The determinant representation of the gauge transformation for the discrete KP hierarchy

0.15 MB·English

by  Liu Shaowei

The determinant representation of the gauge
transformation for the discrete KP hierarchy
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1
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2
n Liu Shaowei†, Cheng Yi†, He Jingsong‡∗,
a †DepartmentofMathematics,UniversityofScienceandTechnologyofChina,Hefei,230026Anhui,P.R.China
J
5 ‡DepartmentofMathematics,NingboUniversity,Ningbo,315211Zhejiang,P.R.China

]
I
S
.
n Abstract
i
l
n
[

2 A successive gauge transformation operator Tn+k for the discrete
v KP(dKP)hierarchyisdefined, whichisinvolvedwithtwotypesofgauge
8 transformations operators. The determinant representation of the T
n+k
6 (n+k)
is established,and then it is used to get a new tau function τ of
8 △
1 the dKP hierarchy from an initial τ . In this process, we introduce a
△
.
4 generalized discrete Wronskian determinant and some useful properties
0 of discrete difference operator.
9
0
: Keywords gauge transformation, dKP hierarchy, tau function
v
i
X
2000 MR Subject Classification 35Q51, 37K10
r
a
∗
Corresponding author, Email:[email protected],[email protected]
1
§1. Introduction
The discrete KP(dKP) hierarchy[1, 2, 3, 4] is an interesting object in the research of the discrete
integrable systems and the discretization of the integrable systems[5, 6, 7, 8]. Naturally, there
exist some similar properties between dKP and KP hierarchy[9], such as tau function[2, 9, 10],
Hamiltonian structures[3, 4] and gauge transformation[1, 11, 12], etc. In particular, Gauge
transformation is one kind of effective way to construct the solution of the integrable sys-
tems, both continuous and discrete. Since Chau et al.[11] have introduced two types of gauge
transformation operators for the KP hierarchy, the similar operator for the constrained KP
hierarchy[13, 14, 15, 16], q-KP hierarchy [17, 18, 19] and discrete KP hierarchy [1] have also
been given. Oevel [1] has given explicitly three types of gauge transforation operators of the
dKP hierarchy, which are called DT, adjoint DT and binary DT. However, DT and adjoint DT
are two elementary gauge transformation operators, because a binary DT consists of a DT and
anadjointDT.HereDTandadjointDTofthedKPhierarchyareregardedasdiscreteanalogues
of the T and T [11] of the KP hierarchy, so we shall denote it by T and T respectively in
D I d i
following sections. Furthermore, in theorem 3 of [1] Oevel has also considered n-fold iteration
of T and binary DT for the dKP hierarchy [1].1 On the other hand, the determinant repre-
d
sentation [12] of the gauge transformation operators provides a simple method to construct the
