Determinantal evaluation of four Wronskian matrices
UDC 517.5 Two determinants of Wronskian matrices are evaluated when the matrix rows are partitioned into $n$ blocks.Analogous formulae are derived for the matrices involving compositions of formal power series as entries.  
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| Дата: | 2021 |
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Institute of Mathematics, NAS of Ukraine
2021
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507001159680000 |
|---|---|
| author | Chu , W. Chu , W. Chu , W. |
| author_facet | Chu , W. Chu , W. Chu , W. |
| author_sort | Chu , W. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-03-31T08:48:07Z |
| description | UDC 517.5
Two determinants of Wronskian matrices are evaluated when the matrix rows are partitioned into $n$ blocks.Analogous formulae are derived for the matrices involving compositions of formal power series as entries.
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| doi_str_mv | 10.37863/umzh.v73i5.340 |
| first_indexed | 2026-03-24T02:02:22Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v73i5.340
UDC 517.5
W. Chu (School Math. and Statist., Zhoukou Normal Univ., China)
DETERMINANTAL EVALUATION OF FOUR WRONSKIAN MATRICES
ДЕТЕРМIНАНТНI ОЦIНКИ ЧОТИРЬОХ МАТРИЦЬ ВРОНСЬКОГО
Two determinants of Wronskian matrices are evaluated when the matrix rows are partitioned into n blocks. Analogous
formulae are derived for the matrices involving compositions of formal power series as entries.
Отримано оцiнки двох детермiнантiв матриць Вронського у випадку, коли рядки матриць розбито на n блокiв.
Аналогiчнi формули запропоновано для матриць з елементами, якi мiстять композицiю формальних степеневих
рядiв.
For the m-times differentiable functions \{ fk(x)\} 0\leq k<m, the corresponding Wronskian determinant
reads as
\mathrm{d}\mathrm{e}\mathrm{t}
0\leq i,j<m
\biggl[
di
dxi
fj(x)
\biggr]
.
One class of these determinants can be evaluated (cf. [2, 4 – 7]) when all the functions are powers of
one fixed function.
Another class concerns formal power series and composite functions. For a fixed unitary formal
power series f(x), let [xk]f(x) stand for the coefficient of xk in f(x) and f \langle \lambda \rangle (x) for the \lambda th com-
posite series of f(x). When matrix entries are replaced by [xi+1]f \langle j\rangle (x), Kedlaya [3] discovered a
product expression for the corresponding determinant, which has been generalized subsequently by
the author [2].
In this paper, we shall investigate the determinantal evaluation when the rows of the matrices just
mentioned are partitioned into n subsets. As preliminaries, a general expansion theorem that express
a determinant as a multiple sum of products of its minors over set-partitions of its row labels will be
proved and then be applied to the Vandermonde determinant in the first section. The second section
will derive two determinant formulae for the Wronskian matrix with its rows being partitioned into
n blocks. Finally, two analogous results will be obtained in the last section for the matrices when the
matrix entries are replaced by coefficients of composite functions of formal power series.
1. Expansion theorem and Vandermonde determinant. For a natural number m, denote
the interval of integers by
[1,m] = (0,m] = \{ k | 1 \leq k \leq m\} .
It can be expressed by the set partition:
[1,m] =
n\biguplus
k=1
(Mk - 1,Mk] ,
where
0 =M0 < M1 < M2 < . . . < Mn = m
with
c\bigcirc W. CHU, 2021
712 ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
DETERMINANTAL EVALUATION OF FOUR WRONSKIAN MATRICES 713
m =Mn =
n\sum
k=1
mk and mk :=Mk - Mk - 1 for 1 \leq k \leq n.
For \sigma \subset [1,m], denote its cardinality, norm and complement to [1,m], respectively, by
| \sigma | =
\sum
\lambda \in \sigma
1, \| \sigma \| =
\sum
\lambda \in \sigma
\lambda and \sigma c = [1,m]\setminus \sigma .
