POLYANALYTIC REPRODUCING KERNELS IN C n

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POLYANALYTIC REPRODUCING KERNELS IN C n

El Hassan Youssfi

To cite this version:

El Hassan Youssfi. POLYANALYTIC REPRODUCING KERNELS IN C n. 2021. �hal-03131190v2�

E. H. YOUSSFI

ABSTRACT. In this note we establish integral formulas for polyanalytic functions in several vari- ables. More precisely, given a positive integerq, we provide explicit expressions for the reproduc- ing kernels of the weighted Bergman spaces ofq-analytic functions on the unit ball inCⁿand that ofq-analytic Fock space inCⁿ.

1. INTRODUCTION AND STATEMENT OF THE MAIN RESULT

Polyanalytic function theory in higher dimensions is still not understood. This is essentially due to the lack of explicit formula for related integral kernels. On the other hand, despite this extensive study, there were only two examples of explicit polyanalytic reproducing kernels; the first one was established by Koshelev [10] in the disc, and the second one was given in [4] and [1], and proved in [7], [8]. It was only in the very recent work by Hachadi and the author [6] that a wide source of examples was constructed. There it is proved that to a given a positive integer qand sequence of orthogonal polynomials associated to a non-negative measure in the half real line[0,+∞[whose support contains at leastq strictly positive points, is naturally associated a Hilbert space ofq-analytic functions and established a general formula for its reproducing kernel.

It is also shown that the formula generalizes to the higher dimensional setting of(q₁, . . . , q_n)- analytic functions , whereq₁,· · · , q_nare positive integers. These functions were considered by Avanissian and Traor [2] and [3]. We recall that a functionf is said to be polyanalytic of order (q1, . . . , qn)or just ((q1, . . . , qn)-analytic) in the domainΩ⊂Cⁿif in this domain it satisfies the generalized Cauchy-Riemann equation

∂^q1+···+qnf

∂z¯₁^q1· · ·∂z¯n^qn

= 0. (1.1)

They can be uniquely expressed as f(z) =

(q1−1,...,qn−1)

X

j=(0,···,0)

z^jφ_j(z) (1.2)

where theφ_j(z)are holomorphic in Ωwhere forj = (j₁, . . . , j_n), k = (k₁, . . . , k_n) ∈ Nⁿ0 and z = (z₁, . . . , z_n) ∈ Cⁿ, the inequality j ≤ k means that j_l ≤ k_l for all l = 1,· · · , n and z^j :=z₁^j1· · ·z_n^jn.

2000Mathematics Subject Classification. Primary 47B35, 32A36, 32A37.

Key words and phrases. Reproducing kernel, Polyanalytic, Bergman space, Fock space.

1

Another notion of polyanalyticity was considered recently by Daghighi [5]. Namley, for a positive integerq, a functionfdefined on an open setΩinCⁿis called analytic of absolute order q(or justq-analytic) if it is of classC^q that satisfies the generalized Cauchy-Riemann equation

∂^αf

∂z^α(z) = 0,

onΩ for all multi-indicesα = (α₁, . . . , α_n) ∈ Nⁿ0 with length |α| := α₁ +· · ·+α_n = q. He proved that these functions f are precisely the slice q-analytic functions in the sense that for all (a, ξ) ∈ Ω× Cⁿ, the one variable functions λ 7→ f(a +λξ) are q-analytic in some open neighborhood of0inC.It is clear that an absoluteq-analytic functionf onBcan be expressed as a sum

f(z) = X

|α|<q

¯

z^αf_α(z), (1.3)

where for all α ∈ Nⁿ with |α| ≤ q − 1 the functions f_α are holomorphic on Ω called the holomorphic components of the polyanalytic functionf.

To the best of our knowledge, no integral representations are available forq-analytic functions inCⁿ.The main goal of this work is to consider this question whenΩis Cⁿ or the unit ball in Cⁿ.

In this note, we letB := {z ∈ Cⁿ : |z| < 1} be the unit ball inCⁿ and denote bydv(z)the Lebesgue measure onCⁿnormalized so thatv(B) = 1.

