Journal of Indian Acad. Math.  
ISSN: 0970-5120  
Vol. 47, No. 1 (2025) pp. 181-189  
Thomas Koshy  
A FAMILY OF GENERALIZED  
GIBONACCI SUMS:  
GRAPH-THEORETIC CONFIRMATION  
Abstract: We confirm a generalized sum of a family of gibonacci  
polynomial squares using graph-theoretic techniques, and its graph-  
theoretic and Pell consequences.  
Keywords: Generalized Gibonacci Polynomials, Pell Polynomials,  
Fibonacci Polynomials, Lucas Polynomials.  
Mathematics Subject Classification (2020) No.: Primary 11B37, 11B39,  
11C08.  
1. Introduction  
Extended gibonacci polynomials zn(x) are defined by the recurrence  
zn2(x) a(x)zn1(x) b(x)zn(x), where x is an arbitrary integer variable;  
a
(x)  
,
b(x)  
,
z0(x), and z1(x) are arbitrary integer polynomials; and n 0  
.
Suppose  
fn , the nth Fibonacci polynomial; and when z0  
ln , the nth Lucas polynomial.  
a
(
x
)
x  
and  
b
(
x
)
1. When z0  
(
x
)
0  
and z1  
(
x
)
1  
,
,
zn  
zn  
(
x
)
)
(
x
)
(
x
)
2 and z1  
(
x
)
x  
(x  
(x)  
They can also be defined by the Binet-like formulas. Clearly, fn(1) Fn  
the nth Fibonacci number; and ln(1) Ln , the nth Lucas number [1, 2].  
,
182  
THOMAS KOSHY  
Pell polynomials pn  
(
x
)
and Pell-Lucas polynomials qn  
, respectively [2, 6].  
(x  
)
are defined by  
pn  
(x  
)
fn(2 and qn ln(2x)  
x
)
(x  
)
In the interest of brevity, clarity, and convenience, we omit the argument in  
the functional notation, when there is no ambiguity; so zn will mean zn(x)  
.
In addition, we let gn fn or ln  
;
bn pn or qn  
;
  x2 4 and 2x    
[6, 7].  
1.1 Fundamental Gibonacci Identities: Gibonacci polynomials satisfy the  
following properties [2, 3, 4, 5, 6, 7]:  
(1)nk 1 fk2,  
(1)nk 2fk2,  
if gn fn  
gnk n k  
g
gn2  
(1)  
(2)  
otherwise;  
nk 1  
(1)  
(1)nk 2fr f2k,  
f f ,  
r 2k  
if gn fn  
gnk r n k  
g
gn k n k r  
g
otherwise;  
1
n k  
[
2l2nr (1)  
l l  
2k r  
]
,
if gn fn  
2  
gnk r n k  
g
gn k n k r  
g
(3)  
nk  
2l2nr (1)  
l l ,  
2k r  
otherwise,  
where  
k
and  
r
are positive integers. These properties can be confirmed using Binet-  
like formulas.  
Consequently, we have  
nk 1  
(1)  
nk  
2l2nr (1)  
[
l l  
2k r  
]
f2k fr, if gn fn  
f2k fr, otherwise.  
(4)  
2
gn2 k rgn2 k gn2 k n k r  
g2  
(1)n k 2  
[
l l  
2k r  
]
n k  
2l2n r (1)  
Again, in the interest of brevity and convenience, we now let  
tk  
A 2l2(2pnt p)k r  
(1)  
tkl2pklr ; and B 2l2(2pnt p)k r  
(
1  
)
l l  
2pk r  
.
It follows identities (1) and (4) that  
A FAMILY OF GENERALIZED GIBONACCI SUMS  
183  
2
(1)tk 1 f ,  
if gn fn  
pk  
g
(2pn t)kg(2pn t 2p)k g(22pnt p)k  
(1) 2fp2k,  
otherwise;  
tk  
(5)  
tk1  
(1)  
Af2pkfr, if gn fn  
2
g(22 pnt)krg(22 pnt2p)k g(22pnt)kg(22pnt2p)k r  
tk  
2
(1) Bf2pkfr, otherwise,  
(6)  
respectively, where k, p, r, and  
t
are positive integers and t 2p [6].  
2. A Telescoping Gibonacci Sum  
Using recursion, we established the following telescoping sum in [6]. In the  
interest of brevity, we omit its proof here.  
Lemma 1: Let k, p, r, t , and  
be positive integers, where t 2p  
.
Then  
gtk r  
gtk  
g(2pn t 2p)k r  
g(2pn t)k r  
r  
.
(7)  
g  
g(2pn t)k  
n 1   
(2pn t 2p)k  
3. A Family of Gibonacci Sums  
This lemma, coupled with identities (5) and (6), played a major role in the  
development of the following theorem. To present it in a concise fashion, we now let:  
1
, if gn fn  
1,  
if gn fn  
µ   
µ  
2  
2, otherwise;  
2, otherwise;  
1, if gn fn  
and  
  
1, otherwise.  