According to the Laplace formula, the determinant of a square matrix of order m can be expanded
along the first \ell rows:
H := \mathrm{d}\mathrm{e}\mathrm{t}
1\leq i,j\leq m
\bigl[
hi,j
\bigr]
=
\sum
\sigma \subset [1,m]
| \sigma | =\ell
( - 1)(
1+\ell
2 )+\| \sigma \| H
\Bigl[
[1, \ell ]| \sigma
\Bigr]
\times H
\Bigl[
(\ell ,m]| \sigma c
\Bigr]
,
where H
\bigl[
\tau | \sigma
\bigr]
stands for the determinant of the submatrix with the rows and columns being indexed,
respectively, by \tau \subset [1,m] and \sigma \subset [1,m], where | \sigma | = | \tau | , of course.
Iterating the last equation for (n - 1)-times, we derive the algebraic identity when the matrix
rows are partitioned into n blocks.
Theorem 1. The following determinant expansion formula holds:
H := \mathrm{d}\mathrm{e}\mathrm{t}
1\leq i,j\leq m
\bigl[
hi,j
\bigr]
=
\sum
\uplus n
k=1\sigma k=[1,m]
| \sigma k| =mk:1\leq k\leq n
\varepsilon (\sigma )
n\prod
k=1
H
\Bigl[
(Mk - 1,Mk]| \sigma k
\Bigr]
,
where the alternating sign \varepsilon (\sigma ) = \pm 1 depends only on the partition function \sigma = (\sigma 1, \sigma 2, . . . , \sigma n).
Now we examine the application of this theorem to the Vandermonde determinant
\Delta (x| [1,m]) = \Delta (x1, x2, . . . , xm) = \mathrm{d}\mathrm{e}\mathrm{t}
1\leq i,j\leq m
\bigl[
xj - 1
i
\bigr]
=
\prod
1\leq i<j\leq m
(xj - xi). (1)
Specifying the kth part of the partition \sigma = (\sigma 1, \sigma 2, . . . , \sigma n) by
\sigma k = \{ \lambda 1(k), \lambda 2(k), . . . , \lambda mk
(k)\} ,
dividing the variables \{ xi\} mi=1 into n subsets
\{ xi\} mi=1 =
n\biguplus
k=1
\{ x1(k), x2(k), . . . , xmk
(k)\} ,
and then applying Theorem 1 to (1), we get the following expression.
Proposition 1 (the Vandermonde determinant).
\prod
1\leq \imath <\jmath \leq n
m\imath \prod
i=1
m\jmath \prod
j=1
\{ xj(\jmath ) - xi(\imath )\}
n\prod
k=1
mk\prod
\kappa =1
x\kappa (k)
\prod
1\leq i<j\leq mk
\{ xj(k) - xi(k)\} =
=
\sum
\uplus n
k=1\sigma k=[1,m]
| \sigma k| =mk:1\leq k\leq n
\varepsilon (\sigma )
n\prod
k=1
\mathrm{d}\mathrm{e}\mathrm{t}
1\leq i,j\leq mk
\Bigl[
x
\lambda j(k)
i (k)
\Bigr]
.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
714 W. CHU
Under the substitutions xi(k) = qixk, the last equality becomes
\prod
1\leq \imath <\jmath \leq n
m\imath \prod
i=1
m\jmath \prod
j=1
\bigl\{
qjx\jmath - qix\imath
\bigr\} n\prod
k=1
(qxk)
(mk+1
2 )
\prod
1\leq i<j\leq mk
\bigl\{
qj - qi
\bigr\}
=
=
\sum
\uplus n
k=1\sigma k=[1,m]
| \sigma k| =mk:1\leq k\leq n
\varepsilon (\sigma )
n\prod
k=1
(qxk)
\| \sigma k\| \Delta (q\sigma k). (2)
Dividing both sides of the last equation by (1 - q)M with M =
\sum n
k=1
\biggl(
mk
2
\biggr)
and then letting
q \rightarrow 1, we find the following limiting equality.
Corollary 1 (algebraic identity).