A first result of this paper establishes an explicit expression of the weighted polyanalytic (in the absolute sense) Bergman kernel of the unit ballB.More precisely, fors >−1,we consider the space A²s,q(B) of all square integrable of q-analytic functions with respect to the measure dν_s(z) := (1− |z|²)^sdv(z).More precisely, we will prove the following

Theorem A. The space A²_s,q(B) is a Hilbert space which coincides with the closure of the q- analytic polynomials inL²(νs)and its reproducing kernel is given by

K_s,q,n(z, w) = ^Γ(q+s+n)_{n!Γ(s+q) (1−hz,wi)}^(1−hw,zi)s+q+n^q−1 P_q−1^(n,s)(1−2|ϕ_z(w)|²),

for allz, w ∈B, whereP_q−1^(n,s)is the classical Jacobi polynomial with parametersn, sand degree q−1,andϕw is the biholomorphic automorphism ofBexchanging0andw.

We point out that whenn = 1,this result was established recently by Hachadi and the author [6]. In this case, the one dimensional weighted polyanalytic Bergman kernel reduces

K_s,q,1(z, w) = (q+s)_(1−hz,wi)^(1−hw,zi)s+q+1^q−1 P_q−1^(1,s)(1−2|ϕ_z(w)|²), (1.4) where

ϕz(w) := z−w 1− hz, wi for all elementsz, wof the unit discDinC.

To state our second result, we consider the Gaussian measureµdefined onCⁿby

dµ(z) :=e^−|z|2dv(z) (1.5)

and denote byF²q(Cⁿ)the weightedq-polyanalytic Fock space onCⁿwhereqis a positive integer.

This is the space of allq-analytic functionsf onCⁿwhich are square integrable with respect to dµ(z).

Theorem B. The space Fq²(Cⁿ) is a Hilbert space which coincides with the closure of the q- analytic polynomials inL²(µ)and its reproducing kernel is given by

K_F,q(z, w) = e^hz,wi

(n−1)!Lⁿ_q−1 |z−w|²

(1.6) for allz, w ∈Cⁿ, whereLⁿ_q−1 is the classical weighted Laguerre polynomial with weightn and degreeq−1.

After submitting this paper, I found a work by Lea-Pacheco, Maximenko and Ramos-Vazquez [12] refering to an earlier version of my work [14]. The authors have found another way to get some of the results presented in this work. Namely, they also compute the reproducing kernel the unit ball.

2. PRELIMINARIES

LetDⁿbe the unit polydisc inCⁿ. This is the open unit ball inCⁿwith respect to the norm kzk∞:= max

1≤j≤n|z_j|, z = (z₁, . . . , z_n)∈Cⁿ.

For a positive integer q, we let A²q(Dⁿ)denote the space of those (q, . . . , q)-analytic functions on Dⁿ which are square integrable with respect to the Lebesgue measure dv(z) on Dⁿ. As a consequence of Theorem D of [6], we have the following

Lemma 2.1. The spaceA²q(Dⁿ)is the closure of the(q, . . . , q)-analytic polynomials in the space L²(Dⁿ)of all square integrable functions onDⁿwith respect to the measuredv(z).Furthermore, its reproducing kernel is given by

K_Dⁿ_,q(z, w) =Qn

j=1K_0,q,1(z_j, w_j)

for allz = (z₁, . . . , z_n), w = (w₁, . . . , w_n)∈ Dⁿ, whereK_0,q,1 is the one dimensional polyana- lytic weighted Bergman kernel of the unit disc given by (1.4) withs= 0.

Lemma 2.2. LetΩbe an open set inCⁿand let%: Ω→]0,+∞[be a continuous weight function.

For each compact subsetCofΩ,and eachα ∈Nⁿ0, there is a positive constantc_α>0such that max

C

∂^αf

∂z¯^α

≤c_αkfk_L²_(Ω,%dv),

for allq-analytic functions inL²(Ω, %dv).In particular, the spaceA²%,q(Ω)of all such functions, equipped theL²(Ω, %dv)inner product is Hilbert space and for allz ∈ Ω,the evaluation func- tionalf 7→f(z)is continuous onA²%,q(Ω).

Proof. It is sufficient to consider compact polydiscsC =a+rDⁿ, fora∈Ωandr >0such that a+ 2rDⁿ ⊂ Ω. Considering this, we letf ∈A²%,q(Ω). Then the functiong(z) :=f(a+ 2rz)is well-defined(q,· · · , q)-analytic function on a neighborhood ofDⁿ. By Lemma 2.1

g(z) = Z

Dⁿ

g(w)K_Dⁿ_,q(z, w)dv(w), from which it follows that

∂^αg(z)

∂z¯^α = Z

Dⁿ

g(w) ∂^α

∂z¯^α (K_Dⁿ_,q(z, w))dv(w), so that by Cauchy-Schwarz inequality we see that

∂^αf

∂z¯^α(a+rz)

≤ kfk_L²_(Ω,%dv) r^|α|

"

Z

Dⁿ

^∂

α

∂¯z^α(K_Dⁿ_,q(z/2, w))

2

%(a+rz) dv(w)

#1/2

. Hence

maxC

∂^αf

∂z¯^α

≤c_αkfk_L²_(Ω,%dv) where

c_α := sup

z∈Dⁿ

"

Z

Dⁿ

^∂

α

∂z¯^α (K_Dⁿ_,q(z/2, w))

2

r^|α|%(a+rz) dv(w)

#1/2

.