These tools served as building blocks in the development of the theorem [6].  
184  
THOMAS KOSHY  
Theorem 1: Let k, p, r, and  
t
be positive integers, where t 2p . Then  
tk  
tk  
(1) µ  
[
2l2(2pn t p)k r (1) l2pklr ]f2pk fr  
gt2k r  
gt2k  
2r  
.
(8)  
[g(2pn t p)k (1)tk µfp2k  
]
2
n 1  
The objective of our discourse is to confirm this result using graph-theoretic  
techniques. To this end, we now present the needed tools.  
4. Graph-Theoretic Tools  
Consider the Fibonacci digraph in Figure 1 with vertices v1 and v2 , where a  
weight is assigned to each edge [2, 5]. It follows from its weighted adjacency  
x
1
0
matrix  
that  
Q   
1
Figure 1: Weighted Fibonacci Digraph  
fn1  
fn  
n
Q   
,
fn  
fn1  
where n 1 [2, 3, 4, 5]. We extend the exponent  
n
to  
0
, which is consistent with  
the Cassini-like formula fn 1 n 1  
f
fn2 (1)n , where f1 1 [2, 5].  
A
walk  
from  
vertex  
vi  
to  
vertex  
vj  
is  
a
sequence  
vi ei vi1 · · ·vj1 ej1 vj of vertices vk and edges ek , where edge ek is  
incident with vertices vk and vk1 . The walk is closed if vi vj ; and open,  
otherwise. The length of a walk is the number of edges in the walk. The weight of a  
walk is the product of the weights of the edges along the walk.  
A FAMILY OF GENERALIZED GIBONACCI SUMS  
185  
The ijth entry of Qn gives the sum of the weights of all walks of length  
n
from vi to vj in the weighted digraph, where 1 i  
,
j n [2, 3, 4]. Consequently,  
the sum of the weights of closed walks of length originating at v1 in the digraph is  
n
fn 1 and that of those originating at v2 is fn 1 . So, the sum of the weights of all  
closed walks of length n in the digraph is fn 1 fn 1 ln [2, 5].  
Let  
A
and  
B
denote sets of walks of varying lengths originating at a vertex  
v
. Then the sum of the weights of the elements (a, b) in the product set A B is  
defined as the product of the sums of weights from each component [3, 4]. This  
definition can be extended to any finite number of component sets. In particular, let  
A, B,C, and  
D
denote the sets of walks of varying lengths originating at a vertex  
v
, respectively. Then the sum of the weights of the elements (a, b, c, d) in the  
product set A B C D is the product of the sums of weights from each  
component [3, 4].  
We now make an interesting observation. Let A {u} and B {v}  
,
where  
u
denotes the closed walk v1 v1 and denotes the closed walk  
v
v1 v2 v1 . The weight of the element (u, u) in A A is  
is 1. Consequently, the sum  
x
2 , and that in B B  
w
of the weights of the elements in  
C(A A)  B B  B B  B B  B B  
is  
given  
by  
w x2 4  2  
.
These tools play a major role in the discourse. With them at our finger tips,  
we are now ready for our pursuit of the graph-theoretic confirmation.  
5. Graph-Theoretic Confirmation  
Let Tndenote the set of closed walks of length  
n
in the digraph originating  
at v1 , and Unthe set of all closed walks of the same length  
n
in the digraph.  
Correspondingly, let Tn denote the sum of the weights of all elements in Tn, and  
Un that of those in Un. Clearly, Tn fn 1 and Un fn 1 fn 1 ln [2, 5].  
With this brief background, we now begin the proof of the gibonacci sum (8) in two  
cases, where k, p, r, t 1 and t 2p  
.
186  
THOMAS KOSHY  
Proof: Case 1: Suppose gn fn . The sum of the weights of the elements in  
the product set  
T2pk 1 T  
is T2pk 1 r 1  
T
f2pk fr ; the sum of those in  
r 1  
T  
T  
is T2  
f(22pnt p)k ; and that of those in  
(2pnt p)k 1  
(2pnt p)k 1  
(2pnt p)k 1  
Tpk 1 Tpk 1 is Tp2k 1 fp2k  
.
We now let  
tk  
(1)  
[
tk 1 2U2(2pn t p)k r (1) U2pkUr  
]
T2pk 1 r 1  
T
Sn  
tk  
2
w
[
T2  
(1) Tp2k 1  
]
(2pn t p)k  
tk  
(1)  
[
tk 1 2l2(2pn t p)k r (1) l2pklr  
]
f2pk fr  
tk  
2
2  
[
f(22pn t p)k (1) fp2k  
]
With identities (3) and (4), and Lemma 1, this yields  
f(22pnt)krf(22pnt2p)k f(22pnt)kf(22pnt2p)k r  
(1)  
tk 1 Af2pk fr  
[
tk  
2
2  
2  
[
f(22pn t p)k (1) fp2k  
]
f(22pnt)kf(22pnt 2p)k  
2
tk  
f(2pn t 2p)k r  
f(22pn t)k r  
(1)  
[Af2pk fr  
f2  
tk  
2
[
f(22pn t p)k (1) fp2k  
]
f(22pn t)k  
n 1  
n 1   
(2pn t 2p)k  
ft2k r  
ft2k  
2r.  
