COMPACTNESS OF GENERALIZED HANKEL OPERATORS WITH UNBOUNDED

COMPACTNESS OF GENERALIZED HANKEL OPERATORS WITH UNBOUNDED

AND ANTI-HOLOMORPHIC SYMBOLS ON GENERALIZED FOCK SPACES

WOLFGANG KNIRSCH and GEORG SCHNEIDER

Letk∈N\{0}andm∈R⁺. Consider the generalized Fock spacesA²(C,|z|^m) :=

{fentire: _C|f(z)|² exp{−|z|^m}dλ(z)<∞}and denote byP the Bergman pro- jection P : L²(C,|z|^m) → A²(C,|z|^m). In a previous paper ([9]) compactness (boundedness) of Hankel operators

H_zk : (Id−P)z^k:A²(C,|z|^m)→A²(C,|z|^m)^⊥

was proved fork < m/2 (k≤m/2).In this paper we investigate how the mentioned result changes,ifwe replace H_zk by the generalized Hankel operator

H_zk: (Id−P1)z^k:A²(C,|z|^m)→A^2,1(C,|z|^m)^⊥,

whereA^2,1 denotes the closure of the linear span of{z^lzⁿ:n, l∈N, l≤1}with orthogonal projectionP1 :L²(C,|z|^m)−→^⊥ A^2,1(C,|z|^m).The main result is the compactness (boundedness) ofH_zk fork < m,(k≤m).

AMS 2000 Subject Classification: Primary 47B35; Secondary 32W05, 32A36.

Key words: Hankel operator, generalized Fock spaces, weighted Bergman space,

∂-Neumann problem.

1. INTRODUCTION

Letk∈N\ {0}and m∈R⁺. Consider the spaces of entire functions A²(C,|z|^m) :=

f :C→Centire :f²_m:=

C|f(z)|²exp{−|z|^m}dλ(z)<∞ with weight functions exp{−|z|^m} and Lebesgue measure dλ in C ∼= R². We call A²(C,|z|^m) generalized Fock spaces and the special case A²(C,|z|²) is the Fock space. Let P = P^(m) : L²(C,|z|^m) −→^⊥ A²(C,|z|^m) be the Bergman projection. Here L²(C,|z|^m) := {f :C → Cmeasurable : f²_m :=

C|f(z)|²exp{−|z|^m}dλ(z)<∞}.Operators of the form Hψ(f) = (Id−P)(ψf), f ∈A²(C,|z|^m)

REV. ROUMAINE MATH. PURES APPL.,52(2007),2, 213–229

are called (Bergman-)Hankel operators with symbolψ:C→Cwhile operators of the form

Tψ(f) =P(ψf), f ∈A²(C,|z|^m),

are called (Bergman-)Toeplitz operators with symbol ψ. Clearly, for ψ ∈ L^∞(C) these operators are bounded with

Tψ ≤ ψ_∞ and Hψ ≤ ψ_∞.

Up to now, investigations of Hankel operators have focused on essentially bouned symbols. Operator-theoretic properties of Hankel operators, like com- pactness have already been extensively investigated – see for example [11].

In this paper we investigate Hankel operators with unbounded and anti- holomorphic symbolψ(z) =z^k.One motivation for investigating Hankel oper- ators with symbolz^kis the following connection with the∂-Neumann problem:

the canonical solution operatorSto∂is a (Bergman-)Hankel operatorHzwith symbolz,ifS is restricted to the Bergman spaceA²(C,|z|^m), i.e.

S = (Id−P)z=Hz :A²(C,|z|^m)→A²(C,|z|^m)^⊥.

Clearly, the Hankel operatorH_zk with symbol z^k corresponds to a solu- tion operator S to ∂^k,where ∂^k = ^∂k

∂z^k. For more details on this we refer the reader to [3], [4], [1], [2], [6], [7], [8], [10] and [9].

In the last years the following results about compactness and other operator-theoretic properties ofHz andH_zk have been investigated.