This completes the proof of the lemma.

Lemma 2.3. Let Ωbe a domain in Cⁿand Da domain in C^p. Suppose that ψ : Ω → Dis a holomorphic map of the form

ψ(z) = g(z) h(z)

where g : Ω → C^p andh : Ω → Care polynomials of respective degreesd_g andd_h such that d_g ≤ d_h andh(z) 6= 0for allz ∈ Ω.Suppose thatq andl are positive integers such thatl ≥q.

Iff is aq-analytic function onD, then the function¯h^l(f ◦ψ)is(ld_h+1)-analytic onΩ.

Proof. Writingf in the formf(w) =P

|α|<qw¯^αf_α(w),it follows for allz ∈D, we have

¯h^l(z)(f ◦ψ)(z) = X

|α|<q

¯h(z)l−|α|

(g(z))^αf_α(ψ(z)) which isq-analytic since for eachα, the function h(z)¯ l−|α|

(g(z))^α is a polynomial of degree at mostld_h+|α|(d_g −d_h)≤ld_h.This completes the proof.

Let Aut(B) be the group of all biholomorphic automorphisms of the unit ball B. Let ϕ ∈ Aut(B)and leta:=ϕ⁻¹(0). Its is well-known (see [13], p28) thatϕcan be written in the form

ϕ=U ϕ_a (2.1)

whereU :Cⁿ→Cⁿis a unitary linear transformation andϕ₀ :=−I and fora6= 0, ϕ_a(z) = a−(1− |a|²)P_az−p

1− |a|²(Q_a(z)

1− hz, ai , for all z ∈B. (2.2) whereP_az :hz, aia/|a|² andQ_a:=I−P_a.

Lemma 2.4. Suppose that ϕ is an automorphism of the unit ball and let J_ϕ be its complex jacobian. If f is a q-analytic function on B, then the function Jϕ

(1−q)/(n+1)

(f ◦ ϕ) is also q-analytic onB.

Proof. Suppose thatϕ∈ Aut(B)and leta:=ϕ⁻¹(0).Then it follows from Theorem 2.2 in [13]

that

J_ϕ(z) =

1− |a|² 1− hz, ai

n+1

, for all z ∈B. (2.3)

In addition, in view of (2.2) we see that there are aC-linear mappingL :Cⁿ →Cⁿand a vector b∈Cⁿsuch that for allz ∈B,

ϕ(z) = L(z) +b

1− hz, ai, for all z ∈B.

Applying Lemma 2.3, withg(z) =L(z) +bandh(z) = 1− hz, aishows that iff isq-analytic function onB,then

J_ϕ(z)(1−q)/(n+1)

(f ◦ϕ)(z) = 1

(1− |a|²)^q−1(¯h(z)^q−1(f ◦ϕ)(z)

which alsoq-analytic. This completes the proof.

3. PROOF OFTHEOREM A

In this section we will prove the following transformation rule stated for the unit ball.

Lemma 3.1. Suppose that ϕ is an automorphism of the unit ball and let J_ϕ be its complex jacobian. Then the Bergman kernelK_q,sofA^s_q(B)follows the transformation rule

K_q,s(z, ξ) =

J_ϕ(z)J_ϕ(ξ)^s+q+1_n+1

J_ϕ(z)J_ϕ(ξ)_n+1^q−1

K_q,s(ϕ(z), ϕ(ξ)) (3.1) for allz, ξ ∈B.

