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Abstract
The Hankel determinant appears in representations of solutions to several integrable systems. An asymptotic expansion of the Hankel determinant thus plays a key role in the investigation of asymptotic analysis of such integrable systems. This paper presents an asymptotic expansion formula of a certain Casorati determinant as an extension of the Hankel case. This Casorati determinant is then shown to be associated with the solution to the discrete hungry Lotka–Volterra (dhLV) system, which is an integrable variant of the famous prey–predator model in mathematical biology. Finally, the asymptotic behavior of the dhLV system is clarified using the expansion formula for the Casorati determinant.
Public Interest Statement
In this paper, we present a powerful new technique for an asymptotic expansion of the Casorati determinant. The Casorati determinant is associated with several difference equations appearing in mathematical physics, and plays a role similar to the Wronskian in the theory of differential equations. Our technique will be useful for asymptotically analyzing not only the discrete hungry Lotka–Volterra system, but also other dynamical systems associated with the Casorati determinant.
1. Introduction
Integrable systems are often classified as nonlinear dynamical systems whose solutions can be explicitly expressed. Such an integrable system is the Toda equation which describes the current–voltage function in an electric circuit. A time discretization, called the discrete Toda equation (Hirota, Citation1981), is simply equal to the recursion formula of the qd algorithm for computing eigenvalues of a symmetric tridiagonal matrix (Henrici, Citation1988; Rutishauser, Citation1990) and singular values of a bidiagonal matrix (Parlett, Citation1995).
Another commonly investigated integrable system is the integrable Lotka–Volterra (LV) system, which is a prey–predator model in mathematical biology (Yamazaki, Citation1987). The discrete LV (dLV) system was shown in Iwasaki and Nakamura (Citation2002) to be applicable to computing for bidiagonal singular values. The hungry Lotka–Volterra (hLV) system is a variant that captures a more complicated prey–predator relationship in comparison with the original LV system (Bogoyavlensky, Citation1988; Itoh, Citation1987). Time discretization of this system leads to the discrete hungry Lotka–Volterra (dhLV) system. It was shown in Fukuda, Ishiwata, Iwasaki, and Nakamura (Citation2009), Fukuda, Ishiwata, Yamamoto, Iwasaki, and Nakamura (Citation2013), Yamamoto, Fukuda, Iwasaki, Ishiwata, and Nakamura (Citation2010) that the dhLV system can generate matrix transformations for computing eigenvalues of a banded totally nonnegative (TN) matrix whose minors are all nonnegative.
The determinant solutions to both the discrete Toda equation and the dLV system can be expressed using the Hankel determinant,(1.1)
(1.1)
where and
correspond to the discrete spatial and discrete time variables, respectively (Tsujimoto, Citation2001). Here, the formal power series
associated with
is assumed to be analytic at
and meromorphic in the disk
. The finite or infinite types of poles
of
are ordered such that
. Then, there exists a nonzero constant
independent of
such that, for some
satisfying
,
(1.2)
(1.2)
as (Henrici, Citation1988). The asymptotic expansion (1.2) as
enables the asymptotic analysis of the discrete Toda equation and the dLV system as in Henrici (Citation1988), Rutishauser (Citation1990) and in Iwasaki and Nakamura (Citation2002), respectively.
A generalization of the Hankel determinant is given in the below determinant of a nonsymmetric square matrix of order
,
(1.3)
(1.3)
which is called the Casorati determinant or Casoratian. The Casorati determinant is useful in the theory of difference equations, particularly in mathematical physics, and plays a role similar to the Wronskian in the theory of differential equations (Vein & Dale, Citation1999). No one wonder here that the formal power series is associated with the Casorati determinant
for each
. The formal power series
differs from
in that not only the superscript, but also the subscript, appears in the coefficients.
