13
n+r+l-i-3 v
and its determinant has value
(-1)
••
' *. That i s , when x = 1, the r-bonacci
numbers
, 0, 1, 1, 2 , • •• , R ^ = R ^ -, + R ^ „ + • • • + R ,
' n+r
n + r - 1 n+r-2
n
have the determinant identity
GENERALIZED FIBONACCI POLYNOMIALS
464
n+r-1
n+r-2
R n+r-2
n+r-3
R
n+1
R
n
n-1
detR =
R
n+1
R
n
R
R
[Dec.
n-1
(-1)
*n-r+3
n-r+2
*n-r+2
n-r+1
(r-l)(2n+l)/2
Notice that Eq. (1.2) gives
det Q2 =
F
n+lWFn-lW "
F
n(x)
=
det R
^
for r = 2. Since we recognize
W x ) F n-l ( x ) " Fn(x)l
a s the characteristic value
a l s , we define
[ 2 ] , [ 3 ] , [4] of sequences arising from the Fibonacci polynomi-
|det R | = 1 as the characteristic value of the sequences arising from the gen-
eralized Fibonacci polynomials,
value
= X
r^.2.
Then, for example, (2.4) gives the characteristic
|(-1)| = 1 for the sequences arising from the Tribonacci polynomials, while (3.2) is
the a r r a y giving the characteristic value
|(-1)
| = 1 for the Quadranacci numbers.
The matrix R just defined has the interesting property that multiplication by Q
duces a matrix of the same form.
To clarify, let R = R
J
with R (x) appearing& in the lower left c o r n e r ,
n
^
R
(4.4)
r , 0n
Q
r
= (r..) be the
r,n
rj
r.. = R , ,., . .(x). Then
ij
= R
r x r
promatrix
n+r+1-i-j
r,n
which is proved by mathematical induction as follows.
Consider the matrix product R
Q
= (p..) for any n, where we observe that the first column of Q contains the multipliers
*3
th r
for the recursion relation for the polynomials R (x). The i
row of R r , n multiplied by
the first column of Q produces
p5. . =
= 7 R . . . . . . (x) x
il
*-^* n+r+1-i-k
k=l
u X)
r+l-k
" = R . .., «(x) = R/ ,-v, ,- . ..(x) ,
n+r+l-i
(n+l)+r+l-i-l
while, when j f 1, since the only non-zero elements of Q
row of R
r,n
times the Jj
p
so that R
ij
R
column of Q
^r
Q = R
,.,, for any
n.
J
r,n^r
r,n+l
R
then
R
n+r+l-i-(j-l)
vproduces
(n+l)+r+l-i-j'
j
2, 3, • • • , r ,
Then, we must have that R nQ = R .,, and, if
n,0^r
r,l
nQ
k 1
r , O^r
R
.th
occur when i = j - 1, the i
= R
. _, ,
r,k-l
Q k = (R n Q k _ 1 ) Q
= R . nQ = R . ,
r , 0A ^ r
r,0^r
^r
r,k-l^r
r,k
1973
]
GENERALIZED FIBONACCI POLYNOMIALS
465
which completes a proof of (4.4) by mathematical induction,
If we equate the elements in the upper left corner of R
and R ^ Q n we obtain the
r,n
r,0^r
identity
V l
( x )
=
V l ( x ) V l ( x ) + Rr_2(x)[xr-2Rn(x)
+x
+
r 3
- R n _ l ( x)
-+EnWx)l
+ Rr_3(x)[xr"3Rn(x) + x r " 4 R n - 1 ( x )
(4.5)
+ ... + R _fx)]
n-r+3
+ R 1 (x)[xR n (x) + B n _ 1 (x)] + R 0 (x)R n (x)
Notice that the matrix Q provides the multipliers for the recursion relation for the
polynomials R (x) but does not depend upon the original values of the polynomials in the
proof of (4.4). L e t H (x) be any sequence of polynomials with r a r b i t r a r y starting values
H 0 (x), H^x), • • • , H r _ 1 (x), and with the same recursion relation as the polynomials R (x).
Form the matrix R* = ( r * ) , r * = H , ,- . .(x).
r,n
rj
IJ
n+r+1-i-j
can derive R* A Q n = R* and thus obtain
r,0^r
rsn
H
n+r-l =
H
r-l(x)Rn+l(x)
+ H
By
used e a r l i e r , we
J the arguments
&
r-2 (x) ^ " X ^
+
+
^"X-l^
• • • + VrHJ^
+ H Q ( x )L[ x r _ 3 R (x) + x r _ 4 R Ax)
r-3
n
n-1
(4 6)
-
+
. . .
+
B
+ , W ]
n-r+3 J
+ Hi(x) [xR n (x) + R n - 1 ( x ) ] + H 0 (x)R n (x)
REFERENCES
1. J o h n l v i e , ,fA General Q-Matrix 9 " Fibonacci Quarterly, Vol. 10, No. 3 , April, 1972, pp.
255-261.
2.
Eugene Levine, "Fibonacci Sequences with Identical Characteristic Values, M Fibonacci
Quarterly, Vol. 6, No. 5, Nov. , 1968, pp. 75-80.
3. Irving Adler, "Sequences with a Characteristic N u m b e r , " Fibonacci Quarterly, Vol. 9,
No. 2 , April, 1971, pp. 147-162.
4.
Brother Alfred Brousseau, Fibonacci and Related Number Theoretic Tables, The Fibonacci Association, 1972.