\prod
1\leq \imath <\jmath \leq n
\{ x\jmath - x\imath \} m\imath m\jmath
n\prod
k=1
x
(mk+1
2 )
k
mk\prod
j=1
(j - 1)! =
\sum
\uplus n
k=1\sigma k=[1,m]
| \sigma k| =mk:1\leq k\leq n
\varepsilon (\sigma )
n\prod
k=1
x
\| \sigma k\|
k \Delta (\sigma k).
2. Wronskian matrices in blocks.
Lemma 1 ([2], Corollaries 4.3 and 4.4). Let f(x) and w(x) be two n-times differentiable func-
tions. The following two Wronskian determinant identities hold:
\mathrm{d}\mathrm{e}\mathrm{t}
1\leq i,j\leq n
\biggl[
di - 1
dxi - 1
\{ wi(x)f
yj (x)\}
\biggr]
=
\biggl\{
f \prime (x)
f(x)
\biggr\} (n2)
\Delta (y)
n\prod
k=1
wk(x)f
yk(x),
\mathrm{d}\mathrm{e}\mathrm{t}
1\leq i,j\leq n
\biggl[
di
dxi
\{ wi(x)f
yj (x)\}
\biggr]
=
\biggl\{
f \prime (x)
f(x)
\biggr\} (n+1
2 )
\Delta (y)
n\prod
k=1
ykwk(x)f
yk(x),
where there is an additional restriction for the second formula that the function wk(x) is a polynomial
of degree < k for 1 \leq k \leq n.
For the m-times differentiable functions fk(x) and wi(x), define the following determinant A
with its rows being divided into n blocks:
A := \mathrm{d}\mathrm{e}\mathrm{t}
1\leq i,j\leq m
\bigl[
ai,j
\bigr]
with ai,j =
di - Mk - 1
dxi - Mk - 1
\bigl\{
wi(x)f
yj
k (x)
\bigr\}
if i \in [Mk - 1,Mk).
Applying Theorem 1 to the last determinant, we get
A :=
\sum
\uplus n
k=1\sigma k=[1,m]
| \sigma k| =mk:1\leq k\leq n
\varepsilon (\sigma )
n\prod
k=1
\mathrm{d}\mathrm{e}\mathrm{t}
1\leq i,j\leq mk
\biggl[
di - 1
dxi - 1
\Bigl\{
wi+Mk - 1
(x)f
y\lambda j(k)
k (x)
\Bigr\} \biggr]
.
Evaluating, by means of the first formula of Lemma 1, the minor
A
\Bigl[
(Mk - 1,Mk]| \sigma k
\Bigr]
= \mathrm{d}\mathrm{e}\mathrm{t}
1\leq i,j\leq mk
\biggl[
di - 1
dxi - 1
\Bigl\{
wi+Mk - 1
(x)f
y\lambda j(k)
k (x)
\Bigr\} \biggr]
=
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
DETERMINANTAL EVALUATION OF FOUR WRONSKIAN MATRICES 715
=
\biggl\{
f \prime k(x)
fk(x)
\biggr\} (mk
2 )
\Delta (y| \sigma k)
mk\prod
i=1
wi+Mk - 1
(x)f
y\lambda i(k)
k (x),
we find the equality
A =
\sum
\uplus n
k=1\sigma k=[1,m]
| \sigma k| =mk:1\leq k\leq n
\varepsilon (\sigma )
n\prod
k=1
\biggl\{
f \prime k(x)
fk(x)
\biggr\} (mk
2 )
\Delta (y| \sigma k)
mk\prod
i=1
wi+Mk - 1
(x)f
y\lambda i(k)
k (x) =
=
m\prod
i=1
wi(x)
n\prod
k=1
\biggl\{
f \prime k(x)
fk(x)
\biggr\} (mk
2 ) \sum
\uplus n
k=1\sigma k=[1,m]
| \sigma k| =mk:1\leq k\leq n
\varepsilon (\sigma )
n\prod
k=1
\Delta (y| \sigma k)
\prod
\lambda \in \sigma k
fy\lambda k (x).