Theorem 1.1. The Hankel operator

Hz:A²(C,|z|²)→L²(C,|z|²) is not compact. For m≥3 the Hankel operator

Hz:A²(C,|z|^m)→L²(C,|z|^m) is compact, but fails to be Hilbert-Schmidt.

Theorem 1.1 is proved in [4]. In [10] is derived an analogue of Theo- rem 1.1 in several dimensions: for n ≥ 2 it is shown that for all m ≥ 0 the Hankel operator Hz_i : A²₁(Cⁿ,|z|^m) → L²(Cⁿ,|z|^m) is not compact for all i∈ {1, . . . , n}. In [9] we can find the following result about Hankel operators with anti-holomorphic symbolsz^k, k ∈N.

Theorem 1.2. The following results hold.

(1) Fork < m/2 the Hankel operator

H_z^k :A²(C,|z|^m)→L²(C,|z|^m) is compact, but fails to be Hilbert-Schmidt.

(2) Fork≤m/2 the Hankel operator H_z^k :L²(C,|z|^m)→L²(C,|z|^m) is bounded, while in case k > m/2 unbounded.

In this paper we consider a slightly different type of Hankel operator, which we call the generalized Hankel operator, namely,

H_z^k = (Id−P1)z^k:A²(C,|z|^m)→A^2,1(C,|z|^m)^⊥, where

A^2,1 :=z^lzⁿ:n, l∈N, l≤1

denotes the closure of the linear span of {z^lzⁿ : n, l ∈ N, l ≤ 1} and P₁ : L² −→^⊥ A^2,1 is the orthogonal projection. In a former paper ([8]) generalized Hankel operators have already been investigated on the Bergman space of the unit disc and on the Fock space.

Theorem 1.3. On the Bergman space of the unit disc D in C the gen- eralized Hankel operator

H_z^k = (Id−P1)z^k:A²(D)→A^2,1(D)^⊥ is compact for all k≥1.

A completely different situation appears on the Fock space. We namely have

Theorem 1.4. On the Fock space the generalized Hankel operator H_z² = (Id−P1)z² :A²(C,|z|²)→A^2,1(C,|z|²)^⊥

is bounded, but fails to be compact. Fork≥3 the generalized Hankel operator H_z^k = (Id−P1)z^k:A²(C,|z|²)→A^2,1(C,|z|²)^⊥

is unbounded.

2. MAIN RESULTS

In this paper we give a generalization of Theorem 1.4 to generalized Fock spacesA²(C,|z|^m). We show that generalized Hankel operatorsH_zkare always compact fork < mand bounded fork≤m.So, this is an improvement on the classical (Bergman-)Hankel operator H_z^k which only is compact for k < ^m₂ and bounded fork≤ ^m₂.

Consequently, in the special case of the Fock space the generalized Han- kel operatorHz is compact. So, this is an improvement on Hz, which is the canonical solution operator S to ∂, and is not compact (see [4], [10] or Theo- rem 1.1).

The following two theorems are our main results.

Theorem 2.1. On generalized Fock spaces the generalized Hankel ope- rator

H_z^k = (Id−P₁)z^k:A²(C,|z|^m)→A^2,1(C,|z|^m)^⊥ is compact fork < m and bounded for k≤m.

The casek≥m is considered in the following theorem.

Theorem 2.2. The following results hold.

(1) Let D := {k, m∈ {1,2, . . .} :k = m}. There exists at most a finite number of (k, m)∈D such that the generalized Hankel operator H_z^k = (Id− P₁)z^k:A²(C,|z|^m)→A^2,1(C,|z|^m)^⊥ is compact.

(2) For k > m we have almost everywhere unboundedness. This means that for every m there exists at most a finite number of k’s such that H_z^k = (Id−P1)z^k:A²(C,|z|^m)→A^2,1(C,|z|^m)^⊥ is not unbounded.

The scheme below visualizes Theorem 2.2.

← Compactness ) m↑ ( Almost unboundedness→ Boundedness

k

In the next section we derive some preparatory results. The main theo- rems will be proved at the end of the paper.