transformed τ function [16, 19, 20, 21] for several special cases of the KP hierarchy and q-KP
hierarchy. In this paper we shall extend these results to discrete KP hierarchy. We combine the
two elementary types of gauge transformation operators (T and T ) for the dKP hierarchy and
d i
let them act n times and k times respectively. So we get the combined gauge operator T and
n+k
(n+k)
its the determinant representation, then use it to construct new tau function τ of the dKP
△
hierarchy from an initial tau function.
The organization of the paper is as follows. In section 2, we give a brief description of the
discrete KP hierarchy and prove some useful properties of the discrete operators. In section 3,
based on the Oevel’s two types elementary gauge transforation operator[1],T and T , we give
d i
the determinant representation of the gauge transformation operator T . In section 4, we
n+k
(n+k)
shall construct the τ function of the dKP after the successive gauge transformations by
△
using previous results. Section 5 is devoted to the conclusions and discussions.
§2. The discrete KP and operators
To be self-contained, we give a brief introduction to dKP hierarchy based on detailed research
in [2]. Here we denote by Γ and △ respectively, the shift and the difference operators acting on
the associative ring F of functions. Where
F = {f(n) = f(n,t ,t ,··· ,t ,···);n ∈ Z,t ∈ R},
1 2 j i
and
Γf(n) = f(n+1),
1The expression of dKP and n-fold iteration of gauge transformation in [1] is different from here, but the
essence is the same.
2
△f(n) = f(n+1)−f(n) = (Γ−I)f(n).
Where I is the identity operator. Define the following operation, for anyj ∈ Z
∞
j j j(j −1)···(j −i+1)
△j ◦f = (△if)(n+j −i)△j−i, = . (2.1)
(cid:18)i(cid:19) (cid:18)i(cid:19) i!
Xi=0
So we obtain an associative ring F(△) of formal pseudo difference operators, with the operation
“+” and “◦”
d
F(△) = R = fj(n)△j,fj(n) ∈ R,n ∈ Z.
 j=X−∞ 
We also denote by R = d f (n)△j, the positive difference part of R and by R =
+ j=0 j −
−1 f (n)△j, the VolterrPa part of R. Also define the adjoint operator to the △ operator by
j=−∞ j
△P∗,
△∗f(n) = (Γ−1−I)f(n) = f(n−1)−f(n).
Where Γ−1f(n) = f(n−1), and the corresponding “◦” operation is
∞
j
△∗◦f = (△∗if)(n+i−j)△∗j−i.
(cid:18)i(cid:19)
Xi=0
Then we obtain the adjoint ring F(△∗) to the F(△), and the formal adjoint to R ∈ F(△) is
R∗ ∈ F(△∗), defined by R∗ = d △∗j◦f (n). The ∗ operation satisfies (F ◦G)∗ = G∗◦F∗
j=−∞ j
for two operators and f(n)∗ =Pf(n) for a function.
The discrete KP-hierarchy [2] is a family of evolution equation in infinitely many variables
t = (t ,t ,···)
1 2
∂L
= [(Li) ,L]. (2.2)
+
∂t
i
Where L is a general first-order pseudo difference operator
∞
L = △+ f (n)△−j. (2.3)
j
Xj=0
Similar to KP, L also can be generate by dressing operator W
∞
W(n;t) = 1+ w (n;t)△−j,
