HJM, Vol. 51, No. 1, 2025

HOUSTON JOURNAL OF MATHEMATICS

Electronic Edition Vol. 51, No. 1, 2025

Editors: D. Bao (San Francisco, SFSU), S. Berhanu (Temple), D. Blecher (Houston), B. G. Bodmann (Houston), M. Gehrke (CNRS), Y. Hattori (Matsue, Shimane), A. Haynes (Houston), W. B. Johnson (College Station), H. Koivusalo (Bristol), T. H. Lê (Mississippi), M. Marsh (Sacramento), M. Ru (Houston), S. W. Semmes (Rice), D. Werner (FU Berlin).
Managing Editors: B. G. Bodmann and A. Haynes (Houston)

Houston Journal of Mathematics

Contents

Xiaoliang Cheng, College of Mathematics and Computer, Jilin Normal University, Siping, 136000, China(chengxiaoliang92@163.com), Kuai Yu, College of Mathematics and Computer, Jilin Normal University, Siping, 136000, China(yk2761219361@gmail.com), Lin Gui, College of Mathematics and Computer, Jilin Normal University, Siping, 136000, China(guilin7777777@163.com), and An Wang, School of Mathematical Sciences, Capital Normal University, Beijing, 100048, China(wangan@cnu.edu.cn).
The Bergman kernel function and the vanishing theorem for Forelli-Rudin construction on the classical domain of second class, pp. 1–29.

ABSTRACT. We investigate the d-boundedness of the Bergman kernel function for Forelli-Rudin construction on the classical domain of second class. We derive the Bergman kernel function using the series method and Hua’s method. In particular, the finite form is derived through this approach. We also obtain that the Bergman kernel function is d-bounded concerning the Bergman metric. Finally, we prove that the L²-cohomology vanishing theorem on this domain is valid.

Asha Barua, School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, 1201 West University Drive, Edinburg, TX 78539-2999, USA (ashabarua@vt.edu), Angelina Chavera, School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, 1201 West University Drive, Edinburg, TX 78539-2999, USA (angelinachavera1@gmail.com), Ivan Djordjevic, School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, 1201 West University Drive, Edinburg, TX 78539-2999, USA (ivan.djordjevic.us@gmail.com), Valerie Manzano, School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, 1201 West University Drive, Edinburg, TX 78539-2999, USA (vmdance94@gmail.com), Sergio Soto Quintero, School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, 1201 West University Drive, Edinburg, TX 78539-2999, USA (ssoto14@stedwards.edu), Mrinal Kanti Roychowdhury, School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, 1201 West University Drive, Edinburg, TX 78539-2999, USA (mrinal.roychowdhury@utrgv.edu), and Hilda Tejeda, School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, 1201 West University Drive, Edinburg, TX 78539-2999, USA (sophia.tejeda.06@gmail.com).
Quantization for the mixtures of overlap probability distributions, pp. 31–54.

ABSTRACT. Optimal quantization for mixed distributions has emerged as a compelling area of study. In this work, we have focused on a mixed distribution formed from two uniform distributions with partially overlapping supports. For this class of distributions, we have examined the structure of optimal sets of $n$ -means and the corresponding $n$ th quantization errors for all positive integers $n$ . Initially, we explicitly determined the optimal sets and quantization errors for $1 ≤ n ≤ 6$ . Subsequently, we established several key lemmas and propositions and proposed an algorithm that facilitates the computation of optimal $n$ -means and quantization errors for all $n ≥ 5$ . Numerical results are also presented to illustrate the application of the algorithm in deriving these quantities. The findings of this study offer valuable insight and serve as a foundation for further research on quantization in the context of mixed distributions with overlapping supports.

Shichuang Wang, Department of Mathematics and Statistics, North China University of Water Resources and Electric Power, Jinshui E Road, Zhengzhou, Henan, 450046, P. R. China; Institute of Mathematics, Henan Academy of Sciences, Zhengzhou, Henan 450046, P. R. China (x20231080769@stu.ncwu.edu.cn), and Feng Zhao, Department of Mathematics and Statistics, North China University of Water Resources and Electric Power, Jinshui E Road, Zhengzhou, Henan, 450046, P. R. China;Institute of Mathematics, Henan Academy of Sciences, Zhengzhou, Henan 450046, P. R. China (zhaofeng@ncwu.edu.cn).
The mean value estimate for Fourier coefficients of automorphic forms, pp. 55–69.

ABSTRACT. Let f be a normalized holomorphic cusp form of even integral weight k for the full modular group Γ = SL(2, ℤ). In this paper, we investigate the mean value estimate of higher moments of Fourier coefficients λ_{f⊗⊗_lf}(n) of the l-fold product of f.

Guodong Hua, School of Mathematics and Statistics, Weinan Normal University, Weinan, Shaaxi Province, China 714099, Research Institute of Qindong Mathematics, Weinan Normal University, Weinan, Shaaxi Province, China 714099 (gdhuasdu@163.com).
The distributions of Fourier coefficients associated to multiple cusp forms on arithmetic progressions, pp. 71–93.