3. PRELIMINARIES AND

A COMPLETE ORTHONORMAL SYSTEM

It is well known that{zⁿ/cn,m:n∈N}is a complete orthonormal system for the generalized Fock space

A²(C,|z|^m) :=

f entire :f²_m :=

C|f(z)|² exp{−|z|^m}dλ(z)<∞

,

wherec²_n,m are the so-called moments, which are given by c²_n,m=zⁿ²_m =

C|zⁿ|² exp{−|z|^m}dλ(z) =

= 2π _∞

0 r²ⁿexp{−r^m}rdr= 2π mΓ

2n+ 2

m .

Frequently, we will abbreviatec²_n,mto c²_n. Furthermore, Γ is the Gamma function, which is defined for allx >0 by

Γ (x) = _∞

0 t^x−1e^−tdt.

In the special casem = 2 of the Fock space, the moments are given by c²_n=πΓ (n+ 1) =πn!.For the generalized Fock spaces we will use Stirling’s formula

Γ (l+ 1)∼l^le^−l√ 2πl

to investigate the asymptotic behavior of the moments c²_n=c²_n,m= 2π

mΓ

2n+ 2

m ∼

2n+ 2 m −1

2n+2m −1

e^{−2n+2m −1}

2n+ 2 m −1. The results below can be found in [8], but we will repeat them for the reader’s convenience.

Lemma 3.1. The set {ψ_0,k, ψ_1,k : k ∈ N} is a complete orthonormal system forA^2,1(C,|z|^m). Here

ψ_0,k = z^k ck

and ψ_1,k = zz^k+1−^c2k+1_c2 k z^k

c²_k+2−^c4k+1_c2 k

.

Next, we express the action of the orthogonal projection P1 on the com- plete orthonormal system {ψ_0,k, ψ_1,k : k ∈ N} in terms of moments. We thus have

Proposition3.2. LetP1 :L²(C,|z|^m)−→^⊥ A^2,1(C,|z|^m) be the orthogo- nal projection. ThenP1(z^kz^n+k) =an,0zⁿ+an,1zzⁿ⁺¹,where the coefficients an,0, an,1 are given by

a_n,1= c²_nc²_n+k+1−c²_n+1c²_n+k

c²_nc²_n+2−c⁴_n+1 , a_n,0= c²_n+k

c²_n −c²_n+1 c²_n a_n,1. Proof. This can be shown as in [8].

After these preparations we can calculate a convenient form of the norm ofH_z^k(z^n+k/cn+k).

Proposition 3.3. For all n∈N we have H_zk

z^n+k

c_n+k ²=an−bnhn,

with

an= c²_n+2k

c²_n+k −c²_n+k

c²_n , bn= c²_n+k+1

c²_n+1 −c²_n+k

c²_n and hn =

c²_n+k+1 c²_n+k −^c2n+1_c2n

c²_n+2

c²_n+1 −^c2n+1_c2n

.

Proof. Clearly,

an,1 = c²_nc²_n+k+1−c²_n+1c²_n+k

c²_nc²_n+2−c⁴_n+1 = c²_n+k c²_n+1

c²_n+k+1 c²_n+k −^c2n+1_c2

n

c²_n+2 c²_n+1 −^c2n+1_c2

n

=: c²_n+k c²_n+1hn

and

an,0 = c²_n+k

c²_n − c²_n+1

c²_n an,1 = c²_n+k

c²_n −c²_n+k

c²_n hn= c²_n+k

c²_n (1−hn). Calculating the norm ofH_z^k

z^n+k yields

H_z^k(z^n+k)²=z^kz^n+k−a_n,0zⁿ−a_n,1zzⁿ⁺¹²=z^kz^n+k2+a²_n,0zⁿ²+ +a²_n,1zzⁿ⁺¹²−2an,0

z^kz^n+k|zⁿ

−2an,1

z^kz^n+k|zzⁿ⁺¹

+2an,0an,1

zⁿ|zzⁿ⁺¹

=c²_n+2k−2a_n,1c²_n+k+1−2a_n,0c²_n+k+a²_n,1c²_n+2+ 2a_n,0a_n,1c²_n+1+a²_n,0c²_n. Consequently,