Proof. It is sufficient to assume that ϕ◦ϕ(z) = z, for allz ∈ D. We recall that the measure

dv(z)

(1−|z|²)ⁿ⁺¹ is invariant under the action of the automorphism group of the unit ball. We also observe that for any fixedξ∈B,the function

z 7→ (J_ϕ(z))^s+q+1n+1

J_ϕ(z)^q−1_n+1

K_q,s(ϕ(z), ξ)

is an element of A^sq(B). By the reproducing property and change of variables formula we see that

(J_ϕ(z))^s+q+1n+1

J_ϕ(z)^q−1_n+1

Kq,s(ϕ(z), ξ) = Z

B

(J_ϕ(w))^s+q+1n+1

J_ϕ(w)_n+1^q−1

Kq,s(ϕ(w), ξ)Kq,s(z, w)dνs(w)

= Z

B

J_ϕ(w)(s+q+1)/(n+1

)

(J_ϕ(w))^(q−1)/2 K_q,s(w, ξ)K_q,s(z, ϕ(w))dν_s(w)

=

J_ϕ(ξ)(s+q+1)/(n+1)

(J_ϕ(ξ))(q−1)/(n+1) K_q,s(z, ϕ(ξ)) Replacingξbyϕ(ξ)the latter equalities yield

J_ϕ(z)J_ϕ(ξ)(s+q+1)/(n+1)

Jϕ(z)Jϕ(ξ)

(q−1)/(n+1) K_q,s(ϕ(z), ϕ(ξ)) =K_q,s(z, ξ).

This completes the proof.

Lemma 3.2. The Bergman kernelK_q,sofA^sq(B)satisfies the equality K_q,s(z, w) = (1− hw, zi)^q−1

(1− hz, wi)^s+n+qK_q,s(ϕ_z(w),0) (3.2) for allz, ξ ∈B.

Proof. Letz, w ∈B.Applying Lemma 3.1 withϕ=ϕ_zandξ=ϕ_z(w)and using (2.3) yields

K_q,s(z, w) =

J_ϕ(z)J_ϕ(ξ)^s+q+1_n+1

Jϕ(z)Jϕ(ξ) ^q−1_n+1

K_q,s(0, ϕ_z(w))

= (1− hw, zi)^q−1

(1− hz, wi)^s+n+qK_q,s(0, ϕ_z(w)).

SinceK_q,s(z, w) = K_q,s(w, z), for allz, w ∈B,the lemma follows.

We shall make use of the classical Jacobi polynomialsPm^(s,d)with parameters(s, d)and degree m.An explicit formula for these polynomials is given by

P_m^(s,d)(x) = 1 2^m

m

X

k=0

s+m k

d+m m−k

(x−1)^m−k(x+ 1)^k. (3.3)

This can be found in [9]. It is well-known by formula (3.96) in ( [11], p. 71) that these polyno- mials verify the equality

P_m^(s,d)(1−2x) = Γ(m+s+ 1) m!

m

X

j=0

(−1)^{j m}_j

Γ(m+j+s+d+ 1)

Γ(m+s+d+ 1)Γ(j+s+ 1) x^j. (3.4) The Jacobi polynomials satisfy the orthogonality condition

Z 1

0

P_m^(s,d)(2x−1)P_m^(s,d)0 (2x−1)x^d(1−x)^sdx=δ_m,m⁰h^s,d_m (3.5) where

h^s,d_m := Γ (s+m+ 1)) Γ (d+m+ 1)

m!Γ (s+d+m+ 1) (s+d+ 2m+ 1). (3.6) Lemma 3.3. For eachz ∈B,we have

K_q,s(z,0) = ^Γ(q+s+n)_n!Γ(q+s)P_q−1^n,s(1−2|z|²). (3.7) Proof. Due to Lemma 3.1 we see that the q-analytic function z 7→ K_q,s(z,0) invariant under unitary linear transformations and hence it is of the form K_q,s(z,0) = Pq−1

j=0c_j|z|^2j for some complex coefficientsc₀,· · · , c_q−1. Set

Q^s,n_q−1(x) :=

q−1

X

j=0

cjx^j, x∈]0,1[. (3.8)

We equip the space P^q−1 of polynomials of degree at most q −1 with the L²-inner product associated to the measure xⁿ(1−x)^sdx in the interval ]0,1[. Integrating in polar coordinates shows that for allj = 0,· · ·, q−2,one has that

Z 1

0

x^jQ^s,n_q−1(x)xⁿ(1−x)^sdx= 2 Z 1

0

x^2jQ^s,n_q−1(x²)x²ⁿ(1−x²)^sdx

= 1 n

Z

B

|z|^2jQ^s,n_q−1(|z|²)|z|²(1− |z|²)^sdv(z)

= 1 n

Z

B

|z|^2j+2K_q−1^s,n(z,0)(1− |z|²)^sdv(z)

= 0,

where the latter equality holds due to the reproducing property at 0 of the q analytic poly- nomial |z|^2j+2. Hence Q^s,n_q−1 is orthogonal in L²(0,1[, xⁿ(1− x)^sdx) to all polynomials with strictly smaller thanq−1so that by (3.5) we see thatQ^s,n_q−1(x)is proportional to the polynomial P_q−1^(s,n)(2x−1)and thus for some complex constantC(s, n, q−1)we have

Q^s,n_q−1(x) =C(s, n, q)(−1)^q−1P_q−1^(s,n)(2x−1) =C(s, n, q)P_q−1^(n,s)(1−2x).

