A matrix approach to the binomial theorem
Motivated by the formula $x^n = \sum_{k=0}^n\left(n \atop k\right) (x - 1)^k$ we investigate factorizations of the lower triangular Toeplitz matrix with $(i, j)$th entry equal to $x^{i-j}$ via the Pascal matrix. In this way, a new computational approach to a generalization of the binomial theorem is...
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| author | Stanimirovic, S. Станіміровіч, С. |
| author_facet | Stanimirovic, S. Станіміровіч, С. |
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| description | Motivated by the formula $x^n = \sum_{k=0}^n\left(n \atop k\right) (x - 1)^k$ we investigate factorizations of the lower triangular Toeplitz matrix
with $(i, j)$th entry equal to $x^{i-j}$ via the Pascal matrix. In this way, a new computational approach to a generalization of the binomial theorem is introduced.
Numerous combinatorial identities are obtained from these matrix relations. |
| first_indexed | 2026-03-24T02:28:18Z |
| format | Article |
| fulltext |
UDC 512.5
S. Stanimirović (Univ. Niš, Serbia)
A MATRIX APPROACH TO THE BINOMIAL THEOREM
МАТРИЧНИЙ ПIДХIД ДО БIНОМIАЛЬНОЇ ТЕОРЕМИ
Motivated by the formula xn =
∑n
k=0
(
n
k
)
(x−1)k, we investigate factorizations of the lower triangular Toeplitz matrix
with (i, j)th entry equal to xi−j via the Pascal matrix. In this way, a new computational approach to a generalization of
the binomial theorem is introduced. Numerous combinatorial identities are obtained from these matrix relations.
На основi формули xn =
∑n
k=0
(
n
k
)
(x − 1)k розглянуто факторизацiї нижньотрикутної матрицi Теплiца, (i, j)-
й елемент якої дорiвнює xi−j , з використанням матрицi Паскаля. Тим самим уведено новий обчислювальний
пiдхiд до узагальнення бiномiальної теореми. Iз використанням цих матричних спiввiдношень отримано численнi
комбiнаторнi тотожностi.
1. Introduction. The Pascal matrix of order n, denoted by Pn[x] = [pi,j [x]], i, j = 1, . . . , n, is a
lower triangular matrix with elements equal to
pi,j [x] =
xi−j
(
i− 1
j − 1
)
, i− j > 0,
0, i− j < 0,
and its inverse Pn[x]−1 = [p′i,j [x]], i, j = 1, . . . , n, has the elements equal to
p′i,j [x] =
(−x)i−j
(
i− 1
j − 1
)
, i− j > 0,
0, i− j < 0.
For the sake of simplicity we denote Pn[1] with Pn. Many properties of the Pascal matrix have
been examined in the recent literature (see for instance [1, 11, 12]). We are particulary interested on
the usage of the Pascal matrix as a powerful tool for deriving combinatorial identities. Precisely,
recalling that a Toeplitz matrix is matrix having constant entries along the diagonals, then the Pascal
matrix can be factorized in a form Pn = TnRn or Pn = LnTn, where Tn denotes the n × n lower
triangular Toeplitz matrix. Usually, the Toeplitz matrix Tn is filled with the numbers from the well-
known sequences. By equalizing the (i, j) th elements of the matrices in these matrix equalities, we
establish correlations between binomial coefficients and the terms from the well-known sequences.
Following this idea, some combinatorial identities via Fibonacci numbers were derived in [4,14],
as well as the identities for the Catalan numbers [9], Bell [10], Bernoulli [13] and the Lucas numbers
[15] were also computed. In [5] the authors derived identities by using the factorizations of the
Pascal matrix via generalized second order recurrent matrix. Some combinatorial identities were also
computed in [2, 3, 6 – 8] by various matrix methods.
