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12 pages, 282 KB

Open AccessArticle

The Cotangent Function as an Avatar of the Polylogarithm Function of Order 0 and Ramanujan’s Formula

by Ruiyang Li, Haoyang Lu and Shigeru Kanemitsu

Axioms 2025, 14(10), 774; https://doi.org/10.3390/axioms14100774 - 21 Oct 2025

In this paper we will be concerned with zeta-symmetry—the functional equation for the (Riemann) zeta-function (equivalents to which are called modular relations)—and reveal the reason why so many results are intrinsic to PFE (Partial Fraction Expansion) for the cotangent function. The hidden reason [...] Read more.

In this paper we will be concerned with zeta-symmetry—the functional equation for the (Riemann) zeta-function (equivalents to which are called modular relations)—and reveal the reason why so many results are intrinsic to PFE (Partial Fraction Expansion) for the cotangent function. The hidden reason is that the cotangent function (as a function in the upper half-plane, say) is the polylogarithm function of order 0 (with complex exponential argument), and therefore it shares properties intrinsic to the Lerch zeta-function of order 0. Here we view the Lerch zeta-function defined in the unit circle as a zeta-function in a wider sense, as a function defined in the upper and lower half-planes. As evidence, we give a plausibly most natural proof of Ramanujan’s formula, including the eta transformation formula as a consequence of the modular relation via the cotangent function, speculating the reason why Ramanujan had been led to such a formula. Other evidence includes the pre-Poisson summation formula as the pick-up principle (which in turn is a generalization of the argument principle). Full article

(This article belongs to the Special Issue Special Functions and Related Topics, 2nd Edition)

15 pages, 307 KB

Open AccessArticle

Fuzzy Treatment for Meromorphic Classes of Admissible Functions Connected to Hurwitz–Lerch Zeta Function

by Ekram E. Ali, Rabha M. El-Ashwah, Abeer M. Albalahi and Rabab Sidaoui

Axioms 2025, 14(7), 523; https://doi.org/10.3390/axioms14070523 - 8 Jul 2025

Viewed by 400

Abstract

Fuzzy differential subordinations, a notion taken from fuzzy set theory and used in complex analysis, are the subject of this paper. In this work, we provide an operator and examine the characteristics of meromorphic functions in the punctured open unit disk that are [...] Read more.

Fuzzy differential subordinations, a notion taken from fuzzy set theory and used in complex analysis, are the subject of this paper. In this work, we provide an operator and examine the characteristics of meromorphic functions in the punctured open unit disk that are related to a class of complex parameter operators. Complex analysis ideas from geometric function theory are used to derive fuzzy differential subordination conclusions. Due to the compositional structure of the operator, some pertinent classes of admissible functions are studied through the application of fuzzy differential subordination. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory, 3rd Edition)

15 pages, 298 KB

Open AccessFeature PaperArticle

The Approximation of Analytic Functions Using Shifts of the Lerch Zeta-Function in Short Intervals

by Antanas Laurinčikas

Axioms 2025, 14(6), 472; https://doi.org/10.3390/axioms14060472 - 17 Jun 2025

Viewed by 462

Abstract

In this paper, we obtain approximation theorems of classes of analytic functions by shifts

L (λ, α, s + i τ)

of the Lerch zeta-function for

τ \in [T, T + H]

where [...] Read more.

In this paper, we obtain approximation theorems of classes of analytic functions by shifts

L (λ, α, s + i τ)

of the Lerch zeta-function for

τ \in [T, T + H]

where

H \in [T^{27 / 82}, T^{1 / 2}]

. The cases of all parameters,

λ, α \in (0, 1]

, are considered. If the set

{\log (m + α) : m \in N_{0}}

is linearly independent over

Q

, then every analytic function in the strip

{s = σ + i t \in C : σ \in (1 / 2, 1)}

is approximated by the above shifts. Full article

23 pages, 340 KB

Open AccessArticle

Third-Order Fuzzy Subordination and Superordination on Analytic Functions on Punctured Unit Disk

by Ekram E. Ali, Georgia Irina Oros, Rabha M. El-Ashwah and Abeer M. Albalahi

Axioms 2025, 14(5), 378; https://doi.org/10.3390/axioms14050378 - 17 May 2025

Viewed by 421

Abstract

This work’s theorems and corollaries present new third-order fuzzy differential subordination and superordination results developed by using a novel convolution linear operator involving the Gaussian hypergeometric function and a previously studied operator. The paper reveals methods for finding the best dominant and best [...] Read more.