ABSTRACT. Let f and g be two distinct normalized primitive holomorphic cusp forms of even integral weights k₁ and k₂ for the full modular group Γ = SL(2, ℤ), respectively. And denote by λ_f(n) and λ_g(n) the nth normalized Fourier coefficients of f and g, respectively. In this paper, we are interested in the average behaviour of the following sum

∑ λ2f(ni)λ2g(nj), n≤x n≡ℓ(modq)

where i,j ≥ 1 are positive integers, and q is a prime with (ℓ,q) = 1.

Huafeng Liu, School of Mathematics and Statistics, Shandong Normal University
Jinan, Shandong 250358, China (huafengliu@sdnu.edu.cn), and Jingyang Lü, School of Mathematics and Statistics, Shandong Normal University
Jinan, Shandong 250358, China (lvjingyang@sdnu.edu.cn).
On power sums and sign changes of Fourier coefficients of cusp forms, pp. 95–129.

ABSTRACT. Let f be a normalized Hecke eigenform of weight k for the full modular group SL(2, ℤ). Denote by λ_f(n) the n-th Fourier coefficient of f. Let Q(x) denote a primitive integral positive definite binary quadratic form (reduced form) given by Q(x) = ax₁² + bx₁x₂ + cx₂², where x = (x₁,x₂) ∈ ℤ²,a,b,c ∈ ℤ with gcd(a,b,c) = 1 and fixed discriminant D = b² − 4ac < 0. We assume that for such discriminant D, the class number h(D) is 1. In this paper, we first establish the asymptotic formulae for the sums of Fourier coefficient λ_f(n) supported on the integers n represented in the form of Q(x). Moreover, we show a quantitative result for the number of sign changes of the sequence of Fourier coefficients λ_f(n) with n represented in the form of Q(x) in the interval (x,2x] for sufficiently large x. Our results improve previous results.

Raju Biswas, Department of Mathematics, Raiganj University, Raiganj, West Bengal 733134, India (rajubiswasjanu02@gmail.com), and Rajib Mandal, Department of Mathematics, Raiganj University, Raiganj, West Bengal 733134, India (rajibmathresearch@gmail.com).
Solutions of systems of certain Fermat-type PDDEs, pp. 131–162.

ABSTRACT. The objective of this paper is to investigate the existence and the forms of the pair of finite order entire and meromorphic solutions of some certain systems of Fermat-type partial differential-difference equations of several complex variables. These results represent some refinements and generalizations of the earlier findings, especially the results due to Xu et al. (J. Math. Anal. Appl. 483(2) (2020)). We provide some examples to support the results.

Lei Mou, School of Mathematical Sciences, Capital Normal University, Beijing 100048, China (moulei@cnu.edu.cn), and Piyu Li, School of Mathematics and Statistics, Xuzhou University of Technology, Xuzhou 221018, China (lpy91132006@aliyun.com.cn).
Star countability of infinite products of subspaces of ordinals, pp. 163–172.

ABSTRACT. For an infinite cardinal κ, a topological space X is called κ-compact if every F ⊆ X with |F|≥ κ has an accumulation point. A space X is said to be star countable (respectively, star Lindelf) if for every open cover 𝒰 of X, there exists a countable subset (respectively, a Lindelf subspace) F of X such that St(F,𝒰) = X. For each i < ω, let A_i be a subspace of an ordinal. We give a characterization when ∏ _i<ωA_i is κ-compact, where κ > ω is regular. We show that the three conditions of ∏ _i<ωA_i being ω₁-compact, star countable and star Lindelf are equivalent. We also prove that for each ordinal λ with cf(λ) = ω and λ > ω₁, the product λ^ω₁ is star Lindelf but not star countable.

Attila Losonczi, Budapest Semesters in Mathematics College International, Hungary 1406 Budapest Bethlen Gábor tér 2 (alosonczi2@gmail.com).
On the cardinality of π(δ), pp. 173–180.

ABSTRACT. We prove that the cardinality of transitive quasi-uniformities in a quasi-proximity class is at least 2^{2^ℵ₀} if there exist at least two transitive quasi-uniformities in the class. The transitive elements of π(δ) are characterized if 𝒱_δ is transitive, and in this case we give a condition when there exists a unique transitive quasi-uniformity in π(δ).

Earnest Akofor, Department of Mathematics and Computer Science, Faculty of Science, University of Bamenda, PO Box 39 Bambili, NW Region, Cameroon (eakofor@gmail.com).
Modulo arithmetic of function spaces: Subset hyperspaces as quotients of function spaces, pp. 181–207.

ABSTRACT. Let X be a (topological) space and Cl(X) the collection of nonempty closed subsets of X. Given a topology on Cl(X), making Cl(X) a space, a (subset) hyperspace of X is a subspace of Cl(X) into which X is embedded by mapping each point to its singleton. In this note, we characterize certain hyperspaces of X as explicit quotient spaces of X-valued function spaces and discuss metrization of associated compact-subset hyperspaces in this setting. In particular, we find that any hyperspace topology containing the Vietoris topology is a quotient of a function space topology containing the topology of pointwise convergence.