On the Fock Kernel for the Generalized Fock Space and Generalized Hypergeometric Series

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Introduction
For any fixed parameter α > 0, we consider where dAðzÞ is the Euclidean area measure on the complex plane ℂ. Here, c m,α is a normalizing constant so that dλ m,α is a probability measure on ℂ.
We call the generalized Fock space F 2 m ðℂÞ ≔ F 2 m,α ðℂÞ the set of all entire functions f in L 2 ðℂ, dλ m ðzÞÞ. It is easy to see that F 2 m ðℂÞ is a Hilbert space with the inner product: Let fϕ j ð·Þ: j ∈ ℕg be the countable orthonormal basis for F 2 m ðℂÞ. Then, the generalized Fock kernel B m ðz, wÞ ≔ B m,α ð z, wÞ for F 2 m ðℂÞ is defined by If m = 2, then F 2 2 ðℂÞ is the usual Fock space. In fact, it is well known that B 2 ðz, wÞ = e αz w for z, w ∈ ℂ. See the detailed properties on the usual Fock space in the book [1] written by Zhu. In fact, the explicit form of B 2 ðz, wÞ is very useful for studying the properties of the Fock space in [2]. In this paper, we focus on the following natural question. Question: compute the Fock kernel B m ðz, wÞ for any positive rational number m.
In the theory of the Bergman kernel, it is difficult to find the closed form of the Bergman kernel for a general domain. Instead, in the case of a complex ellipsoid or similar domains, one can see the expression of the Bergman kernel in terms of the hypergeometric series in [3,4].
The generalized hypergeometric series p F q ða 1 , ⋯, a p ; b 1 , ⋯, b q ; xÞ is defined by where ðaÞ k is the Pochhammer symbol defined by If p = q + 1, then the series converges for |x | <1 and diverges for |x | >1. If p < q + 1, then the series converges for all x. If p > q + 1, then the series converges only at x = 0. It is well known that the Bergman kernel for the complex ellipsoid is closely connected with 2 F 1 and its higher dimensional hypergeometric series (Appell hypergeometric series or Lauricella hypergeometric series). Using the theory of the hypergeometric series, new formulas of the Bergman kernel have been computed in [3,[5][6][7].
Recently, new interesting generalized Fock spaces have been studied. In [8], Gonessa investigated the duality on the generalized Fock space with respect to the minimal norm. In [9], one can see the boundedness of the Bergman projection on the generalized Fock-Sobolev space with respect to dλ m ðzÞ. But they did not obtain the explicit forms of the integral kernel. In [10], Cho et al. computed the Fock kernel for the space with respect to dμ α ðzÞ = c α jzj 2α e −jzj 2 dV ðzÞ. For α > 0, the kernel K α ðz, wÞ is represented by K α ðz, wÞ= 1 F 1 ðn, n + α ; hz, wiÞ for z, w ∈ ℂ n .
The main theorem of this paper is the following. At first, we consider the case when m is a positive integer. (i) If m is even, then (ii) If m is odd, then Now, we generalize to the case when m is a positive rational number. (i) If m = 2p/q, where 2p and q are relatively prime, then where (ii) If m = ð2p + 1Þ/q, where 2p + 1 and q are relatively prime, then where In particular, if m = 4, then there is a close connection between B 4 ðz, wÞ and the error function.
where erf ðxÞ is the error function denoted by In general, it is difficult to find the closed form of the generalized hypergeometric series p F q . Using the hypergeometric series in Theorems 1 and 2, we obtain the following closed forms for m = 1, 2/3, 1/2.
as z approaches the strongly pseudoconvex boundary point p ∈ bD.
Using the properties of the incomplete gamma function, we can obtain the similar result also for the generalized Fock space.
Theorem 5. Let m be any positive even integer. Then, Remark 6. The usual Fock kernel B 2 ðz, wÞ = e αz w is very simple but plays an important role in the research of the function theoretic properties of the Fock space F 2 2,α ðℂÞ. Theorems 1 and 2 in this paper are the first result on the generalized Fock space F 2 m,α ðℂÞ for any m ≠ 2. Also, we hope that the explicit formulas in Theorems 3 and 4 can give a clue on studying optimal pointwise estimates for B m ðz, wÞ for some m.