To the best of our knowledge, from the viewpoint of the formal power series , the asymptotic analysis for the Casorati determinant
has not yet been discussed in the literature. The first purpose of this paper is to present an asymptotic expansion of the Casorati determinant
as
. The asymptotic behavior of the dhLV system was discussed in Fukuda et al. (Citation2009, Citation2013) in the case where the discretization parameter
is restricted to be positive. However, it was suggested in Yamamoto et al. (Citation2010) that the choice
in the dhLV system yields a convergence acceleration of the
transformations. The discrete time evolution in the dhLV system with
corresponds to a reverse of the continuous-time evolution in the hLV system. It is interesting to note that such artificial dynamics are useful for computing eigenvalues of a TN matrix. The second purpose of this paper is to provide an asymptotic analysis for the dhLV system without being limited by the sign of
.
The remainder of this paper is organized as follows. In Section 2, we first observe that the entries in can be expressed using poles of
. We then give an asymptotic expansion of the Casorati determinant in terms of the poles of
as
by expanding the theorem analyticity for the Hankel determinant given in Henrici (Citation1988). In Section 3, we find the determinant solution to the dhLV system through relating the dhLV system to a three-term recursion formula. With the help of the resulting theorem for the Casorati determinant
, we explain in Section 4 that the determinant solution to the dhLV system can be rewritten using the Casorati determinant
, and we clarify the asymptotic behavior of the solution to the dhLV system. Finally, we give concluding remarks in Section 5.
2. An asymptotic expansion of the Casorati determinant
In this section, we first give an expression of the entries of the Casorati determinant in terms of poles of the formal power series
associated with
. Referring to the theorem on analyticity for the Hankel determinant given in Henrici (Citation1988), we present an asymptotic expansion of the Casorati determinant
as
using the poles of
. We also describe the case where some restriction is imposed on the poles of
.
Let , which is the formal power series associated with
for
be analytic at
and meromorphic in the disk
. Moreover, let
denote the poles of
such that
. By extracting the principal parts in
, we derive
(2.1)
(2.1)
where is an arbitrary positive integer,
are some nonzero constants, and
, which contains the terms with respect to
, satisfies
(2.2)
(2.2)
for some nonzero positive constants and
with
. The proof of (2.2) is given in Henrici (Citation1988) utilizing the Cauchy coefficient estimate. We now give a lemma for an expression of
appearing in
.
Lemma 2.1
Let us assume that the poles of
are not multiple. Then,
can be expressed using
as
(2.3)
(2.3)
where are some nonzero constants.
Proof
The crucial element is the replacement in (2.1), namely,
(2.4)
(2.4)
Since each in (2.4) can be regarded as the summation of a geometric series, we obtain
which implies (2.3).
Similarly to the asymptotic expansion as of the Hankel determinant
given in Henrici (Citation1988), we have the following theorem for the Casorati determinant
.
Theorem 2.2
Let us assume that the poles of
are not multiple. Then there exists some constant
independently of
such that, as
,
(2.5)
(2.5)
where denotes the mapping from
to
and
is some constant such that
.
Proof
By applying Lemma 2.1 and the addition formula of determinants to the Casorati determinant , we derive
(2.6)
(2.6)
where in the first summation
and in the second summation denotes a determinant of the same form as
except that the
th column is replaced with
for at least one of
. Evaluating the first summation in (2.6), we obtain
(2.7)
(2.7)
where(2.8)
(2.8)
To estimate the second summation in (2.6), for example, we consider the case where the st column is replaced with
. It immediately follows from (2.2) that
It is also easy to check if the
th column is replaced with
. Similarly, by examining the case where some columns are replaced with some of
, we can see that
(2.9)
(2.9)
Thus, from (2.7)–(2.9), we obtain (2.5).
Now, let us consider the restriction in
. Then, by replacing
with
in (2.3), we easily obtain
(2.10)
(2.10)
As a specialization of Theorem 2.2, we derive the following theorem for an asymptotic expansion of the Casorati determinant with restricted
as
.