The starting point of the present paper is one of the most beautiful formulas in mathematics, a
particular case of the binomial theorem
xn =
n∑
k=0
(
n
k
)
(x− 1)k. (1.1)
c© S. STANIMIROVIĆ, 2012
1578 ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 11
A MATRIX APPROACH TO THE BINOMIAL THEOREM 1579
The essential observation is that the binomial coefficient in (1.1) may be considered as the element
of the Pascal matrix Pn, and the left-hand side of (1.1) may be considered as the element of the lower
triangular Toeplitz matrix with (i, j) th entry equal to xi−j , which is denoted by Zhang [11] with
Sn[x]. Therefore, our goal is to factorize the matrix Sn[x] via the Pascal matrix Pn and to give some
combinatorial identities via this computational method. Some of our results represent generalizations
of some well-known identities, such as the binomial theorem (1.1). These identities involve the
binomial coefficients and the hypergeometric function 2F1. Recall that the hypergeometric function
2F1(a, b, c; z) is defined by
2F1(a, b, c; z) =
∞∑
k=0
(a)k(b)k
(c)k
zk
k!
,
where
(a)n =
a(a+ 1) . . . (a+ n− 1), n > 0,
1, n = 0,
is the well-known rising factorial symbol.
2. The results. First we find a matrix Rn[x] which establishes the relation between matrices
Sn[x] and Pn.
Theorem 2.1. The matrix Rn[x] = [ri,j [x]], i, j = 1, . . . , n, x ∈ R, whose entries are de-
fined by
ri,j [x] =
(−1)i−j
(
i− 1
j − 1
)
2F1(1, j − i; j;x), i > j,
0, i < j,
satisfies
Sn[x] = PnRn[x]. (2.1)
Proof. Our goal is to prove Rn[x] = P−1n Sn[x]. Let us denote the sum
∑n
k=1
p′i,ksk,j [x] by
mi,j [x]. It is easy to show that mi,j [x] = 0 = ri,j [x] for i < j. On the other hand, in the case i > j
we have
mi,j [x] =
i∑
k=j
p′i,ksk,j [x] =
i∑
k=j
(−1)i−k
(
i− 1
k − 1
)
xk−j =
i−j∑
k=0
(−1)i−j+k
(
i− 1
j + k − 1
)
xk.
After applying the transformations
i−j∑
k=0
(−1)i−j+k
(
i− 1
j + k − 1
)
xk = (−1)i−j
i−j∑
k=0
(i− 1)! (−x)k
(i− j − k)! (j + k − 1)!
=
= (−1)i−j (i− 1)!
(j − 1)! (i− j)!
∞∑
k=0
(1)k (j − i)k
(j)k
xk
k!
=
= (−1)i−j
(
i− 1
j − 1
)
2F1(1, j − i; j;x)
we show that mi,j [x] = ri,j [x] in the case i > j, which was our original attention.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 11
1580 S. STANIMIROVIĆ
Corollary 2.1. For positive integers i and j satisfying i > j and real x, the following identity
is valid (
i− 1
j − 1
) i−j∑
k=0
(−1)k
(
i− j
k
)
2F1(1,−k; j;x) = xi−j . (2.2)
Proof. From matrix relation (2.1) we obtain si,j [x] =
∑i
k=j
pi,k rk,j [x], or in an expanded form
xi−j =
i∑
k=j
(
i− 1
k − 1
)
(−1)k−j
(
k − 1
j − 1
)
2F1(1, j − k; j;x).
Making use of the formula for the binomial coefficients
(
r
m
)(
m
l
)
=
(
r
l
)(
r − l
m− l
)
, together with
the substitution k 7→ k + j, we finish the proof.
Remark 2.1. By putting j = 1 in equality (2.2), we obtain the binomial theorem (1.1).
In the following identity we establish the relation between hypergeometric functions 2F1.
Corollary 2.2. The following identity is valid for arbitrary nonnegative integers i and j satis-
fying i > j and x ∈ R(
i
j
)
+ x
(
i− 1
j
)
2F1(1, j − i+ 1; 1− i;x) =
(
i
j
)
2F1(1, j − i;−i;x). (2.3)
Proof. It is straightforward to show that
(
Sn[x]−1
)
i,j
=
1, i = j,
−x, i = j + 1,
0, otherwise.