j
Xj=1
and
L = W ◦△◦W−1. (2.4)
There are wave function w(n;t,z) and adjoint wave function w∗(n;t,z),
∞
w(n;t,z) = W(n;t)(1+z)nexp( t zi)
i
Xi=1
3
∞
w (n;t) w (n;t)
= (1+ 1 + 2 +···)(1+z)nexp( t zi) (2.5)
i
z z2
Xi=1
and
∞
w∗(n;t,z) = (W−1(n−1;t))∗(1+z)−nexp( −t zi)
i
Xi=1
w∗(n;t) w∗(n;t) ∞
= (1+ 1 + 2 +···)(1+z)−nexp( −t zi). (2.6)
i
z z2
Xi=1
Also a tau function τ = τ(n;t) for dKP exists [2], which satisfies that
△
∞
τ(n;t−[z])
w(n;t,z) = (1+z)nexp( t zi) (2.7)
i
τ(n;t)
Xi=1
and
∞
τ(n;t+[z])
w∗(n;t,z) = (1+z)−nexp( −t zi), (2.8)
i
τ(n;t)
Xi=1
where [z] = (1, 1 , 1 ,···).
z 2z2 3z3
By comparing (2.5) with (2.7) one finds that w are expressed by τ(n;t),and further obtains
i
dressing operator W(n;t) as following form
1 1
W(n;t) = 1−( ∂ τ(n;t))△−1+( (∂2 −∂ )τ(n;t))△−2+···. (2.9)
τ(n;t) t1 2τ(n;t) t1 t2
And taking it back into (2.4), then comparing (2.4) and (2.3), all of dynamical variables {f (n)}
i
can be expanded by τ(n;t). The first few of {f } can be written out explicitly below,
i
f (n) = w (n)−w (n+1) = △∂ lnτ(n;t), (2.10)
0 1 1 t1
f (n) = w (n+1)w (n)−w (n+1)−w2(n)+w (n)−(△w )(n)
−1 1 1 2 1 2 1
= (△w (n))(w (n)−1)−△w (n)
1 1 2
1
= △∂ lnτ(n;t) ∂ lnτ(n;t)+1 −△ (∂2 −∂ )τ(n;t) (2.11)
t1 t1 2τ(n;t) t1 t2
(cid:0) (cid:1)(cid:0) (cid:1) (cid:0) (cid:1)
.
.
.
On the other hand, from the Sato equation of the dKP hierarchy, we can get an important
formula on residue of Ln,i.e.
∂2 lnτ(n;t) = resLi, i ≥ 1, (2.12)
t1ti
which leads to another useful forms of f through τ(n;t).For examples,
i
f (n) = ∂2 lnτ(n;t), (2.13)
−1 t1
4
∂2 lnτ(n;t) = △f (n)+f (n+1)+f (n)f (n)+f (n)f (n−1)+f (n)(,2.14)
t1,t2 −1 −2 0 −1 −1 0 −2
.
.
.
Note that,in general, (2.12) implies recursion relation of f (n) if i ≥ 2, one can not get explicit
−i
form of f form it except f . This is a crucial difference of dKP hierarchy from KP hierarchy.
−i −1
Now we prove some useful properties for the operators which are used later.
Lemma 2.1 For f ∈ F and △, Γ as above, the following identities hold.
(1) △◦Γ = Γ◦△ (2.15)
(2) △∗ = −△◦Γ−1 (2.16)
(3) (△−1)∗ = (△∗)−1 = −Γ◦△−1 (2.17)
(4) f ◦△−1 = △−i−1◦△i(Γf) (2.18)
Xi≥0
(5) △−1◦f ◦△−1 = (△−1f)◦△−1−△−1◦Γ(△−1f) (2.19)
Proof: For ∀g(n) ∈ F,
(1)
(△◦Γ)g(n) = g(n+2)−g(n+1) = Γ(g(n+1)−g(n)) = (Γ◦△)g(n)
(2)
△∗g(n) = −(g(n)−g(n−1)) = (−△◦Γ−1)g(n)
(3)
I = (△∗)−1◦△∗ = (−Γ◦△−1)◦(−△◦Γ−1)
(4)
∞
−i−1
△−i−1◦△i(Γf) = (△i+jf)(n−i−j)◦△−i−j−1(by (2.1))
(cid:18) j (cid:19)
Xi≥0 Xi≥0Xi=0
∞ l
j −l−1
= (△lf)(n−l)◦△−l−1(let l = i+j)
(cid:18) j (cid:19)
Xl=0Xj=0
= f ◦△−1.
l l
j −l−1 l
(Since when l ≥ 1 = (−1)j = 0)
(cid:18) j (cid:19) (cid:18)j(cid:19)
X0 X0
(5)We define2
Γ−1 ∞
△−1 = (Γ−I)−1 = = (Γ−1)i. (2.20)
I −Γ−1
Xi=1
2This definition is coincide with (2.1)
5
And thus
∞ ∞
( f(n−i))( g(n−j))
Xi=1 Xj=1
∞ ∞ ∞ ∞
= Γ−i(f(n) g(n−m))+ Γ−j(g(n) f(n−m+1)). (2.21)