Theorem 2.3
Let us assume that the poles of
are not multiple. Then there exists some constant
independently of
such that, for
, as
,
(2.11)
(2.11)
Proof
Replacing in (2.8) gives
(2.12)
(2.12)
Thus, by taking into account that only in the case where
are distinct to each other, we can simplify (2.7) as
(2.13)
(2.13)
where denotes the bijection from
to
. It is noted here that the bijection
is equal to the mapping
with
. Moreover, there exists a constant
, which is not equal to one in Theorem 2.2, such that
. This is because
and
do not always satisfy
even if
in Theorem 2.2. Thus, (2.9) becomes
(2.14)
(2.14)
Therefore, from (2.13) and (2.14), we obtain (2.11).
Theorem 2.3 covers an asymptotic expansion of the Hankel determinant . Theorems 2.2 and 2.3 should be useful for the asymptotic analysis of dynamical systems with solutions expressed in terms of the Casorati determinant
.
3. The dhLV system and its determinant solution
In this section, similarly to work in Tsujimoto and Kondo (Citation2000), Spiridonov and Zhedanov (Citation1997), we derive the dhLV system from a three-term recursion formula, and then clarify the determinant expression of an auxiliary variable in the solution to the dhLV system through investigating the three-term recursion formula.
Let us consider a three-term recursion formula with respect to the polynomials at the discrete time
,
(3.1)
(3.1)
where is a positive integer and
do not depend on
. Accordingly,
are all monic. Moreover, let us prepare a time evolution from
to
,
(3.2)
(3.2)
where . Then, by replacing
with
in (3.1) and using (3.2), we obtain
(3.3)
(3.3)
By using (3.1) again for deleting except for terms with respect to and
in (3.3), we derive
Thus, it is observed that(3.4)
(3.4)
(3.5)
(3.5)
Let us introduce a new variable such that
(3.6)
(3.6)
(3.7)
(3.7)
Then, it follows from (3.5)–(3.7) that(3.8)
(3.8)
Moreover, from (3.6) and (3.8), we see that(3.9)
(3.9)
It is obvious from (3.7) that the right-hand side of (3.9) is equal to . This implies that
in (3.8) also satisfies (3.4). Consequently, by combining (3.6) and (3.8), noting that
and replacing
with
, we have the discrete system
(3.10)
(3.10)
Equation (3.10) can be regarded as a discretization of the hLV system which differs from the simple LV system in that more than one food exists for each species. Thus, (3.10) is the dhLV system and corresponds to the number of the species of foods for each species. Clearly, from the definition, (3.10) with
is simply equal to the dLV system. The dhLV system (3.10) is essentially equal to the dhLV system in Fukuda et al. (Citation2009),
(3.11)
(3.11)
This is because (3.11) is derived by replacing with
and
with
for
in (3.10).
Let be polynomials satisfying a three-term recursion formula,
(3.12)
(3.12)
where do not depend on
. It is obvious from (3.12) that
,
,
are also all monic. Moreover, let us introduce a linear functional (form)
,
(3.13)
(3.13)
where is a weight function. The linear functional
with
is equivalent to that in Chihara (Citation1978). Further,
with arbitrary
is a specialization of a linear function appearing in Maeda, Miki, and Tsujimoto (Citation2013). Since it follows from (3.1), (3.12), and (3.13) that
and
, we easily derive
(3.14)
(3.14)
Let for
. From (3.12), it turns out that
is expressed as
where
are some constants at each
and each
. Since it is clear from (3.1) that
can be given as the summation of
and the linear combination of
,
,
,
, we see from (3.13) that
. Thus, it follows that
By combining the above with (3.13), we derive a system of linear equations(3.15)
(3.15)
Since are uniquely determined, the coefficient matrix in (3.15) is nonsingular. This suggests that (3.15) can be transformed into
(3.16)
(3.16)
where the hat denotes cofactors of the coefficient matrix in (3.15) and(3.17)
(3.17)
It is of significance to note that . Thus, by examining the last row for both sides of (3.16), we find
(3.18)
(3.18)
Equations (3.14) and (3.18) therefore lead to(3.19)
(3.19)
Since we can easily obtain the solution to the dhLV system (3.10), by combining (3.6) with (3.19), the determinant expression of is important for the asymptotic analysis of the dhLV system (3.10) in the next section.