When yj = y + j for j \in [1,m], the last expression results in
m\prod
i=1
wi(x)
n\prod
k=1
fymk
k (x)
\biggl\{
f \prime k(x)
fk(x)
\biggr\} (mk
2 ) \sum
\uplus n
k=1\sigma k=[1,m]
| \sigma k| =mk:1\leq k\leq n
\varepsilon (\sigma )
n\prod
k=1
\Delta (\sigma k)f
\| \sigma k\|
k (x).
The above multiple sum can be expressed, in view of Corollary 1, as a compact product. Summing
up, we have proved the following theorem.
Theorem 2. For each k \in [1, n] and i \in [1,m], let fk(x) and wi(x) be the m-times differen-
tiable functions. Define the determinant \bfA by higher derivatives
\bfA := \mathrm{d}\mathrm{e}\mathrm{t}
1\leq i,j\leq m
\bigl[
\bfa i,j
\bigr]
with \bfa i,j =
di - Mk - 1
dxi - Mk - 1
\Bigl\{
wi(x)f
j+y
k (x)
\Bigr\}
,
where j \in [1,m] and i \in [Mk - 1,Mk) for k \in [1, n]. Then we have the determinantal evaluation
\bfA =
m\prod
i=1
wi(x)
\prod
1\leq \imath <\jmath \leq n
\{ f\jmath (x) - f\imath (x)\} m\imath m\jmath \times
\times
n\prod
k=1
f
(1+y)mk
k (x)
\bigl\{
f \prime k(x)
\bigr\} (mk
2 )
mk\prod
j=1
(j - 1)!.
When y = - 1 and wi(x) \equiv 1 for 1 \leq i \leq m, the following particular cases are worthy of
mention.
Case 1: m1 = m2 = . . . = mn = 1. The Vandermonde determinant.
Case 2: n = 1. The determinant discovered by Mina [4] (see also van der Poorten [5])
\mathrm{d}\mathrm{e}\mathrm{t}
1\leq i,j\leq n
\biggl[
di - 1
dxj
f j - 1(x)
\biggr]
=
\bigl\{
f \prime (x)
\bigr\} (n2) n\prod
k=1
(k - 1)!.
It has been generalized by Wilf et al. [6, 7] to the formula
\mathrm{d}\mathrm{e}\mathrm{t}
1\leq i,j\leq n
\biggl[
di - 1
dxi - 1
fyj (x)
\biggr]
=
\biggl\{
f \prime (x)
f(x)
\biggr\} (n2)
\Delta (y)
n\prod
k=1
fyk(x).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
716 W. CHU
Case 3: n = 2. The problem proposed by Aharonov and Elias [1], which may be reproduced as
follows: for the square matrix of order m+ n defined by
ai,j(x) =
\left\{
di - 1
dxi - 1
f j - 1(x), 1 \leq i \leq m,
di - m - 1
dxi - m - 1
gj - 1(x), m < i \leq m+ n,
its determinant is evaluated as follows (where the product expression has been corrected by removing
an extra factor):
\mathrm{d}\mathrm{e}\mathrm{t}
1\leq i,j\leq m+n
\bigl[
ai,j(x)
\bigr]
=
\bigl\{
f \prime (x)
\bigr\} (m2 ) \bigl\{ g\prime (x)\bigr\} (n2)\times
\times \{ g(x) - f(x)\} mn
m - 1\prod
i=1
i!
n - 1\prod
j=1
j!.
Following exactly the same proving procedure, we can evaluate, by means of the second formula
of Lemma 1, another Wronskian determinant with the rows being partitioned into n blocks.
Theorem 3. For each k \in [1, n] and i \in (Mk - 1,Mk], let fk(x) be a m-times differentiable
function and wi(x) a polynomial of degree < i - Mk - 1. Define the determinant \bfB by higher
derivatives
\bfB := \mathrm{d}\mathrm{e}\mathrm{t}
1\leq i,j\leq m
\bigl[
bi,j
\bigr]
with \bfb i,j =
di - Mk - 1
dxi - Mk - 1
\Bigl\{
wi(x)f
j+y
k (x)
\Bigr\}
,
where j \in [1,m] and i \in (Mk - 1,Mk] for k \in [1, n]. Then we have the determinantal evaluation
\bfB =
m\prod
i=1
(y + i)wi(x)
\prod
1\leq \imath <\jmath \leq n
\{ f\jmath (x) - f\imath (x)\} m\imath m\jmath \times
\times
n\prod
k=1
fymk
k (x)
\bigl\{
f \prime k(x)
\bigr\} (mk+1
2 )
mk\prod
j=1
(j - 1)!.