Xi=1 mX=1 Xj=1 mX=1
By using (2.20), equation (2.21) implies
((△−1f)◦△−1)g(n) = (△−1◦f ◦△−1)g(n)+(△−1◦Γ(△−1f))g(n),
which finishes the proof because g(n) is an arbitrary function. (cid:3)
§3. The determinant representation
In this section, we shall construct the determinant representation of the gauge transformations.
To this end, we start with two elementary types of gauge transformation operators of the dKP
hierarchy[1]:
△φ
T (φ) = Γ(φ)◦△◦φ−1 = △− , (3.1)
d
φ
△(Γ−1(ψ))
T (ψ) = Γ−1(ψ−1)◦△−1◦ψ = (△+ )−1. (3.2)
i
ψ
Where φ and ψ are called eigenfunctions and adjoint eigenfunctions of L for gauge transforma-
tion, which satisfy the following dynamic equation of dKP system
∂ φ = (Ln) φ,
tn +
∂ ψ = −(Ln)∗ψ.
tn +
Here the definition of adjoint eigenfunction are different from (2.8).
Set {φ ,φ ,··· ,φ } and {ψ ,ψ ,··· ,ψ } be two sets of n different eigenfunctions and n dif-
1 2 n 1 2 n
ferent adjoint eigenfunctions of the dKP hierarchy associated with L. Then under the transfor-
mation (3.1), the corresponding transformations of {L,φ,ψ } are
L → L(1) = T (φ )◦L◦T (φ )−1,
d 1 d 1
φ → φ(1) = T (φ )φ = Γφ−(Γφ )φ−1φ, (3.3)
d 1 1 1
Γ(△−1(φ ψ))
ψ → ψ(1) = (T (φ )∗)−1ψ = 1 . (3.4)
d 1
Γφ
1
Similarly under the transformation (3.2), the corresponding transformations of {L,φ,ψ } are
L → L(1) = T (ψ )◦L◦T (ψ )−1,
i 1 i 1
6
△−1(φψ )
φ → φ(1) = T (ψ )φ = 1 , (3.5)
i 1 Γ−1(ψ )
1
ψ → ψ(1) = (T (ψ )∗)−1ψ = −ψψ−1Γ−1(ψ )+Γ−1(ψ). (3.6)
i 1 1 1
Where φ(1),ψ(1) are corresponding eigenfunction and adjoint eigenfunction to L(1). Note T and
d
T will annihilate their generation function,i.e.,
i
T (φ )φ = 0, (T−1(ψ ))∗ψ = 0, (3.7)
d 1 1 i 1 1
which is a crucial fact to find the representation of the T below.
n+k
Now we consider the successive gauge transformation including the two type transformations.
In [1], only n-fold T or n-fold binary DT was considered. However we shall study a general
d
gauge transformation operator T which consists of n-fold T and k-fold T in the following.
n+k d i
We define the operator as following,
T = T (ψ(n+k−1))◦···◦T (ψ(n))◦T (φ(n−1))◦···◦T (φ(1))◦T (φ ), (3.8)
n+k i k i 1 d n d 2 d 1
in which
(i−1) (i−2) (i−3) (1)
φ = T (φ )◦T (φ )···T (φ )◦T (φ )·ψ ,
i d i−1 d i−2 d 2 d 1 i
ψ(n) = (T (φ(n−1))−1)∗···(T (φ(1))−1)∗◦(T (φ )−1)∗·ψ ,
j d n d 2 d 1 j
ψ(n+l) = (T (ψ(n+l−1))−1)∗···(T (ψ(n+1))−1)∗◦(T (ψ(n))−1)∗·ψ(n).
j i l i 2 i 1 j
That means under transformation (3.7), we have
L −T−d−(φ−1→) L(1) −T−d−(φ−(2−1→)) L(2) → ··· → L(n−1) −T−d−(φ−n(−n−−1−→)) L(n)
−T−i−(ψ−1(−n→)) L(n+1) −T−i(−ψ−2(n−+−1→)) L(n+2) → ··· → L(n+k−1) −T−i(−ψ−k(n−+−k−−−1→)) L(n+k). (3.9)
and {φ(n),ψ(n)} are corresponding eigenfunctions and adjoint eigenfunctions of L(n).
j j
NowweshowthattheoperatorT hasadeterminantrepresentation, whichgivestheexplicit