Let us define the time evolution of the linear functional from to
by
(3.20)
(3.20)
where is an arbitrary polynomial. Then, it is easy to check that
and
are orthogonal to each other with respect to
. Equation (3.20) yields a time evolution with respect to
’s,
(3.21)
(3.21)
Noting (3.1) and (3.12), we find that with
can be expressed as the linear combination of
,
,
,
. Thus, by combining it with (3.13), we derive
(3.22)
(3.22)
Similarly, in the case where with
, we have
(3.23)
(3.23)
Taking into account that the sequence with (3.21) is a specialization of the sequence
appearing in the previous section, we may replace
with
in the following discussion. Thus, we can rewrite
as
where is an
-by-
diagonal matrix with the relationship concerning the evolution from
to
,
(3.24)
(3.24)
4. Asymptotic analysis of the dhLV system
This section begins by explaining that the auxiliary variable in the dhLV system can be rewritten in terms of the Casorati determinant. By using Theorem 2.2, we clarify the asymptotic behavior of the dhLV variables as .
The st,
nd,
,
th row and column blocks in
are
-by-
matrices, but the
th row and column blocks are
-by-
matrices. The following lemma gives the representation of
in terms of the
appearing in Section 1.
Lemma 4.1
The auxiliary variable is expressed as
(4.1)
(4.1)
(4.2)
(4.2)
Proof
Let us introduce a new determinant of a square matrix of order ,
(4.3)
(4.3)
We begin by showing that can be transformed into a block diagonal determinant with respect to
. By interchanging the
nd,
rd,
th rows and columns with the
th,
th,
,
th rows and columns in
, we observe that the same form of
appears in the
st diagonal block of
. The entries in the
st,
nd,
,
th rows and columns in
are simultaneously all
, except for those in the diagonal block section. Permutations similar to the above provide the forms of
as the
nd,
rd,
,
th blocks in
. Thus,
can be expressed in terms of
as
(4.4)
(4.4)
Similarly, can be transformed into the determinant of a block diagonal matrix whose
blocks are
and
. Thus, it follows that
(4.5)
(4.5)
The cases where and
in (3.19) become
(4.6)
(4.6)
By combining them with (4.4) and (4.5), we obtain(4.7)
(4.7)
The entries in the th row of
are given by the linear combination
for
. By multiplying the
th row by
and then adding it to the
th, we get row
as the new
th row. Similarly, for the
th,
th,
,
nd rows, it follows that
It is worth noting here that the subscript can be regarded as be transformed into the superscript
. Thus,
in (4.3) is equal to the Casorati determinant
in (1.3). Then, by accounting for it in (4.6) and (4.7), we have (4.1) and (4.2).
Lemma 4.1 with Theorem 2.2 leads to the following theorem for asymptotic behavior of as
.
Theorem 4.2
The auxiliary variable converges to some constant
as
.
Proof
Let be the mapping from
to
where
are positive integers such that
. Then, it follows from Theorem 2.2 that
(4.8)
(4.8)
It is of significance to note the relationship between and
is derived from (3.24),
(4.9)
(4.9)
Equation (4.9) implies that the poles of and
are equal to each other, namely,
,
,
. Thus, by combining them with Theorem 2.2, we derive
(4.10)
(4.10)
Since (4.8) and (4.10) imply that as
, we can conclude that
as
.
By considering the positivity of ,
,
,
, we derive the following theorem for the asymptotic behavior of
,
,
,
as
.
Theorem 4.3
Let us assume that ,
,
,
for
. Then
,
,
,
converge to
as
.
Proof
From the Jacobi determinant identity (Hirota, Citation2003), it follows that(4.11)
(4.11)
Equation (4.11) allows us to simplify as
(4.12)
(4.12)
From (4.8), we derive(4.13)
(4.13)
Thus, by combining (4.13) and ,
,
,
with (4.12), we have
(4.14)
(4.14)
Therefore, by taking into account that ,
,
,
in (4.14), we find that
,
,
,
as
.