H_z^k

z^n+k

c_n+k ² =h²_n

c²_n+kc²_n+2

c⁴_n+1 − c²_n+k c²_n

−2hn

c²_n+k+1

c²_n+1 − c²_n+k c²_n

+ +c²_n+2k

c²_n+k −c²_n+k c²_n =h²_n

c²_n+kc²_n+2

c⁴_n+1 −c²_n+k c²_n

−2hnbn+an. With

c²_n+kc²_n+2

c⁴_n+1 − c²_n+k

c²_n = c²_n+k c²_n+1

c²_n+2

c²_n+1 −c²_n+1 c²_n

=

= c²_n+k c²_n+1

c²_n+2

c²_n+1 −^c2n+1_c2n

c²_n+k+1 c²_n+k −^c2n+1_c2n

c²_n+k+1

c²_n+k −c²_n+1 c²_n

=h⁻¹_n

c²_n+k+1

c²_n+1 −c²_n+k c²_n

=h⁻¹_n bn

we get

H_z^k

z^n+k

c_n+k ²=an−hnbn. This completes the proof.

4. PROOF OF THE MAIN RESULT

It follows from Proposition 3.3 that we have to investigate the asymptotic behaviour of expressions of the form

c²_n+l

c²_n+j −c²_n+r c²_n+s,

where, fortunately,l−j =^! r−s, that we will essentially use later on. First, we investigate the asymptotic behavior of quotients of moments.

Proposition 4.1. Let l, j ∈ N with l > j. The asymptotic behavior of quotients of moments is given by

c²_n+l c²_n+j ∼

2

m(n+l+ 1−m/2)

_m²_(l−j)

e^−m2(l−j)γn(l, j), where

γn=γn(l, j) =

1 +^l+1−m/2_n 1 + ^j+1−m/2_n

²ⁿ

m

1 +^l+1−m/2_n 1 +^j+1−m/2_n

^4(j+1)−m

2m

.

Proof. We have c²_n+l=c²_n+l,m∼

2(n+l) + 2

m −1

2(n+l)+2 m −1

e^{−2(n+ml)+2−1}

2(n+l) + 2

m −1

and so c²_n+l c²_n+j ∼

1+^l+1−m/2_n 1+^j+1−m/2_n

_m²_(n+1)−1

1+^l+1−m/2_n 1+^j+1−m/2_n

^2j_m 2

m(n+l+1−m/2)

_m²_(l−j)

×e^−m2(l−j)

1 +^l+1−m/2_n 1 + ^j+1−m/2_n . Now, the proposition follows from

γn(l, j) =

1+^l+1−m/2_n 1+^j+1−m/2_n

^2(n+1)−m

m

1+^l+1−m/2_n 1+^j+1−m/2_n

^2j

m

1+^l+1−m/2_n 1 +^j+1−m/2_n =

=

1 +^l+1−m/2_n 1 + ^j+1−m/2_n

²ⁿ_m

1 + ^l+1−m/2_n 1 +^j+1−m/2_n

^4(j+1)−m_2m

.

Clearly, γn(l, j)→const(l, j) forn→ ∞,but for our purpose this is not enough. We also have to investigate how fast γn tends to const(l, j). To de- scribe the limiting behaviour we use the so-called Landau symbols Ok :=O ₁

n^k

.

Proposition 4.2. Let l, j ∈N withl > j. Then γn =γn(l, j) =

= exp

(l−j)

c₁+d₁ n

1+c₂(d²−c²)1 n+

K+d₁d²−c² 2

1 n²+O

1 n³

, where c₁ = _m², c₂ = _m¹, d₁ = d₁(j) = ^4(j+1)−m_2m , c = c(l) := l+ 1−m/2, d=d(j) :=j+ 1−m/2 andK =K(l, j) = ^2(c_3m^3−d3) +^(d2_2m^−c₂²⁾².