To compute the constant, we apply the reproducing property to the constant function1and get Z 1

0

Q^s,n_q−1(x)xⁿ⁻¹(1−x)^sdx= 2 Z 1

0

Q^s,n_q−1(x²)x²ⁿ⁻¹(1−x²)^sdx

= 1 n

Z

B

Q^s,n_q−1(|z|²)|z|²(1− |z|²)^sdv(z)

= 1 n

Z

B

K_q−1^s,n(z,0)(1− |z|²)^sdv(z)

= 1 n, and thus

1

n =C(s, n, q)(−1)^q−1 Z 1

0

P_q−1^(s,n)(2x−1)xⁿ⁻¹(1−x)^sdx.

To compute the constantC(s, n, q),let I(s, n−1, q−1) :=

Z 1

0

P_q−1^(s,n)(2x−1)xⁿ⁻¹(1−x)^sdx.

The the orthogonality property of Jacobi polynomials, combined with integration by parts, yields I(s, n−1, q−1) =−q+s+n

n I(s+ 1, n, q−2) so that by induction we get

I(s, n−1, q−1) = (−1)^q−1(n−1)!Γ(2q+s+n−1)

Γ(q+s+n)Γ(n+q−1)I(s+q−1, n+q−2,0).

Since

I(s+q−1, n+q−2,0) = Γ(q+s)Γ(n+q−1) Γ(2q+s+n−1) it follows that

I(s, n−1, q−1) = (−1)^q−1(n−1)!Γ(q+s) Γ(q+s+n) showing that

C(s, n, q) = Γ(q+s+n) n!Γ(q+s) and hence

Q^s,n_q−1(x) = Γ(q+s+n)

n!Γ(q+s) P_q−1^(n,s)(1−2x).

This completes the proof of the lemma.

Proof of Theorem A. Combining Lemmas 3.2 and 3.3 gives the expression of the reproducing kernelK_q,s.To show that the q-analytic polynomials are dense in A^sq(B)it suffices to observe that for allα, β ∈Nⁿ0,and anyq-analytic functionf ∈A^sq(B)we have

∂^α+βf

∂z^α∂z^β(0) =hf, Pi

where

P(w) :=

∂^α+β

∂z^α∂z^βK_q,s(·, w)

z=0

.

On the other hand, a little computing shows that P is a q-analytic polynomial provided that

|β| < q. Therefore, iff is orthogonal to q-analytic polynomials inA^sq(B), thenf must vanish everywhere, showing thatq-analytic polynomials are dense inA^sq(B)

4. THE POLYANALYTICFOCK SPACE

We consider the Gaussian measureµgiven by (1.5) and the corresponding Fock spaceF²_q(Cⁿ) of all q-analytic functions f on Cⁿ which are square integrable with respect to dµ(z). From Lemma 2.2, we see thatFq²(Cⁿ)furnished with theL²(µ)scalar product turns out to be a Hilbert space for which the evaluation functionalf 7→f(z)is continuous for allz ∈Cⁿ.HenceF²q(Cⁿ) admits a reproducing kernelK_F,(·,·).

On the other hand, by the change of variable formula, it is clear that ifa∈Cⁿandf ∈Fq²(Cⁿ), then the functionz 7→e^−hz,aif(z+a)remains inFq²(Cⁿ).

Lemma 4.1. The reproducing kernelK_F,q ofFq(Cⁿ)follows the transformation rule

e^−hz,aiK_F,q(U z+a, ξ) =e^ha,ξiK_F,q(z, U ξ−U a) (4.1) for allz, ξ ∈Cⁿ,and all unitary transformationsU :Cⁿ →Cⁿ.In particular,

K_F,q(z, w) = e^hz,wiK_F,q(z−w,0) (4.2) and

K_F,q(U z, U w) = K_F,q(z, w) (4.3) for allz, ξ ∈Cⁿand all unitary transformationsU :Cⁿ →Cⁿ.