We are now in a position to write the inverse of the Rn[x] as Rn[x]
−1 = Sn[x]−1Pn. In this way we
obtain
(
Rn[x]
−1)
i,j
=
1, i = j,(
i− 1
j − 1
)
− x
(
i− 2
j − 1
)
, i > j,
0, i < j.
From relation Pn = Sn[x]Rn[x]
−1 we get identity(
i− 1
j − 1
)
=
i∑
k=j+1
xi−k
((
k − 1
j − 1
)
− x
(
k − 2
j − 1
))
+ xi−j
valid for all positive integers i > j. After the replacement (i, j) 7→ (i+ 1, j + 1), we get equality(
i
j
)
=
i−j−1∑
k=0
xk
((
i− k
j
)
− x
(
i− k − 1
j
))
+ xi−j
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 11
A MATRIX APPROACH TO THE BINOMIAL THEOREM 1581
valid for all nonnegative integers i > j. Our problem now reduces to show the following two
relations:
i−j−1∑
k=0
xk
(
i− k
j
)
= −xi−j +
(
i
j
)
2F1(1, j − i;−i;x), (2.4)
i−j−1∑
k=0
xk
(
i− k − 1
j
)
=
(
i− 1
j
)
2F1(1, j − i+ 1; 1− i;x). (2.5)
In order to prove (2.4), we start from its left-hand side and transform it into
i−j−1∑
k=0
xk
(
i− k
j
)
= −xi−j +
i−j∑
k=0
xk
(
i− k
j
)
= −xi−j + i!
j! (i− j)!
i−j∑
k=0
(1)k (j − i)k
(−i)k
xk
k!
,
and (2.4) now immediately follows. The reader may establish (2.5) in a similar way, and the proof is
therefore completed.
Remark 2.2. Some pedestrian manipulations yields that in the case j = 1, relation (2.3) reduces
to the well-known identity
∑i−1
k=0
xk =
xi − 1
x− 1
.
In the rest of this section we investigate another factorization of the matrix Sn[x] via the Pascal
matrix, analogical to (2.1).
Theorem 2.2. The matrix Ln[x] = [li,j [x]], i, j = 1, . . . , n, x ∈ R, whose entries are defined
by
li,j [x] =
xi
(1 + x)j
+
(−1)i−j
x
(
i
j − 1
)
2F1
(
1, i+ 1; i− j + 2;−1
x
)
, i > j,
0, i < j,
satisfies
Sn[x] = Ln[x]Pn. (2.6)
Proof. We prove Ln[x] = Sn[x]P−1n . Let us denote the sum
∑n
k=1
si,k[x]p
′
k,j by ti,j [x]. We
have ti,j [x] = 0 = li,j [x] for i < j, while in the case i > j,
ti,j [x] =
i∑
k=j
si,k[x]p
′
k,j =
i−j∑
k=0
xi−j−k (−1)k
(
j − 1 + k
j − 1
)
=
=
∞∑
k=0
xi−j−k (−1)k
(
j − 1 + k
j − 1
)
−
∞∑
k=i−j+1
xi−j−k (−1)k
(
j − 1 + k
j − 1
)
. (2.7)
Now it suffices to prove the following two identities
∞∑
k=0
xi−j−k (−1)k
(
j − 1 + k
j − 1
)
=
xi
(1 + x)j
, (2.8)
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 11
1582 S. STANIMIROVIĆ
∞∑
k=i−j+1
xi−j−k (−1)k
(
j − 1 + k
j − 1
)
= −(−1)i−j
x
(
i
j − 1
)
2F1
(
1, i+ 1; i− j + 2;−1
x
)
. (2.9)
In order to prove (2.8), we use the binomial theorem and obtain
∞∑
k=0
xi−j−k (−1)k
(
j − 1 + k
j − 1
)
=
xi
xj
∞∑
k=0
(j)k
k!
(
−1
x
)k
=
=
xi
xj
∞∑
k=0
(
−j
k
)(
1
x
)k
=
xi
xj
(
1 + x
x
)−j
.