When y = 0 and wi(x) \equiv 1 for 1 \leq i \leq m, three cases are given below.
Case 1: m1 = m2 = . . . = mn = 1. It is equivalent to the Vandermonde determinant.
Case 2: n = 1. We get the determinant
\mathrm{d}\mathrm{e}\mathrm{t}
1\leq i,j\leq n
\biggl[
di
dxj
f j(x)
\biggr]
= \{ f \prime (x)\} (
n+1
2 )
n\prod
k=1
k!.
This is a variant of Mina’s determinant which was extended by the author [2] (Proposition 2.4) to the
following one:
\mathrm{d}\mathrm{e}\mathrm{t}
1\leq i,j\leq n
\biggl[
di
dxi
fyj (x)
\biggr]
=
\biggl\{
f \prime (x)
f(x)
\biggr\} (n+1
2 )
\Delta (y)
n\prod
k=1
ykf
yk(x).
Case 3: n = 2. A variant of the determinant by Aharonov and Elias [1]. For the square matrix
of order m+ n defined by
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
DETERMINANTAL EVALUATION OF FOUR WRONSKIAN MATRICES 717
bi,j(x) =
\left\{
di
dxi
f j(x), 1 \leq i \leq m,
di - m
dxi - m
gj(x), m < i \leq m+ n,
there exists the determinantal evaluation
\mathrm{d}\mathrm{e}\mathrm{t}
1\leq i,j\leq m+n
\bigl[
bi,j(x)
\bigr]
= (m+ n)!
\bigl\{
f \prime (x)
\bigr\} (m+1
2 ) \bigl\{ g\prime (x)\bigr\} (n+1
2 )\times
\times \{ g(x) - f(x)\} mn
m - 1\prod
i=1
i!
n - 1\prod
j=1
j!.
Proof of Theorem 3. Recalling Theorem 1, we can express the determinant in question as the
multiple sum
\bfB =
\sum
\uplus n
k=1\sigma k=[1,m]
| \sigma k| =mk:1\leq k\leq n
\varepsilon (\sigma )
n\prod
k=1
\mathrm{d}\mathrm{e}\mathrm{t}
1\leq i,j\leq mk
\biggl[
di
dxi
\Bigl\{
wi+Mk - 1
(x)f
y+\lambda j(k)
k (x)
\Bigr\} \biggr]
.
According to the second formula of Lemma 1, we can evaluate the minor
\bfB
\Bigl[
(Mk - 1,Mk]| \sigma k
\Bigr]
= \mathrm{d}\mathrm{e}\mathrm{t}
1\leq i,j\leq mk
\biggl[
di
dxi
\Bigl\{
wi+Mk - 1
(x)f
y+\lambda j(k)
k (x)
\Bigr\} \biggr]
=
=
\biggl\{
f \prime k(x)
fk(x)
\biggr\} (mk+1
2 )
\Delta (\sigma k)
mk\prod
i=1
wi+Mk - 1
(x) \{ y + \lambda i(k)\} fy+\lambda i(k)
k (x),
which leads us to the equality
\bfB =
\sum
\uplus n
k=1\sigma k=[1,m]
| \sigma k| =mk:1\leq k\leq n
\varepsilon (\sigma )
n\prod
k=1
\biggl\{
f \prime k(x)
fk(x)
\biggr\} (mk+1
2 )
\Delta (\sigma k)\times
\times
mk\prod
i=1
wi+Mk - 1
(x) \{ y + \lambda i(k)\} fy+\lambda i(k)
k (x) =
=
m\prod
i=1
(y + i)wi(x)
n\prod
k=1
fymk
k (x)
\biggl\{
f \prime k(x)
fk(x)
\biggr\} (mk+1
2 )
\times
\times
\sum
\uplus n
k=1\sigma k=[1,m]
| \sigma k| =mk:1\leq k\leq n
\varepsilon (\sigma )
n\prod
k=1
\Delta (\sigma k)f
\| \sigma k\|
k (x).