Proof. Note that γn= exp{lnγn}=

= exp

2n m ln

1+^l+1−_n^m/2 1+^j+1−_n^m/2

(I)

·exp

4(j+1)−m 2m ln

1+^l+1−m/2_n 1+^j+1−m/2_n

=:(II)

.

Setc=c(l) :=l+ 1−m/2,d=d(j) :=j+ 1−m/2. Using the equation ln

1 + 1

n = 1 n− 1

2n² + 1 3n³ +O

1 n⁴ one can write

(I) = exp 2n

m

ln

1 + c n

−ln

1 + d n

= exp

2(c−d)

m + d²−c²

m n + 2c³−d³ 3m n² +O

1 n³

= exp

2(c−d) m

×

×

1+

d²−c² m

1

n+2c³−d³ 3m n²

+ 1 2

d²−c² m

1

n+ 2c³−d³ 3m n²

₂ +O

1 n³

= exp

2(l−j) m

1+d²−c² m

1 n+

2(c³−d³)

3m +(d²−c²)² 2m²

1 n²+O

1 n³

=: exp

2(l−j)

m 1 +d²−c² m

1

n+K 1 n² +O

1 n³

.

Setd₁ :=d₁(j) = ^4(j+1)−m_2m .Then it is easy to see that (II) = exp

d₁

ln

1 + c

n −ln

1 +d

n

= exp

d1

c n− c²

2n² +O 1

n³ − d

n− d² 2n² +O

1 n³

= exp

d1

c−d

n +d²−c² 2n² +O

1 n³

= exp

d₁(c−d) n

exp

d₁d²−c² 2n² +O

1 n³

= exp

d1(l−j)

n 1 +d1d²−c² 2n² +O

1 n³

.

Consequently, we deduce that (I) (II) = exp

2(l−j)

m +d1(l−j) n

×

×

1 +d²−c² m

1

n+K 1 n² +O

1 n³

1 +d₁d²−c² 2n² +O

1 n³

= exp

(l−j)

c₁+d₁

n 1+d²−c² m

1 n+

K+d₁d²−c² 2

1 n²+O

1 n³

.

This completes the proof.

For further calculations of the moments we use the formula below, which we call the exact Stirling’s formula. This formula consists of an asymptotic termA, which corresponds to Stirling’s formula, and the so-called error term E,which we have to simplify.

EXACT STIRLING’S FORMULA ([5]). We have

c²_n+l=

2(n+l) + 2

m −1

2(n+l)+2 m −1

e^{−2(n+l)+2m −1}

2(n+l) + 2

m −1×

×



1 + 1 12

1

2(n+l)+2

m −1+ 1

288

$ 1

2(n+l)+2

m −1

%₂ +O 1

n³





=: [Asymptotic term]_l×[Error term]_l=:Al×El=A×E.

Calculation of the error termE. Letc=c(l) := 1 +η:=l+ 1−m/2.

Then forn large enough we have 1

2(n+l)+2

m −1 = m

2

1

n+l+ 1−m/2 = m 2n

1

1 +^l+1−m/2_n =: m 2n

1 1 + _n^c

= m 2n

)∞ j=0

−c n

_j

= m 2n

1− c

n+ c² n² +O

1 n³ and

$ 1

2(n+l)+2

m −1

%₂ = m² 4n²

1−2c

n +3c² n² +O

1 n³ . Consequently,

E= 1 + m 24n

1− c

n+ c² n² +O

1

n³ + m² 1152n²

1−2c

n +3c² n² +O

1 n³

= 1 + m 24

1 n +

m²

1152 −mc 24

1 n² +

mc²

24 −m²c 576

1 n³ +O

1 n⁴

= 1 + m 24

1 n +

m²

1152 −mc 24

1 n² +O

1 n³ .

Now, we shall use the exact Stirling’s formula to determine the limiting behavior ofc²_n+l/c²_n+j. First, a preparatory result.