(9)  
We now turn to the next case.  
Case 2: Let gn ln . Recall that the sum  
w
of the weights of the elements  
in Cis given byw x2 4  2 , and that of the elements in the product set  
CT2k 1 Tr1 is given by wT2pk 1 r 1  
T
 2f2pk fr . The sum of the weights of  
the  
elements  
in  
the  
product  
set  
U(2pnt p)k U(2pnt p)k  
is  
U(22pnt p)k l(22pn t p)k ; and that of those in T2pk 1 Tpk 1 is T22pk 1 f22pk  
.
A FAMILY OF GENERALIZED GIBONACCI SUMS  
As above, we now let  
187  
tk  
(1)tk 1w  
[
2U2(2pn t p)k r (1) U2pkUr  
]
T2pk 1 r 1  
T
Sn  
2
[
U(22pn t p)k (1)tkwTp2k 1  
]
(1)tk 12  
[
2l2(2pn t p)k r (1)tkl2pklr  
]f2pk fr  
.
tk  
2
[
l(22pn t p)k (1) 2fp2k  
]
It then follows by identities (3) and (4), and Lemma 1 that  
l(22pnt)kl(22pnt 2p)k r l(22pnt)k rl(22pnt2p)k  
(1)tk 12Bf2pk fr  
tk  
2
[
l(22pnt p)k (1) 2fp2k  
]
l(22pnt)kl(22pnt2p)k  
2
(1)tk 12Bf2pk fr  
l(2pn t 2p)k r  
l(22pn t)k r  
l2  
tk  
2
[l(22pn t p)k (1) 2fp2k  
]
l(22pn t)k  
n 1  
n 1   
(2pn t 2p)k  
lt2k r  
lt2k  
2r.  
(10)  
This equation, coupled with equation (9), yields Theorem 1, as desired.  
Interestingly, equation (9) can be rewritten in terms of graph-theoretic tools.  
To realize this goal, we define T0 1, U0 2; Hn Tn or Un;  
1  
1,  
if Hn Tn  
, if Hn Tn  
w  
µ   
µ  
w, otherwise;  
w, otherwise;  
1, if Hn Tn  
1  
if Hn Tn  
  
  
1, otherwise;  
1  
otherwise;  
188  
THOMAS KOSHY  
1, if Hn Tn  
'   
0, otherwise.  
With these new tools, and integers k, p, r, and  
the graph-theoretic version of equation (8):  
t
as before, we now present  
tk  
tk  
2U2(2pn t p)k r (1) U2pkUr  
(1) µ  
[
]
T2pk 1 r 1  
T
Ht2k r '  
Ht2k '  
2r  
.
tk  
2
[
H(22pn t p)k (1) µTp2k 1  
]
n 1  
(11)  
Next, we turn to the Pell implications of the graph-theoretic techniques.  
6. Pell Consequence  
With the gibonacci-Pell relationship bn(x) gn(2x), we can construct the  
graph-theoretic proof of the Pell version of Theorem 1 independently by changing the  
weight of the loop at v1 from  
x
to 2x . We encourage gibonacci enthusiasts to  
explore this path.  
7. Acknowledgment  
The author would like to thank Z. Gao for a careful reading of the article.  
REFERENCES  
[1] M. Bicknell (1970): A Primer for the Fibonacci Numbers: Part VII, The Fibonacci  
Quarterly, Vol. 8(4), pp. 407-420.  
[2] T. Koshy (2019): Fibonacci and Lucas Numbers with Applications, Volume II, Wiley,  
Hoboken, New Jersey.  
[3] T. Koshy (2019): A Recurrence for Gibonacci Cubes with Graph-theoretic Confirmations,  
The Fibonacci Quarterly, 57(2), pp. 139-147.  
[4] T. Koshy (2021): Graph-theoretic Confirmations of Four Sums of Gibonacci Polynomial  
Products of Order 4, The Fibonacci Quarterly, Vol. 59(2), pp. 167-175.  
[5] T. Koshy (2023): Sums Involving Gibonacci Polynomial Squares: Graph-theoretic  
Confirmations, The Fibonacci Quarterly, Vol. 61(2), pp. 119-128.  
A FAMILY OF GENERALIZED GIBONACCI SUMS  
189  
[6] T. Koshy (2024): Sums Involving A Family of Gibonacci Polynomial Squares:  
Generalizations, The Fibonacci Quarterly, Vol. 62(1), pp. 75-83.  
[7] T. Koshy, A Family of Gibonacci Sums: Generalizations and Consequences, Journal of  
the Indian Academy of Mathematics, Vol. 2, pp. 195.  
Prof. Emeritus of Mathematics,  
(Received, October 3, 2024)  
Framingham State University,  
Framingham, MA01701-9101, USA  
E-mail: tkoshy@emeriti.framingham.edu