The identity (2.9) can be verified by applying the following transformations:
∞∑
k=i−j+1
xi−j−k (−1)k
(
j − 1 + k
j − 1
)
=
∞∑
k=0
x−k−1 (−1)i−j+1+k
(
i+ k
j − 1
)
=
= −(−1)i−j
x
i!
(j − 1)! (i− j + 1)!
∞∑
k=0
(1)k (i+ 1)k
(i− j + 2)k
(−1/x)k
k!
.
Since formulas (2.8) and (2.9) are valid, we apply them on (2.7), and the proof is completed.
Corollary 2.3. For integers i > j > 0 and real x we have
i∑
k=j
(−1)i−k
(
i− j + 1
k − j
)
2F1
(
1, i+ 1; i− k + 2;−1
x
)
=
=
(
x
1 + x
)i+1
2F1
(
1, i+ 1; i− j + 2;
1
1 + x
)
. (2.10)
Proof. From (2.6) we obtain
xi−j =
i∑
k=j
xi
(1 + x)k
(
k − 1
j − 1
)
+
+xi
i∑
k=j
(−1)i−k
xi+1
(
i
k − 1
)(
k − 1
j − 1
)
2F1
(
1, i+ 1; i− k + 2;−1
x
)
. (2.11)
An argument similar to the one used to prove (2.8) and (2.9) can be employed to prove the following
relation:
i∑
k=j
1
(1 + x)k
(
k − 1
j − 1
)
= x−j − 1
(1 + x)i+1
(
i
j − 1
)
2F1
(
1, i+ 1; i− j + 2;
1
1 + x
)
. (2.12)
The proof is finished after applying (2.12), in a conjunction with the transformation formula for the
binomial coefficients, on (2.11).
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 11
A MATRIX APPROACH TO THE BINOMIAL THEOREM 1583
Corollary 2.4. The identity
i∑
k=1
(−1)i−k
(
i
k − 1
)
2F1
(
1, i+ 1; i− k + 2;−1
x
)
=
(
x
1 + x
)i
(2.13)
is valid for every positive integer i and real x.
Proof. The proof follows from the previous corollary after some calculations in the case j = 1,
since the elements in the first column of the Pascal matrix are equal to pi,1 = 1.
Corollary 2.5. For i > j > 0 and x ∈ R the following is satisfied(
i
j
)
+
1
x
(
i
j − 1
)
2F1
(
1, 1− j; i− j + 2;−1
x
)
=
(
i
j
)
2F1
(
1,−j; i− j + 1;−1
x
)
. (2.14)
Proof. From relation Ln[x]−1 = PnSn[x]−1 we verify that the matrix Ln[x]−1 has the elements
equal to
(
Ln[x]−1
)
i,j
=
(
i− 1
j − 1
)
− x
(
i− 1
j
)
, i > j,
0, i < j.
Now we exploit the relation Pn = Ln[x]−1Sn[x], and obtain(
i
j
)
=
i−j∑
k=0
((
i
i− k
)
− x
(
i
i− k + 1
))
xi−j−k.
The proof is finished after verifying the following two identities:
i−j∑
k=0
(
i
i− k
)
xi−j−k =
(1 + x)i
xj
− 1
x
(
i
j − 1
)
2F1
(
1, 1− j; i− j + 2;−1
x
)
,
i−j∑
k=0
(
i
i− k + 1
)
xi−j−k+1 =
(1 + x)i
xj
−
(
i
j
)
2F1
(
1,−j; i− j + 1;−1
x
)
,
analogously as in Corollary 2.2.
Corollary 2.6. For integer n > 0 and x ∈ R, we have
n∑
k=1
2k−1(−1)n−k
(
n
k − 1
)
2F1
(
1, n+ 1;n− k + 2;−1
x
)
=
x
x− 1
((
2x
1 + x
)n
− 1
)
. (2.15)
Proof. Let En = [1, 1, . . . , 1]T . Since Sn[x]En = Ln[x]PnEn, we have
1
(x2 − 1)/(x− 1)
(x3 − 1)/(x− 1)
...
