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19 pages, 294 KB

Open AccessArticle

Abel-Type Transformations and Telescoping Structures in Reciprocal Series of Second-Order Linear Recurrences

by Kunle Adegoke, Robert Frontczak and Taras Goy

AppliedMath 2026, 6(9), 141; https://doi.org/10.3390/appliedmath6090141 - 25 Aug 2026

Viewed by 245

We develop a unified method for transforming and evaluating infinite series involving products of terms of second-order linear recurrences in the denominator. The approach is based on a discrete Abel-type summation formula (summation by parts), which converts reciprocal series with three or more factors into expressions exhibiting a partial telescoping structure. As a consequence, we obtain general transformation formulas for series of the form

\sum_{k = 1}^{\infty} \frac{{(\pm 1)}^{k}}{w_{r k + l} w_{m k + s} w_{m (k + 1) + s}}

, together with extensions to products of four or more terms. These formulas provide a systematic framework that unifies and extends many known identities for Fibonacci and Lucas numbers. In addition, the method leads to explicit evaluations and identities involving several classical combinatorial sequences, including Catalan numbers, harmonic numbers, and Stirling numbers of both kinds. A key feature of the approach is that it naturally distinguishes between even and odd values of the parameter m, leading to structurally different representations. The results show that summation by parts is an effective and flexible tool for reducing multi-factor reciprocal sums to simpler forms. Full article

(This article belongs to the Section Deterministic Mathematics)

16 pages, 293 KB

Open AccessArticle

A Half-Discrete Hardy–Mulholland-Type Inequality Involving One Multiple Upper Limit Function and One Partial Sum

by Bicheng Yang, Shanhe Wu and Jianquan Liao

Mathematics 2025, 13(15), 2497; https://doi.org/10.3390/math13152497 - 3 Aug 2025

Viewed by 911

Abstract

In this paper, by using the techniques of real analysis, with the help of the Euler–Maclaurin summation formula, Abel’s summation by parts formula, and the differentiation mid-value theorem, we establish a half-discrete Hardy–Mulholland-type inequality involving one multiple upper limit function and one partial sum. Based on the obtained inequality, we characterize the condition of the best possible constant factor related to several parameters. At the end of the paper, we illustrate that some new half-discrete Hardy–Mulholland-type inequalities can be deduced from the special values of the parameters. Our results enrich the current results in the study of half-discrete Hardy–Mulholland-type inequalities. Full article

(This article belongs to the Special Issue Advances in Convex Analysis and Inequalities)

13 pages, 286 KB

Open AccessArticle

A Weighted Generalization of Hardy–Hilbert-Type Inequality Involving Two Partial Sums

by Bicheng Yang and Shanhe Wu

Mathematics 2023, 11(14), 3212; https://doi.org/10.3390/math11143212 - 21 Jul 2023

Cited by 3 | Viewed by 1479

Abstract

In this paper, we address Hardy–Hilbert-type inequality by virtue of constructing weight coefficients and introducing parameters. By using the Euler–Maclaurin summation formula, Abel’s partial summation formula, and differential mean value theorem, a new weighted Hardy–Hilbert-type inequality containing two partial sums can be proven, which is a further generalization of an existing result. Based on the obtained results, we provide the equivalent statements of the best possible constant factor related to several parameters. Also, we illustrate how the inequalities obtained in the main results can generate some new Hardy–Hilbert-type inequalities. Full article

(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)

13 pages, 293 KB

Open AccessArticle

A Reverse Hardy–Hilbert’s Inequality Containing Multiple Parameters and One Partial Sum

by Bicheng Yang, Shanhe Wu and Xingshou Huang

Mathematics 2022, 10(13), 2362; https://doi.org/10.3390/math10132362 - 5 Jul 2022

Viewed by 2549

Abstract

In this work, by introducing multiple parameters and utilizing the Euler–Maclaurin summation formula and Abel’s partial summation formula, we first establish a reverse Hardy–Hilbert’s inequality containing one partial sum as the terms of double series. Then, based on the newly proposed inequality, we characterize the equivalent conditions of the best possible constant factor associated with several parameters. At the end of the paper, we illustrate that more new inequalities can be generated from the special cases of the reverse Hardy–Hilbert’s inequality. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Models and Applications)

26 pages, 427 KB

Open AccessArticle

Boundedness of Some Paraproducts on Spaces of Homogeneous Type

by Xing Fu

Mathematics 2021, 9(20), 2591; https://doi.org/10.3390/math9202591 - 15 Oct 2021

Viewed by 2238

Abstract

Let

(X, d, μ)

be a space of homogeneous type in the sense of Coifman and Weiss. In this article, the author develops a partial theory of paraproducts

{Π_{j}}_{j = 1}^{3}

defined via approximations [...] Read more.

Let

(X, d, μ)

be a space of homogeneous type in the sense of Coifman and Weiss. In this article, the author develops a partial theory of paraproducts

{Π_{j}}_{j = 1}^{3}

defined via approximations of the identity with exponential decay (and integration 1), which are extensions of paraproducts defined via regular wavelets. Precisely, the author first obtains the boundedness of

Π_{3}

on Hardy spaces and then, via the methods of interpolation and the well-known

T (1)

theorem, establishes the endpoint estimates for

{Π_{j}}_{j = 1}^{3}

. The main novelty of this paper is the application of the Abel summation formula to the establishment of some relations among the boundedness of

{Π_{j}}_{j = 1}^{3}

, which has independent interests. It is also remarked that, throughout this article,

μ

is not assumed to satisfy the reverse doubling condition. Full article

(This article belongs to the Special Issue Recent Developments of Function Spaces and Their Applications I)

13 pages, 271 KB

Open AccessArticle

A More Accurate Half-Discrete Hilbert-Type Inequality Involving One upper Limit Function and One Partial Sum

by Xianyong Huang, Shanhe Wu and Bicheng Yang

Symmetry 2021, 13(8), 1548; https://doi.org/10.3390/sym13081548 - 23 Aug 2021

Cited by 2 | Viewed by 2358

Abstract

In this paper, by virtue of the symmetry principle, we construct proper weight coefficients and use them to establish a more accurate half-discrete Hilbert-type inequality involving one upper limit function and one partial sum. Then, we prove the new inequality with the help of the Euler–Maclaurin summation formula and Abel’s partial summation formula. Finally, we illustrate how the obtained results can generate some new half-discrete Hilbert-type inequalities. Full article

(This article belongs to the Special Issue Symmetry in the Mathematical Inequalities)

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