This work’s theorems and corollaries present new third-order fuzzy differential subordination and superordination results developed by using a novel convolution linear operator involving the Gaussian hypergeometric function and a previously studied operator. The paper reveals methods for finding the best dominant and best subordinant for the third-order fuzzy differential subordinations and superordinations, respectively. The investigation concludes with the assertion of sandwich-type theorems connecting the conclusions of the studies conducted using the particular methods of the theories of the third-order fuzzy differential subordination and superordination, respectively. Full article

(This article belongs to the Special Issue Advances in Geometric Function Theory and Related Topics)

14 pages, 293 KB

Open AccessArticle

Fuzzy Subordination Results for Meromorphic Functions Associated with Hurwitz–Lerch Zeta Function

by Ekram E. Ali, Georgia Irina Oros, Rabha M. El-Ashwah, Abeer M. Albalahi and Marwa Ennaceur

Mathematics 2024, 12(23), 3721; https://doi.org/10.3390/math12233721 - 27 Nov 2024

Cited by 2 | Viewed by 993

Abstract

The notion of the fuzzy set was incorporated into geometric function theory in recent years, leading to the emergence of fuzzy differential subordination theory, which is a generalization of the classical differential subordination notion. This article employs a new integral operator introduced using [...] Read more.

The notion of the fuzzy set was incorporated into geometric function theory in recent years, leading to the emergence of fuzzy differential subordination theory, which is a generalization of the classical differential subordination notion. This article employs a new integral operator introduced using the class of meromorphic functions, the notion of convolution, and the Hurwitz–Lerch Zeta function for obtaining new fuzzy differential subordination results. Furthermore, the best fuzzy dominants are provided for each of the fuzzy differential subordinations investigated. The results presented enhance the approach to fuzzy differential subordination theory by giving new results involving operators in the study, for which starlikeness and convexity properties are revealed using the fuzzy differential subordination theory. Full article

(This article belongs to the Special Issue Advanced Research in Complex Analysis Operators and Special Classes of Analytic Functions)

23 pages, 539 KB

Open AccessArticle

On Convoluted Forms of Multivariate Legendre-Hermite Polynomials with Algebraic Matrix Based Approach

by Mumtaz Riyasat, Amal S. Alali, Shahid Ahmad Wani and Subuhi Khan

Mathematics 2024, 12(17), 2662; https://doi.org/10.3390/math12172662 - 27 Aug 2024

Viewed by 1045

Abstract

The main purpose of this article is to construct a new class of multivariate Legendre-Hermite-Apostol type Frobenius-Euler polynomials. A number of significant analytical characterizations of these polynomials using various generating function techniques are provided in a methodical manner. These enactments involve explicit relations [...] Read more.

The main purpose of this article is to construct a new class of multivariate Legendre-Hermite-Apostol type Frobenius-Euler polynomials. A number of significant analytical characterizations of these polynomials using various generating function techniques are provided in a methodical manner. These enactments involve explicit relations comprising Hurwitz-Lerch zeta functions and

λ

-Stirling numbers of the second kind, recurrence relations, and summation formulae. The symmetry identities for these polynomials are established by connecting generalized integer power sums, double power sums and Hurwitz-Lerch zeta functions. In the end, these polynomials are also characterized Svia an algebraic matrix based approach. Full article

(This article belongs to the Section E: Applied Mathematics)

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12 pages, 1495 KB

Open AccessArticle

Geometric Features of the Hurwitz–Lerch Zeta Type Function Based on Differential Subordination Method

by Faten F. Abdulnabi, Hiba F. Al-Janaby, Firas Ghanim and Alina Alb Lupaș

Symmetry 2024, 16(7), 784; https://doi.org/10.3390/sym16070784 - 21 Jun 2024

Cited by 1 | Viewed by 1659

Abstract

The interest in special complex functions and their wide-ranging implementations in geometric function theory (GFT) has developed tremendously. Recently, subordination theory has been instrumentally employed for special functions to explore their geometric properties. In this effort, by using a convolutional structure, we combine [...] Read more.