Evaluating the last multiple sum by Corollary 1, we complete the proof of the determinant formula
displayed in Theorem 3.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
718 W. CHU
3. Matrices involving formal power series. In this section, two analogous determinants with
matrix entries involving formal power series will be evaluated. For subsequent application, we
reproduce first the following determinant identity.
Lemma 2 ([2], Theorem 3.3). Let F be a unitary formal power series defined by F (x) :=
:= x+ \phi x2 + . . . . The following determinant identity on composition series holds:
\mathrm{d}\mathrm{e}\mathrm{t}
1\leq i,j\leq n
\Bigl[
[xi+1]F \langle yj\rangle (x)
\Bigr]
= \phi (
n+1
2 )
\prod
1\leq i<j\leq n
(yj - yi)
n\prod
k=1
yk.
Let Fk(x) be the unitary formal power series with the two initial terms of Fk being given
explicitly by
[x]Fk(x) = 1 and [x2]Fk(x) = \phi k. (3)
Define the following determinant C with its rows being divided into n blocks:
C := \mathrm{d}\mathrm{e}\mathrm{t}
1\leq i,j\leq m
\bigl[
ci,j
\bigr]
with ci,j = [x1+i - Mk - 1 ]\beta
yj
k F
\langle yj\rangle
k (x) if i \in (Mk - 1,Mk],
where \{ \beta k\} nk=1 are constants independent of x. Applying Theorem 1 to the last determinant, we get
C :=
\sum
\uplus n
k=1\sigma k=[1,m]
| \sigma k| =mk:1\leq k\leq n
\varepsilon (\sigma )
n\prod
k=1
\mathrm{d}\mathrm{e}\mathrm{t}
1\leq i,j\leq mk
\biggl[
[xi+1]\beta
y\lambda j(k)
k F
\langle y\lambda j(k)\rangle
k (x)
\biggr]
.
Evaluating, by means of Lemma 2, the minor of order mk
C
\Bigl[
(Mk - 1,Mk]| \sigma k
\Bigr]
= \mathrm{d}\mathrm{e}\mathrm{t}
1\leq i,j\leq mk
\biggl[
[xi+1]F
\langle y\lambda j(k)\rangle
k (x)
\biggr] \prod
\lambda \in \sigma k
\beta y\lambda k =
= \phi
(mk+1
2 )
k \Delta (y| \sigma k)
\prod
\lambda \in \sigma k
y\lambda \beta
y\lambda
k ,
we find the equality
C =
\sum
\uplus n
k=1\sigma k=[1,m]
| \sigma k| =mk:1\leq k\leq n
\varepsilon (\sigma )
n\prod
k=1
\phi
(mk+1
2 )
k \Delta (y| \sigma k)
\prod
\lambda \in \sigma k
y\lambda \beta
y\lambda
k =
=
m\prod
i=1
yi
n\prod
k=1
\phi
(mk+1
2 )
k
\sum
\uplus n
k=1\sigma k=[1,m]
| \sigma k| =mk:1\leq k\leq n
\varepsilon (\sigma )
n\prod
k=1
\Delta (y| \sigma k)
\prod
\lambda \in \sigma k
\beta y\lambda k .
When yj = y + j for j \in [1,m], the expression displayed above reads as
m\prod
i=1
(y + i)
n\prod
k=1
\beta ymk
k \phi
(mk+1
2 )
k
\sum
\uplus n
k=1\sigma k=[1,m]
| \sigma k| =mk:1\leq k\leq n
\varepsilon (\sigma )
n\prod
k=1
\Delta (\sigma k)\beta
\| \sigma k\|
k .