(xn − 1)/(x− 1)
= Ln[x] ·
1
2
4
...
2n−1
,
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 11
1584 S. STANIMIROVIĆ
that is
xn − 1
x− 1
=
xn
1 + x
n∑
k=1
(
2
1 + x
)k−1
+
+
n∑
k=1
2k−1
x
(−1)n−k
(
n
k − 1
)
2F1
(
1, n+ 1;n− k + 2;−1
x
)
.
Now we apply
∑n
k=1
(
2
1 + x
)k−1
= −x+ 1
x− 1
((
2
1 + x
)n
− 1
)
, and the proof is completed.
3. Conclusion. By observing the fact that the binomial theorem xn =
∑n
k=0
(
n
k
)
(x−1)k can be
written in the matrix mode, in this note we investigate factorizations of the matrix with (i, j) th entry
equal to xi−j via the Pascal matrix. Later, we use these factorizations to derive numerous combinato-
rial identities. Some of them are especially interesting, like (2.2) which represents a generalization of
the Binomial theorem, as well as (2.3) which represents a generalization of the well-known identity∑i−1
k=0
xk =
xi − 1
x− 1
. We leave for the future research deriving more combinatorial identities from
various matrix factorizations.
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Received 15.01.12
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 11
|
| id | umjimathkievua-article-2684 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:28:18Z |
| publishDate | 2012 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/99/1877c6c8f7b9dc3dd7587839a2174199.pdf |
| spelling | umjimathkievua-article-26842020-03-18T19:32:37Z A matrix approach to the binomial theorem Матричний пiдхiд до бiномiальної теореми Stanimirovic, S. Станіміровіч, С. Motivated by the formula $x^n = \sum_{k=0}^n\left(n \atop k\right) (x - 1)^k$ we investigate factorizations of the lower triangular Toeplitz matrix with $(i, j)$th entry equal to $x^{i-j}$ via the Pascal matrix. In this way, a new computational approach to a generalization of the binomial theorem is introduced. Numerous combinatorial identities are obtained from these matrix relations. На основi формули$x^n = \sum_{k=0}^n\left(n \atop k\right) (x - 1)^k$ розглянуто факторизацiї нижньотрикутної матрицi Теплiца, $(i, j)$-й елемент якої дорiвнює $x^{i-j}$, з використанням матрицi Паскаля. Тим самим уведено новий обчислювальний пiдхiд до узагальнення бiномiальної теореми. Iз використанням цих матричних спiввiдношень отримано численнi комбiнаторнi тотожностi. Institute of Mathematics, NAS of Ukraine 2012-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2684 Ukrains’kyi Matematychnyi Zhurnal; Vol. 64 No. 11 (2012); 1578-1584 Український математичний журнал; Том 64 № 11 (2012); 1578-1584 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2684/2126 https://umj.imath.kiev.ua/index.php/umj/article/view/2684/2127 Copyright (c) 2012 Stanimirovic S. |
| spellingShingle | Stanimirovic, S. Станіміровіч, С. A matrix approach to the binomial theorem |
| title | A matrix approach to the binomial theorem |
| title_alt | Матричний пiдхiд до бiномiальної теореми |
| title_full | A matrix approach to the binomial theorem |
| title_fullStr | A matrix approach to the binomial theorem |
| title_full_unstemmed | A matrix approach to the binomial theorem |
| title_short | A matrix approach to the binomial theorem |
| title_sort | matrix approach to the binomial theorem |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2684 |
| work_keys_str_mv | AT stanimirovics amatrixapproachtothebinomialtheorem AT stanímírovíčs amatrixapproachtothebinomialtheorem AT stanimirovics matričnijpidhiddobinomialʹnoíteoremi AT stanímírovíčs matričnijpidhiddobinomialʹnoíteoremi AT stanimirovics matrixapproachtothebinomialtheorem AT stanímírovíčs matrixapproachtothebinomialtheorem |