By recalling the relationship of the dhLV variable to the auxiliary variable
in (3.6), we have the following theorem concerning an asymptotic convergence of
as
.
Theorem 4.4
As , the dhLV variable
converges to some nonzero constant
, and
,
,
,
go to
, provided that
satisfy
for
and the limit of
as
exists.
Proof
The proof is given by induction for . Without loss of generality, let us assume that
where
denotes some constant. From (3.6), it holds that
(4.15)
(4.15)
By taking the limit as of both sides of (4.15) with
and using
as
, we obtain
(4.16)
(4.16)
where . By considering Theorem 4.2 with (4.16) in the case where
in (4.15), we successively check that
as
.
Let us assume that and
as
. Equation (4.15) with
becomes
(4.17)
(4.17)
It is clear that the denominator on the right-hand side of (4.17) converges to as
under this assumption. By combining it with
as
, we observe that
as
. Moreover, it follows that
(4.18)
(4.18)
since and
as
.
The convergence theorem concerning the dhLV system (3.10) in Fukuda et al. (Citation2009) is restricted to the case where the dhLV variable is positive and the discretization parameter
is fixed positive for every
. Theorem 4.4 claims that the
th species survives and the
th,
th,...,
th species vanish as
even in the case where
is a changeable negative for each
. Although the case of negative
is not longer recognized as a valid biological model, we note that the convergence is not different from the positive case if the values of
are suitable for
.
To observe the asymptotic convergence numerically, we consider two cases where and
in the dhLV system (3.10). The initial values are set as
for
in the dhLV system (3.10) with
and
. Figure shows the behavior of
for
in the case where
and
. This figure demonstrates that
tends to
as
grows larger even if
. We also see that the case where
has a superior convergence speed in comparison with the case where
. Similarly, the asymptotic behavior of
can be seen to follow Theorem 4.4.
5. Concluding remarks
In this paper, we associated a formal power series with the Casorati determinant
, and gave asymptotic expansions of the Casorati determinants as
in Theorems 2.2 and 2.3. By making use of Theorem 2.2, we then clarified the asymptotic behavior of the dhLV variables as
in Theorem 4.4.
Theorems 2.2 and 2.3 may contribute to asymptotic analysis for other discrete integrable systems. One possible application is the discrete hungry Toda (dhToda) equation derived from the numbered box and ball system through inverse ultra-discretization (Tokihiro, Nagai, & Satsuma, Citation1999). The dhToda equation has a relationship of variables to the dhLV system whose solution is given in the Casorati determinant (Fukuda, Yamamoto, Iwasaki, Ishiwata, & Nakamura, Citation2011). The Casorati determinant directly appears in, for example, the solution to the discrete Darboux–Pöschl–Teller equation which is a discretization of a dynamical system concerning a special class of potentials for the -dimensional Schrödinger equation (Gaillard & Matveev, Citation2009).
It was proved in Fukuda et al. (Citation2013) that the dhLV system (3.10) with a fixed positive is associated with the
transformation for a TN matrix. The paper (Yamamoto et al., Citation2010) also suggested that the dhLV system (3.10) with changeable negative
generates the shifted
transformation for a TN matrix. Eigenvalues of an
-by-
TN matrix correspond to the constants
,
,
,
in Theorem 4.4. Theorems 4.2–4.4 will be useful for investigating the convergence of the sequence of the shifted
transformations based on the dhLV system (3.10) in the changeable negative case of
.
Acknowledgements
The authors would like to thank Prof. S. Tsujimoto for helpful discussions on the determinant expression. The authors also thank the reviewers for their careful reading and insightful suggestions.
Additional information
Funding
Notes on contributors
Masato Shinjo
Masato Shinjo is a doctoral student in the Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University. He studies asymptotic analysis of nonlinear dynamical systems known as integrable systems.
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