In view of Corollary 1, the last multiple sum admits a closed expression. Therefore, we establish the
following theorem.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
DETERMINANTAL EVALUATION OF FOUR WRONSKIAN MATRICES 719
Theorem 4. For the unitary formal power series Fk(x) subject to condition (3), define the
determinant \bfC by coefficients of composite series
\bfC := \mathrm{d}\mathrm{e}\mathrm{t}
1\leq i,j\leq m
\bigl[
\bfc i,j
\bigr]
with \bfc i,j = [x1+i - Mk - 1 ]
\Bigl\{
\beta y+j
k F
\langle y+j\rangle
k (x)
\Bigr\}
,
where j \in [1,m] and i \in (Mk - 1,Mk] for k \in [1, n]. Then we have the determinantal evaluation
\bfC =
m\prod
i=1
(y + i)
\prod
1\leq \imath <\jmath \leq n
\{ \beta \jmath - \beta \imath \} m\imath m\jmath
n\prod
k=1
\beta ymk
k
\bigl(
\phi k\beta k
\bigr) (mk+1
2 )
mk\prod
j=1
(j - 1)!.
It is curious to notice that the last determinant vanishes for n > 1 if all the \{ \beta k\} nk=1 are identical.
When n = 1 and y = 0, Theorem 4 recovers the following determinant identity discovered by
Kedlaya [3]:
\mathrm{d}\mathrm{e}\mathrm{t}
1\leq i,j\leq n
\Bigl[
[xi+1]F \langle j\rangle (x)
\Bigr]
= \phi (
n+1
2 )
n\prod
k=1
k!,
where F (x) is a unitary formal power series F (x) := x+ \phi x2 + . . . .
If the ith row of the matrix in Lemma 2 is replaced by the extraction of ith (instead of (i+1)th)
coefficients of composition series, we have the following stronger result.
Lemma 3 ([2], Theorem 4.6). Let F (x), G(x) and Wi(x) be three formal power series with
the initial coefficients of F and G being given explicitly by
[x]F (x) = 1, [x2]F (x) = \phi , and [x]G(x) = \psi .
Then the following determinant identity holds:
\mathrm{d}\mathrm{e}\mathrm{t}
1\leq i,j\leq n
\Bigl[
[xi]
\Bigl\{
Wi(x)F
\langle yj\rangle [G(x)]
\Bigr\} \Bigr]
= \phi (
n
2)\psi (
n+1
2 )
\prod
1\leq i<j\leq n
(yj - yi)
n\prod
k=1
Wk(0).
By following the same proof as that for Theorem 4 and invoking Lemma 3, we can evaluate
another similar determinant involving formal power series in the matrix entries.
Theorem 5. Let Fk(x), Gk(x) and Wi(x) be the formal power series with the initial terms of
Fk and Gk being given by
[x]Fk(x) = 1, [x2]Fk(x) = \phi k, and [x]Gk(x) = \psi k.
For the constants \{ \beta k\} nk=1 independent of x, define the determinant \bfD by coefficients of composite
series
\bfD := \mathrm{d}\mathrm{e}\mathrm{t}
1\leq i,j\leq m
\bigl[
\bfd i,j
\bigr]
with di,j = [xi - Mk - 1 ]\beta y+j
k
\Bigl\{
Wi(x)F
\langle y+j\rangle
k
\bigl[
Gk(x)
\bigr] \Bigr\}
,
where j \in [1,m] and i \in (Mk - 1,Mk] for k \in [1, n]. Then we have the determinantal evaluation
\bfD =
m\prod
i=1
Wi(0)
\prod
1\leq \imath <\jmath \leq n
\{ \beta \jmath - \beta \imath \} m\imath m\jmath
n\prod
k=1
\bigl(
\beta 1+y
k \psi k
\bigr) mk
\bigl(
\phi k\psi k\beta k
\bigr) (mk
2 )
mk\prod
j=1
(j - 1)!.