n+k
form of T and is useful for direct computation. First we introduce a discrete Wronskian
n+k
determinant. Define
φ φ ··· φ
1 2 n
(cid:12) △φ △φ ··· △φ (cid:12)
1 2 n
Wn△(φ1,··· ,φn) = (cid:12)(cid:12) .. .. .. (cid:12)(cid:12), (3.10)
(cid:12) . . ··· . (cid:12)
(cid:12) (cid:12)
(cid:12)△n−k−1φ △n−k−1φ ··· △n−k−1φ (cid:12)
1 2 n
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
and more general
7
△−1ψ φ △−1ψ φ ··· △−1ψ φ
k 1 k 2 k n
. . .
(cid:12) . . . (cid:12)
(cid:12) . . ··· . (cid:12)
(cid:12) (cid:12)
(cid:12)△−1ψ1φ1 △−1ψ1φ2 ··· △−1ψ1φn(cid:12)
IWn△+k(ψk,··· ,ψ1;φ1,··· ,φn) = (cid:12)(cid:12)(cid:12) φ1 φ2 ··· φn (cid:12)(cid:12)(cid:12), (3.11)
(cid:12) △φ △φ ··· △φ (cid:12)
(cid:12) 1 2 n (cid:12)
(cid:12) .. .. .. (cid:12)
(cid:12) . . ··· . (cid:12)
(cid:12) (cid:12)
(cid:12)△n−k−1φ △n−k−1φ ··· △n−k−1φ (cid:12)
(cid:12) 1 2 n(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
which shall be used in the following theorem.
Theorem 3.1 If n > k, T and T−1 has the following determinant representation:
n+k n+k
△−1ψ φ ··· △−1ψ φ △−1◦ψ
k 1 k n k
. . .
(cid:12) . . . (cid:12)
(cid:12) . ··· . . (cid:12)
(cid:12) (cid:12)
1 (cid:12)(cid:12)△−1ψ1φ1 ··· △−1ψ1φn △−1◦ψ1(cid:12)(cid:12)
Tn+k = IW△ (ψ ,··· ,ψ ;φ ,··· ,φ ) ·(cid:12)(cid:12) φ1 ··· φn 1 (cid:12)(cid:12), (3.12)
n+k k 1 1 n (cid:12)(cid:12) △φ1 ··· △φn △ (cid:12)(cid:12)
(cid:12) .. .. .. (cid:12)
(cid:12) . ··· . . (cid:12)
(cid:12) (cid:12)
(cid:12) △n−kφ ··· △n−kφ △n−k (cid:12)
(cid:12) 1 n (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
φ ◦△−1 Γ(△−1ψ φ ) ··· Γ(△−1ψ φ ) Γ(φ ) ··· Γ(△n−k−2φ )
1 k 1 1 1 1 1
(cid:12)φ ◦△−1 Γ(△−1ψ φ ) ··· Γ(△−1ψ φ ) Γ(φ ) ··· Γ(△n−k−2φ )(cid:12)
2 k 2 1 2 2 2
Tn−+1k = (cid:12)(cid:12) .. .. .. .. .. (cid:12)(cid:12)
(cid:12) . . ··· . . ··· . (cid:12)
(cid:12) (cid:12)
(cid:12)φ ◦△−1 Γ(△−1ψ φ ) ··· Γ(△−1ψ φ ) Γ(φ ) ··· Γ(△n−k−2φ )(cid:12)
n k n 1 n n n
(cid:12) (cid:12)
(cid:12)(cid:12) (−1)n−1 (cid:12)(cid:12)
· . (3.13)
Γ(IW△ (ψ ,··· ,ψ ;φ ,··· ,φ )
n+k k 1 1 n
here the determinant of T expand by the last column and the functions are on the left hand
n+k
with the action ”◦”. As for T−1 the determinant expand by the first column and the functions
n+k
are on the right hand with the action ”◦” too.
Proof: For n > k, we know the highest order item of T is △n−k. So we can assume T
n+k n+k
has the following form
n−k −1
T = a ◦△i+ a ◦△−1◦ψ , a = 1, (3.14)
n+k i i |i| n−k
Xi=0 iX=−k
n
T−1 = φ ◦△−1◦b . (3.15)
n+k j j
Xj=1
8
It is easy to find
T ·φ = 0, i = 1,2,··· ,n, (3.16)
n+k i
(T−1 )∗·ψ = 0, j = 1,2,··· ,k, (3.17)
n+k j
from (3.7), which give the following algebraic system on a and b ,
i j
a △−1(ψ φ )+···+a △−1(ψ φ )+a φ +···+a △n−k−1φ = −△n−kφ (3.18)
−k k i −1 1 i 0 i n−k−1 i i
i = 1,2,··· ,n,