Proposition 4.3. Forl > j we have c²_n+l

c²_n+j = 2

m(n+l+η)

_m²_(l−j)

×

×

1 +α1+ (l−j)d1

n +α1(l−j)d1+α2+ ((l−j)d1)²

n² +O

1 n³ , where

α₁ =α₁(l, j) = j²−l²+ 2(j−l)η and m

α₂ =α₂(l, j) = m

24(j−l) +K+d₁j²−l²+ 2(j−l)η

2 .

Furthermore, d₁ =d₁(j) and K = K(l, j) are defined in Proposition 4.2 and η=η(m) = 1−m/2.

Proof. From Proposition 4.1 we have (†) c²_n+l

c²_n+j ∼ 2

m(n+l+η) ²

m(l−j)

e^−m2(l−j)γn(l, j)≡ Al

Aj.

Now, we use Proposition 4.2, the exact Stirling’s formula and the fact thatc₁ = _m² to see that

c²_n+l

c²_n+j = Al×El

Aj×Ej (†)=

2

m(n+l+η)

_m²_(l−j)

e^−m2(l−j)γn(l, j) × El

Ej Prop.4.2

=

2

m(n+l+η)

_m²_(l−j)

e^−m2(l−j)exp

(l−j)

c₁+d1

n ×

×

1 +c₂(d²−c²)

n +

K+d1d²−c² 2

1 n²+O

1

n³ ×El

Ej Exact Stirling

=

2

m(n+l+η) ²

m(l−j)

exp

(l−j)d₁ n

×

×

1 + c₂(d²−c²)

n +

K+d₁d²−c² 2

1 n² +O

1 n³ ×

× 1 +₂₄^m1_n+

$ m²

1152 −^mc(l)₂₄ %

1

n² +O ₁

n³

1 + ^m₂₄¹_n+

$ m²

1152 −^mc(j)₂₄ %

1

n² +O ₁

n³

≡

2

m(n+l+η) ²

m(l−j)

exp

(l−j)d₁ n

× P.

It is routine to check that P =α₀+α₁1

n+α₂ 1 n² +O

1

n³ = 1 +

j²−l²+ 2(j−l)η m

1 n+ +

m

24(j−l) +K+d₁j²−l²+ 2(j−l)η 2

1 n² +O

1 n³ . Next, we only have to compute

exp

(l−j)d1

n

× P =

1 +(l−j)d1

n +

(l−j)d1

n

2

+O3

×

×

1 +α₁1

n+α₂ 1 n² +O

1 n³ ,

which yields

exp

(l−j)d1

n

× P= 1 +α1+ (l−j)d1

n +

+α1(l−j)d1+α2+ ((l−j)d1)²

n² +O

1 n³ . This completes the proof.

Remark. In the following we will use the generalized binomial formula:

forα∈Rand 0≤y <1,we have (1 +y)^α =

)∞ k=0

α k y^k =

α

0 +

α 1 y+

α

2 y²+· · · , where

α

k := α(α−1)· · ·(α−k+ 1)

k! and

α

0 := 1.

Now, we can finally determine the limiting behavior of c²_n+l/c²_n+j. The following proposition yields the desired result.

Proposition 4.4. Let l > j and let n be large enough. Then we have c²_n+l

c²_n+j =C(l−j, m)n^2(l−j)m ×

1 +β1(l, j)

n +β2(l, j) n² +O

1 n³ , where C(l−j, m) = ₂

m

^2(l−j)

m . Furthermore,

β1(l, j) =

_2(l−j)

m

1 (l+η) +α1(l, j) + (l−j)d1(j) and

β₂(l, j) =

_2(l−j)

m

2 (l+η)²+α₁(l, j)(l−j)d₁(j) +α₂(l, j) + ((l−j)d₁)²+ +

_2(l−j)

m

1 (l+η) [α₁(l, j) + (l−j)d₁(j)],

whereα₁(l, j),andα₂(l, j)are defined in Proposition 4.3whiled₁(j)is defined in Proposition 4.2with η=η(m) = 1−m/2.