If Gk(x) \equiv x and Wi(x) \equiv 1, this theorem reduces to the following one which may be consi-
dered as a counterpart of Theorem 4.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
720 W. CHU
Corollary 2. Let Fk(x) be the unitary formal power series subject to condition (3). Define the
determinant T by
T := \mathrm{d}\mathrm{e}\mathrm{t}
1\leq i,j\leq m
\Bigl[
ti,j
\Bigr]
with ti,j = [xi - Mk - 1 ]
\Bigl\{
\beta y+j
k F
\langle y+j\rangle
k (x)
\Bigr\}
,
where j \in [1,m] and i \in (Mk - 1,Mk] for k \in [1, n]. Then the following determinant identity holds:
T =
\prod
1\leq \imath <\jmath \leq n
\{ \beta \jmath - \beta \imath \} m\imath m\jmath
n\prod
k=1
\beta
(1+y)mk
k
\bigl(
\phi k\beta k
\bigr) (mk
2 )
mk\prod
j=1
(j - 1)!.
References
1. D. Aharonov, U. Elias, Problem 11608, Amer. Math. Monthly, 118, № 10 (2011).
2. W. Chu, The Faà di Bruno formula and determinant identities, Linear and Multilinear Algebra, 54, № 1, 1 – 25
(2006).
3. K. S. Kedlaya, Another combinatorial determinant, J. Combin. Theory. Ser. A, 90, 221 – 223 (2000).
4. L. Mina, Formule generali delle successive d’una funzione espresse mediante quelle della sua inversa, Giornale Mat.,
43, 196 – 212 (1905).
5. A. J. van der Poorten, Some determinants that should be better known, J. Austr. Math. Soc. Ser. A, 21, 78 – 288
(1976).
6. H. S. Wilf, V. Strehl, Five surprisingly simple complexities, J. Symbolic Comput., 20, № 5/6, 725 – 729 (1995).
7. H. S. Wilf, A combinatorial determinant (1998), arXiv:math/9809120v3.
Received 20.09.18
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
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| id | umjimathkievua-article-340 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:02:22Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/75/65a86be9e44dabe257e1fc0978b72375.pdf |
| spelling | umjimathkievua-article-3402025-03-31T08:48:07Z Determinantal evaluation of four Wronskian matrices Determinantal evaluation of four Wronskian matrices Determinantal evaluation of four Wronskian matrices Chu , W. Chu , W. Chu , W. Wronskian matrix, Formal power series, Composite function, Laplace expansion, Vandermonde determinant Wronskian matrix, Formal power series, Composite function, Laplace expansion, Vandermonde determinant UDC 517.5 Two determinants of Wronskian matrices are evaluated when the matrix rows are partitioned into $n$ blocks.Analogous formulae are derived for the matrices involving compositions of formal power series as entries. &nbsp; UDC 517.5 Детермiнантнi оцiнки чотирьох матриць Вронського Отримано оцінки двох детермінантів матриць Вронського у випадку, коли рядки матриць розбито на $n$ блоків.Аналогічні формули запропоновано для матриць з елементами, які містять композицію формальних степеневих рядів. Institute of Mathematics, NAS of Ukraine 2021-05-24 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/340 10.37863/umzh.v73i5.340 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 5 (2021); 712 - 720 Український математичний журнал; Том 73 № 5 (2021); 712 - 720 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/340/9021 Copyright (c) 2021 WENCHANG CHU |
| spellingShingle | Chu , W. Chu , W. Chu , W. Determinantal evaluation of four Wronskian matrices |
| title | Determinantal evaluation of four Wronskian matrices |
| title_alt | Determinantal evaluation of four Wronskian matrices Determinantal evaluation of four Wronskian matrices |
| title_full | Determinantal evaluation of four Wronskian matrices |
| title_fullStr | Determinantal evaluation of four Wronskian matrices |
| title_full_unstemmed | Determinantal evaluation of four Wronskian matrices |
| title_short | Determinantal evaluation of four Wronskian matrices |
| title_sort | determinantal evaluation of four wronskian matrices |
| topic_facet | Wronskian matrix Formal power series Composite function Laplace expansion Vandermonde determinant Wronskian matrix Formal power series Composite function Laplace expansion Vandermonde determinant |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/340 |
| work_keys_str_mv | AT chuw determinantalevaluationoffourwronskianmatrices AT chuw determinantalevaluationoffourwronskianmatrices AT chuw determinantalevaluationoffourwronskianmatrices |