Open Journal of Mathematical Analysis (OMA)

Inspired by a problem proposal recently published in the journal The Fibonacci Quarterly we offer a generalization consisting of two combinatorial identities involving three complex parameters. These identities turn out to be immensely rich. We demonstrate this by providing basic applications to four different fields: polynomial identities, trigonometric identities, identities involving Horadam numbers, and combinatorial identities. Many of our findings will generalize existing results.

This paper establishes the existence of traveling wave solutions in a Leslie-Gower predator-prey model featuring nonlocal dispersal and multiple time delays in both diffusion and reaction terms. The model captures realistic ecological effects such as spatial movement and delayed species responses. Due to the competitive nature of the interaction, the reaction terms satisfy only a partial monotonicity condition. We establish the existence of traveling waves. This is done by construction upper and lower solutions and developing an iterative scheme whose convergence is ensured by Schauder’s fixed point theorem. The approach is extended to accommodate a relaxed class of super and sub-solutions. Explicit examples, and numerical illustrations are provided.

In this work, we seek conditions for the existence or nonexistence of solutions for nonlinear Riemann-Liouville fractional boundary value problems of order \(\alpha + 2n\), where \(\alpha \in (m-1, m]\) with \(m \geq 3\) and \(m, n \in \mathbb{N}\). The problem’s nonlinearity is continuous and also depends on a positive parameter upon which our constraints are established. Our approach involves constructing a Green’s function by combining the Green’s functions of a lower-order fractional boundary value problem and a right-focal boundary value problem \(n\) times. Leveraging the properties of this Green’s function, we apply Krasnosel’skii’s Fixed Point Theorem to establish our results. Several examples are presented to illustrate the existence and nonexistence regions.

Let \(f(z) = \sum\limits_{k=0}^{\infty} f_k z^k\) be an entire transcendental function, and let \((\lambda_n)\) be a sequence of positive numbers increasing to \(+\infty\). Suppose that the series \(A(z) = \sum\limits_{n=1}^{\infty} a_n f(\lambda_n z)\) is regularly convergent in \(\mathbb{D} = \{ z : |z| < 1 \}\), i.e., \(\mathfrak{M}(r, A) := \sum\limits_{n=1}^{\infty} |a_n| M_f(r \lambda_n) < +\infty\) for all \(r \in [0, 1)\). For a positive function \(l\) continuous on \([0, 1)\), the function \(A\) is said to be of bounded \(l\)–\(\mathfrak{M}\)-index if there exists \(N \in \mathbb{Z}_+\) such that \[\frac{\mathfrak{M}(r, A^{(n)})}{n! \, l^n(r)} \leq \max \left\{ \frac{\mathfrak{M}(r, A^{(k)})}{k! \, l^k(r)} : 0 \leq k \leq N \right\},\] for all \(n \in \mathbb{Z}_+\) and all \(r \in [0, 1)\). The growth of bounded \(l\)–\(\mathfrak{M}\)-index functions is studied. In particular, under the conditions \(a_n \geq 0\) and \(f_k \geq 0\), it is proved that the function \(A\) is of bounded \(l\)–\(\mathfrak{M}\)-index with \(l(r) = p(1 – r)^{-(p+1)}\), \(p > 0\), if and only if \[\lim\limits_{r \uparrow 1} (1 – r)^p \ln \mathfrak{M}(r, A) < +\infty.\] This condition is satisfied if and only if \[\lim\limits_{k \to \infty} k^{-p} \left( \ln^+ (f_k \mu_D(k)) \right)^{p+1} < +\infty,\] where \(\mu_D(k) = \max\{ a_n \lambda_n^k : n \geq 1 \}\).

We introduce Littlewood Paley functions defined in terms of a reparameterization of the Ornstein-Uhlenbeck semigroup obtaining that these operators are bounded in \(L^p\), \(1<p<\infty\), with respect to the unidimensional gaussian measure, by means of singular integrals theory. In addition, we study the Abel summability of the Fourier Hermite expansions considering their pointwise convergence and their convergence in the \(L^p\) sense, obtaining a version of Tauber’s theorem.

The main focus of this paper is to define the Wigner transform on Chébli-Trimèche hypergroups of exponential growth and to present several related results. Next, we introduce a new class of pseudo-differential operators ℒ_ψ1, ψ2(σ), called localization operators, which depend on a symbol σ and two admissible functions ψ₁ and ψ₂. We provide criteria, in terms of the symbol σ, for their boundedness and compactness. We also show that these operators belong to the Schatten–von Neumann class S^p for all p ∈ [1, +∞], and we derive a trace formula.

A Young subgroup of the symmetric group \(\mathcal{S}_{N}\) with three factors, is realized as the stabilizer \(G_{n}\) of a monomial \(x^{\lambda}\) ( \(=x_{1}^{\lambda_{1}}x_{2}^{\lambda_{2} }\cdots x_{N}^{\lambda_{N}}\)) with \(\lambda=\left( d_{1}^{n_{1}},d_{2}^{n_{2}},d_{3}^{n_{3}}\right)\) (meaning \(d_{j}\) is repeated \(n_{j}\) times, \(1\leq j\leq3\)), thus is isomorphic to the direct product \(\mathcal{S}_{n_{1}}\times\mathcal{S}_{n_{2}}\times\mathcal{S}_{n_{3}}\). The orbit of \(x^{\lambda}\) under the action of \(\mathcal{S}_{N}\) (by permutation of coordinates) spans a module \(V_{\lambda}\), the representation induced from the identity representation of \(G_{n}\). The space \(V_{\lambda}\) decomposes into a direct sum of irreducible \(\mathcal{S}_{N}\)-modules. The spherical function is defined for each of these, it is the character of the module averaged over the group \(G_{n}\). This paper concerns the value of certain spherical functions evaluated at a cycle which has no more than one entry in each of the three intervals \(I_{j}=\left\{ i:\lambda_{i}=d_{j}\right\} ,1\leq j\leq3\). These values appear in the study of eigenvalues of the Heckman-Polychronakos operators in the paper by V. Gorin and the author (arXiv:2412:01938v1). The present paper determines the spherical function values for \(\mathcal{S}_{N}\)-modules \(V\) of two-row tableau type, corresponding to Young tableaux of shape \(\left[ N-k,k\right]\). The method is based on analyzing the effect of a cycle on \(G_{n}\)-invariant elements of \(V\). These are constructed in terms of Hahn polynomials in two variables.

Hardy-Hilbert-type integral inequalities are among the classics of mathematical analysis. In particular, this includes well-known variants involving homogeneous power-max kernel functions. In this article, we extend the theory by studying the non-homogeneous case using specially designed power-max kernel functions. Additionally, we explore different integration domains to increase the flexibility of our results in a variety of mathematical contexts. We also establish several equivalences, modifications and generalizations of our main integral inequalities. The proofs are detailed and self-contained. To support the theory, we provide numerical examples together with the corresponding implementation codes for transparency and reproducibility.

In this paper, we extend the one-dimensional Gabor transform discussed to the Weinstein harmonic analysis setting. We obtain the expected properties of extended Gabor transform such as inversion formula and Calderon’s reproducing formula.

This paper shows how a family of function spaces, coined as Assiamoua spaces, plays a fundamental role in the Fourier analysis of vector-valued functions on compact groups. These spaces make it possible to transcribe the classical results of Fourier analysis in the framework of analysis of vector-valued functions and vector measures. The construction of Sobolev spaces of vector-valued functions on compact groups rests heavily on the members of the aforementioned family.