Proof. SetC:=C(l−j, m) :=₂

m

^2(l−j)_m

and letnbe large enough. Then from Proposition 4.3 we get

c²_n+l c²_n+j =

2

m(n+l+η) ²

m(l−j)

×

×

1 +α1+ (l−j)d1

n +α1(l−j)d1+α2+ ((l−j)d1)²

n² +O

1 n³

=C

n

1 +l+η n

2(l−j)

m ×

×

1 +α₁+ (l−j)d₁

n +α₁(l−j)d₁+α₂+ ((l−j)d₁)²

n² +O

1 n³

=C(l−j, m)n^2(l−j)m

1+

_2(l−j)

m

1

l+η n +

_2(l−j)

m

2

l+η n

2

+O 1

n³ ×

×

1 +α₁+ (l−j)d₁

n +α₁(l−j)d₁+α₂+ ((l−j)d₁)²

n² +O

1 n³ . An elementary computation completes the proof.

We will use the following convenient definition for power series coeffi- cients.

Definition 4.1. Let P = *

kckx^k be a formal power series. Then [P]_k denotes the coefficientck of x^k.In the following we consider x= ¹_n.

Now, we shall apply the formula from Proposition 4.4 to compute

an= c²_n+2k

c²_n+k −c²_n+k

c²_n , bn= c²_n+k+1

c²_n+1 −c²_n+k

c²_n , hn= h^[ab]n

h^[bel]n

:=

c²_n+k+1 c²_n+k −^c2n+1_c2

n

c²_n+2

c²_n+1 −^c2n+1_c2 n

.

from Proposition 3.3. We thus have

Corollary 4.5. Let n be large enough and an, bn, h^[ab]n and h^[bel]n as above. Then we have

an=C(k, m)n^2km ×

β₁(2k, k)−β₁(k,0)

n + β₂(2k, k)−β₂(k,0)

n² +O

1 n³

=:C(k, m)n^2km × [P]₁

n +[P]₂ n² +O

1

n³ =:C(k, m)n^2km ×P,

bn=C(k, m)n^2km ×

β1(k+1,1)−β₁(k,0)

n +β2(k+1,1)−β₂(k,0)

n² +O

1 n³

=:C(k, m)n^2km × [T]₁

n +[T]₂ n² +O

1

n³ =:C(k, m)n^2km ×T, h^[ab]_n =C(1, m)n^m2 ×

β₁(k+1, k)−β₁(1,0)

n +β₂(k+1, k)−β₂(1,0)

n² +O

1 n³

=:C(1, m)n^2km × [H]₁

n + [H]₂ n² +O

1

n³ =:C(1, m)n^2km ×H, and

h^[bel]_n =C(1, m)n^m2 ×

β₁(2,1)−β₁(1,0)

n +β₂(2,1)−β₂(1,0)

n² +O

1 n³

=:C(1, m)n^2km × [S]₁

n +[S]₂ n² +O

1

n³ =:C(1, m)n^2km ×S.

Finally, we can now determine the limiting behavior ofH_z^k(z^n+k/cn+k)². The following result holds.

Corollary 4.6. Let C(k, m) :=₂

m

^2k_{m 2k}2

m . Then H_z^k

z^n+k

c_n+k ² =an−bnhn=C(k, m)n^2km ×γ,

whereγ is a power series in1/nwith[γ]₀ = [γ]₁ = 0and[γ]₂ = [P]₂−^[T]_[S]^2[H]₂ ². Proof. We recall that

β1(l, j) =

_2(l−j)

m

1 (l+η) +α1(l, j) + (l−j)d1(j) and so forl, j, r, s with l−j=τ =r−swe get

β₁(l, j)−β₁(r, s) = 2τ

m(l−r) + (α₁(l, j)−α₁(r, s)) +τ(d₁(j)−d₁(s)). With

α1(l, j) = j²−l²+ 2(j−l)η

m = j²−l²+ 2τ η m and

d1(j) = 4(j+ 1)−m 2m we get

β1(l, j)−β1(r, s) = 2τ

m[(l−r) + (j−s)] +(j²−l²)−(s²−r²)

m ,