The interest in special complex functions and their wide-ranging implementations in geometric function theory (GFT) has developed tremendously. Recently, subordination theory has been instrumentally employed for special functions to explore their geometric properties. In this effort, by using a convolutional structure, we combine the geometric series, logarithm, and Hurwitz–Lerch zeta functions to formulate a new special function, namely, the logarithm-Hurwitz–Lerch zeta function (LHL-Z function). This investigation then contributes to the study of the LHL-Z function in terms of the geometric theory of holomorphic functions, based on the differential subordination methodology, to discuss and determine the univalence and convexity conditions of the LHL-Z function. Moreover, there are other subordination and superordination connections that may be visually represented using geometric methods. Functions often exhibit symmetry when subjected to conformal mappings. The investigation of the symmetries of these mappings may provide a clearer understanding of how subordination and superordination with the Hurwitz–Lerch zeta function behave under different transformations. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

13 pages, 297 KB

Open AccessArticle

The Generalized Eta Transformation Formulas as the Hecke Modular Relation

by Nianliang Wang, Takako Kuzumaki and Shigeru Kanemitsu

Axioms 2024, 13(5), 304; https://doi.org/10.3390/axioms13050304 - 2 May 2024

Cited by 1 | Viewed by 1829

Abstract

The transformation formula under the action of a general linear fractional transformation for a generalized Dedekind eta function has been the subject of intensive study since the works of Rademacher, Dieter, Meyer, and Schoenberg et al. However, the (Hecke) modular relation structure was [...] Read more.

The transformation formula under the action of a general linear fractional transformation for a generalized Dedekind eta function has been the subject of intensive study since the works of Rademacher, Dieter, Meyer, and Schoenberg et al. However, the (Hecke) modular relation structure was not recognized until the work of Goldstein-de la Torre, where the modular relations mean equivalent assertions to the functional equation for the relevant zeta functions. The Hecke modular relation is a special case of this, with a single gamma factor and the corresponding modular form (or in the form of Lambert series). This has been the strongest motivation for research in the theory of modular forms since Hecke’s work in the 1930s. Our main aim is to restore the fundamental work of Rademacher (1932) by locating the functional equation hidden in the argument and to reveal the Hecke correspondence in all subsequent works (which depend on the method of Rademacher) as well as in the work of Rademacher. By our elucidation many of the subsequent works will be made clear and put in their proper positions. Full article

(This article belongs to the Section Algebra and Number Theory)

26 pages, 401 KB

Open AccessArticle

Results on Minkowski-Type Inequalities for Weighted Fractional Integral Operators

by Hari Mohan Srivastava, Soubhagya Kumar Sahoo, Pshtiwan Othman Mohammed, Artion Kashuri and Nejmeddine Chorfi

Symmetry 2023, 15(8), 1522; https://doi.org/10.3390/sym15081522 - 2 Aug 2023

Cited by 8 | Viewed by 2212

Abstract

This article considers a general family of weighted fractional integral operators and utilizes this general operator to establish numerous reverse Minkowski inequalities. When it comes to understanding and investigating convexity and inequality, symmetry is crucial. It provides insightful explanations, clearer explanations, and useful [...] Read more.

This article considers a general family of weighted fractional integral operators and utilizes this general operator to establish numerous reverse Minkowski inequalities. When it comes to understanding and investigating convexity and inequality, symmetry is crucial. It provides insightful explanations, clearer explanations, and useful methods to help with the learning of key mathematical ideas. The kernel of the general family of weighted fractional integral operators is related to a wide variety of extensions and generalizations of the Mittag-Leffler function and the Hurwitz-Lerch zeta function. It delves into the applications of fractional-order integral and derivative operators in mathematical and engineering sciences. Furthermore, this article derives specific cases for selected functions and presents various applications to illustrate the obtained results. Additionally, novel applications involving the Digamma function are introduced. Full article

(This article belongs to the Special Issue Asymmetric and Symmetric Study on Number Theory and Cryptography)

2 pages, 156 KB

Open AccessEditorial

Editorial Conclusion for the Special Issue “Applications of Symmetric Functions Theory to Certain Fields”

by Serkan Araci and Ayhan Esi

Symmetry 2023, 15(2), 402; https://doi.org/10.3390/sym15020402 - 3 Feb 2023

Viewed by 1358

Abstract

In this Special Issue, the recent advances in the applications of symmetric functions for mathematics and mathematical physics are reviewed, including many novel techniques in analytic functions, transformation methods, economic growth models, and Hurwitz–Lerch zeta functions that were developed to provide reliable solutions [...] Read more.