Computation of B m ðz, wÞ
Consider dλ m ðzÞ = c m,α e −αjzj m dAðzÞ, where c m,α is a normalizing constant so that dλ m ðzÞ is a probability measure on ℂ. In fact, we can obtain c m,α from the following lemma.

Lemma 7.
For any nonnegative integers k, we have where Γð·Þ is the usual gamma function. In particular, we have Proof. Recall that the usual gamma function Γ is defined by Using the polar coordinate change, we have If we can substitute s = αr m , then by (19) It completes the proof. ☐ It follows that the reproducing kernel B m ðz, wÞ is written as Throughout this paper, we are focusing on computing the function Then, we have Remark 8. If m = 2, then G 2 ðζÞ = ∑ ∞ k=0 ζ k /k! = e ζ . In this case, which is just the usual Fock kernel. Now, we investigate the relation between G m ðζÞ and generalized hypergeometric series for any positive rational number m.

Proof of Theorem 1
In this section, we express the Fock kernel B m ðz, wÞ in terms of the suitable hypergeometric series p F q when m is a positive integer. The crucial term for computing the form of B m ðz, wÞ is Γðð2k + 2Þ/mÞ.

Proof of Theorem 1 (i). Assume that m is an even integer.
Let m = 2p for some p ∈ ℕ. Then, we have Theorem 1 (i) can be easily proven by the following proposition using (24).

Journal of Function Spaces
Proposition 9. Let m be any even positive integer, and let ζ ≔ α 2/m z w. Then, we have where Φða ; b ; xÞ: = 1 F 1 ða ; b ; xÞ is the confluent hypergeometric series.
Proof. Note that there exist unique integers ℓ and r such that Note that It follows that which completes the proof. ☐

Proof of Theorem 1 (ii).
Assume that m is an odd integer. Let m = 2p + 1 for some p ∈ ℕ. Then, Theorem 1 (ii) can be easily proven by the following proposition using (24).

Proposition 10.
Let m be any odd positive integer, and let ζ ≔ α 2/m z w. Then, Proof. Note that there exist unique integers ℓ and r such that k = ð2p + 1Þℓ + r with 0 ≤ r ≤ 2p. Then, Now, we will use the identity for any nonnegative integer ℓ and t ∈ ℝ. In fact, the identity (34) can be proven by Then, by (34), we have

Proof of Theorem 2
In this section, we focus on computing G m when m is a positive rational number.

Proof of Theorem 2 (i): Even Numerator.
Let m = 2p/q, where 2p and q are relatively prime. Then, we have where k = pℓ + r with 0 ≤ r ≤ p − 1.
Lemma 11. The gamma function Γ satisfies the identity Using the above lemma, we can prove the following.
Proof. We will prove it in two different methods. Using the property Γðx + 1Þ = xΓðxÞ, we have Then, there exists x, y ∈ ℤ such that i = qj + y with 0 ≤ j 4 Journal of Function Spaces It can be proven also using Lemma 11. Note that ☐ Now, we prove Theorem 2 (i) using Lemma 12.
Theorem 13 (Theorem 2 (i) again). Let m = 2p/q, where 2p and q are relatively prime. Then, where Thus, we have Proof. By Lemma 12, we have By the definition (4), we see that It follows that If we use (24), then it completes the proof.

Concluding Remarks
In fact, we can consider the more generalized Fock space. Let dλ ϕ ðzÞ = c ϕ e −ϕðzÞ dAðzÞ, where dAðzÞ is the Euclidean area measure on the complex plane ℂ. We assume that ϕðrÞ is radial and increasing on ½0, ∞Þ with lim r⟶∞ ϕðrÞ = ∞. We call the (generalized) Fock space F 2 ϕ ðℂÞ as the set of all entire functions f in L 2 ðℂ, dλ ϕ Þ. Another simple example is ϕðrÞ = ln r. In this case, we can show that the Fock kernel can be written in terms of the Meijer-G function. It will be interesting that one finds the relation between the other hypergeometric series and the new Fock kernel with respect to ϕ.

Data Availability
No data were used to support this study.

Conflicts of Interest
The author declares that he has no conflicts of interest.