b Γ(△−1(ψ φ ))+b Γ(△−1(ψ φ ))+···+b Γ(△−1ψ φ )) = 0 (3.19)
1 j 1 2 j 2 n j n
j = 1,2,··· ,k.
Then with the help of (3.14), by Gramer rule the solution of (3.18) is given by
−1
a = ·
−i IW△ (ψ ,··· ,ψ ;φ ,··· ,φ )
n+k k 1 1 n
△−1ψ φ ··· △−1ψ φ △n−kφ △−1ψ φ ··· △−1ψ φ φ ··· △n−k−1φ
k 1 i+1 1 1 i−1 1 1 1 1 1
(cid:12) .. .. .. .. .. .. .. (cid:12),
(cid:12) . ··· . . . ··· . . ··· . (cid:12)
(cid:12) (cid:12)
(cid:12)△−1ψkφn ··· △−1ψi+1φn △n−kφn △−1ψi−1φn ··· △−1ψ1φn φn ··· △n−k−1φn(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)(3.20)
(cid:12) (cid:12)
−1
a = ×
i
IW△ (ψ ,··· ,ψ ;φ ,··· ,φ )
n+k k 1 1 n
△−1ψ φ ··· △−1ψ φ φ ··· △i−1φ △n−kφ △i+1φ ··· △n−k−1φ
k 1 1 1 1 1 1 1 1
(cid:12)(cid:12) ... ··· ... ... ··· ... ... ... ··· ... (cid:12)(cid:12).(3.21)
(cid:12) (cid:12)
(cid:12)△−1ψkφn ··· △−1ψ1φn φn ··· △i−1φn △n−kφn △i+1φn ··· △n−k−1φn(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
So (cid:12)we obtain the representation of T as (3.12). (cid:12)
n+k
Next we prove the determinant representation of T−1 . Because T ◦ T−1 = 1, For the
n+k n+k n+k
rationality of the definition of T−1 in (3.15) we have to verify
n+k
(T ◦ T−1 ) = I = 0.
n+k n+k − −
Taking (3.14) and (3.15) into the left hand side ,then
n−k n −1 n
(T ◦T−1 ) = a (△iφ )◦△−1◦b + a ◦△−1◦ψ φ ◦△−1◦b . (3.22)
n+k n+k − i j j i |i| j j
Xi=0 Xj=1 iX=−kXj=1
Furthermore, we can rewrite the second term in (3.22) from (2.19)
n−k n
(T ◦T−1 ) = a (△iφ )◦△−1◦b
n+k n+k − i j j
Xi=0 Xj=0
9
n n
+ a ◦(△−1ψ φ )◦△−1◦b − a ◦△−1◦Γ(△−1ψ φ )b
−k k j j −k k j j
Xj=0 Xj=0
+ ···
n n
+ a ◦(△−1ψ φ )◦△−1◦b − a ◦△−1◦Γ(△−1ψ φ )b .
−1 1 j j −1 1 j j
Xj=0 Xj=0
(3.23)
Rearranging the right hand side, and using with (3.18) and (3.19), then (3.23) becomes
n
(T ◦T−1 ) = T (φ )◦△−1◦b (3.24)
n+k n+k − n+k j j
Xj=1
k
− a ◦△−1◦(T−1 )∗(ψ ) (3.25)
i n+k i
Xi=1
= 0, (3.26)
due to (3.16) and (3.17). On the other hand, by lemma 2.1, T−1 has the following form
n+k
n−k−2 n
T−1 = △−1−i◦Γ(△iφ )b
n+k j j
Xi=0 Xj=1
n ∞ n
+ △−n+k ◦ Γ(△n−k−1φ )b + △−1−i◦Γ(△iφ )b , (3.27)
j j j j
Xj=0 i=Xn−kXj=0
and the T−1 may be expressed as
n+k
T−1 = △−n+k +(lower oder item), (3.28)
n+k
according to the T ◦T−1 = I and the form of T . Then we get the following algebraic
n+k n+k n+k
equations on b ,
i
Γ(△−1(ψ φ ))b +Γ(△−1(ψ φ ))b +···+Γ(△−1ψ φ ))b = 0
k 1 1 k 2 2 k n n
 .
.
 .



Γ(△−1(ψ1φ1))b1+Γ(△−1(ψ1φ2))b2+···+Γ(△−1ψ1φn))bn = 0


Γ(φ1)b1+Γ(φ2)b2+···+Γ(φn)bn = 0 . (3.29)

Γ(△φ1)b1+Γ(△φ2)b2+···+Γ(△φn)bn = 0
.
.
 .


Γ(△n−k−2φ1)b1+Γ(△n−k−2φ2)b2+···+Γ(△n−k−2φn)bn = 0

Γ(△n−k−1φ1)b1+Γ(△n−k−1φ2)b2+···+Γ(△n−k−1φn)bn = 1



Notethefirstk equationsaregivenby(3.19)andthelastn−k equationsaregivenbycomparing
the corresponding terms in (3.27) with (3.28). Solving (3.29) gives
1
b = ·
i
Γ(IW△ (ψ ,··· ,ψ ;φ ,··· ,φ ))
n+k k 1 1 n
10