Proof. If a, z, ξ ∈ Cⁿ, then applying twice the reproducing formula and a change of variable yield

e^−hz,aiK_F,q(U z+a, ξ) = Z

Cⁿ

e^{−hU w,ai}K_F,q(U w+a, ξ)K_F,q(z, w)dµ(w)

= Z

Cⁿ

e^−ha,wiK_F,q(z, U w−U a)K_F,q(ξ, w)dµ(w)

=e^−ha,wiK_F,q(z, U ξ−U a).

This completes the proof.

In view of the latter lemma, in order compute the Fock reproducing kernel it is sufficient to calculate K_F,q(z,0) for all z ∈ Cⁿ. To do so, we shall make use of the classical weighted Laguerre polynomialsL^m_d of degreedand weightm. These polynomials satisfy,

Z +∞

0

L^m_d (x)L^m_d0 (x)x^me^−xdx= Γ (d+m+ 1)

d! δ_d,d⁰. (4.4)

They have the following explicit representation L^m_d (x) =

d

X

l=0

(n+m)!

l!Γ (d+m+ 1−l)

(−x)^n−l

(n−l)!. (4.5)

Lemma 4.2. For allz∈Cⁿ, we have

K_F,q(z,0) = 1

n!Lⁿ_q−1 |z|²

. (4.6)

Proof. In virtue of (4.3) we see thatz 7→K_F,q(z,0)is invariant under unitary linear transforma- tions and hence it is of the form

K_F,q(z,0) =

q−1

X

j=0

c_j|z|^2j

for some complex coefficients c₀,· · · , cq−1. To compute the coefficients c_j, we consider the space P^q−1 of polynomials with degree at most q −1 and equip it with the L²-inner product associated to the measuree^−xdxin the interval[0,+∞[.Set

Rq(x) :=

q−1

X

j=0

cjx^j, x≥0.

Integrating in polar coordinates shows that for allj = 0,· · · , q−2,one has that Z +∞

0

x^jR_q(x)xⁿe^−xdx= 1 n

Z

Cⁿ

|z|^2j+2Rq−1(|z|²)dµ(z)

= 1 n

Z

Cⁿ

|z|^2j+2K_F,q(z,0)dµ(z)

= 0,

where the latter equality holds due to the reproducing property at0of theqanalytic polynomial

|z|^2j+2.HenceRq−1(x)is orthogonal inL²(µ)to all polynomials with strictly smaller thanq−1 so that by (4.4 ) we see that Rq−1 is proportional to the polynomial Lⁿ_q−1 and thus for some complex constantc(n, q)we have

Rq−1(x) =c(n, q)Lⁿ_q−1(x).

To compute the constant, we apply the reproducing property to the constant function1and get Z +∞

0

Rq−1(x)xⁿ⁻¹e^−xdx= 1 n

Z

Cⁿ

K_F,q(z,0)dµ(z)

= 1 n, and thus

1

n =c(n, q) Z +∞

0

Lⁿ_q−1(x)xⁿ⁻¹e^−xdx.

To compute the constantc(n, q),let I(n, q) :=

Z +∞

0

Lⁿ_q−1(x)xⁿ⁻¹e^−xdx.

Using the observation that d

dx e^−xLⁿ⁻¹_q (x)

=−e^−xLⁿ_q−1(x) integrations by parts shows that

I(n, q) = (n−1)!I(0, q+n−1) = (n−1)!.

The lemma follows.

Proof of Theorem B. Combining Lemmas 4.1 and 4.2 gives the expression of the reproducing kernelK_F,q.To show that theq-analytic polynomials are dense inFq(Cⁿ)it suffices to observe that for allα, β ∈Nⁿ0,and anyq-analytic functionf ∈Fq²(Cⁿ)we have

∂^α+βf

∂z^α∂z^β(0) =hf, Pi where

P(w) :=

∂^α+β

∂z^α∂z^βK_F,q(·, w)

z=0

.

On the other hand, a little computing shows that P is a q-analytic polynomial provided that

|β| < q.Therefore, if f is orthogonal toq-analytic polynomials inF²q(Cⁿ), thenf must vanish everywhere, showing thatq-analytic polynomials are dense inF²_q(Cⁿ).

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YOUSSFI: I2M, U.M.R. C.N.R.S. 7373, CMI, UNIVERSIT_{E D}´ ’AIX-MARSEILLE, 39 RUEF-JOLIOT-CURIE, 13453 MARSEILLECEDEX13, FRANCE

E-mail address:el-hassan.youssfi@univ-amu.fr