In this Special Issue, the recent advances in the applications of symmetric functions for mathematics and mathematical physics are reviewed, including many novel techniques in analytic functions, transformation methods, economic growth models, and Hurwitz–Lerch zeta functions that were developed to provide reliable solutions to combinatorial problems [...] Full article

(This article belongs to the Special Issue Applications of Symmetric Functions Theory to Certain Fields)

12 pages, 312 KB

Open AccessArticle

Joint Approximation of Analytic Functions by Shifts of Lerch Zeta-Functions

by Antanas Laurinčikas, Toma Mikalauskaitė and Darius Šiaučiūnas

Mathematics 2023, 11(3), 752; https://doi.org/10.3390/math11030752 - 2 Feb 2023

Cited by 3 | Viewed by 1205

Abstract

In this paper, we consider the simultaneous approximation of tuples of analytic functions by tuples of shifts of Lerch zeta-functions with arbitrary parameters. We prove that there exists a closed set of tuples of functions analytic in the right-hand side of the critical [...] Read more.

In this paper, we consider the simultaneous approximation of tuples of analytic functions by tuples of shifts of Lerch zeta-functions with arbitrary parameters. We prove that there exists a closed set of tuples of functions analytic in the right-hand side of the critical strip, which is approximated by the above tuples of shifts. Further, a generalization for some compositions of tuples of Lerch zeta-functions is given. Full article

(This article belongs to the Special Issue Analytic Methods in Number Theory and Allied Fields)

7 pages, 251 KB

Open AccessArticle

On the Order of Growth of Lerch Zeta Functions

by Jörn Steuding and Janyarak Tongsomporn

Mathematics 2023, 11(3), 723; https://doi.org/10.3390/math11030723 - 1 Feb 2023

Viewed by 1758

Abstract

We extend Bourgain’s bound for the order of growth of the Riemann zeta function on the critical line to Lerch zeta functions. More precisely, we prove L(λ, α, 1/2 + it) ≪ t^13/84+ϵ as t → [...] Read more.

We extend Bourgain’s bound for the order of growth of the Riemann zeta function on the critical line to Lerch zeta functions. More precisely, we prove L(λ, α, 1/2 + it) ≪ t^13/84+ϵ as t → ∞. For both, the Riemann zeta function as well as for the more general Lerch zeta function, it is conjectured that the right-hand side can be replaced by t^ϵ (which is the so-called Lindelöf hypothesis). The growth of an analytic function is closely related to the distribution of its zeros. Full article

(This article belongs to the Special Issue Analytic Methods in Number Theory and Allied Fields)

21 pages, 377 KB

Open AccessArticle

Unification of Chowla’s Problem and Maillet–Demyanenko Determinants

by Nianliang Wang, Kalyan Chakraborty and Shigeru Kanemitsu

Mathematics 2023, 11(3), 655; https://doi.org/10.3390/math11030655 - 28 Jan 2023

Cited by 1 | Viewed by 1293

Abstract

Chowla’s (inverse) problem (CP) is to mean a proof of linear independence of cotangent-like values from non-vanishing of

L (1, χ)

= \sum_{n = 1}^{\infty} \frac{χ (n)}{n}

. On the other hand, we refer to [...] Read more.

Chowla’s (inverse) problem (CP) is to mean a proof of linear independence of cotangent-like values from non-vanishing of

L (1, χ)

= \sum_{n = 1}^{\infty} \frac{χ (n)}{n}

. On the other hand, we refer to determinant expressions for the (relative) class number of a cyclotomic field as the Maillet–Demyanenko determinants (MD). Our aim is to develop the theory of discrete Fourier transforms (DFT) with parity and to unify Chowla’s problem and Maillet–Demyanenko determinants (CPMD) as different-looking expressions of the relative class number via the Dedekind determinant and the base change formula. Full article

(This article belongs to the Special Issue Analytic Methods in Number Theory and Allied Fields)

13 pages, 318 KB

Open AccessArticle

On Discrete Approximation of Analytic Functions by Shifts of the Lerch Zeta Function

by Audronė Rimkevičienė and Darius Šiaučiūnas

Mathematics 2022, 10(24), 4650; https://doi.org/10.3390/math10244650 - 8 Dec 2022

Cited by 2 | Viewed by 1112

Abstract

The Lerch zeta function is defined by a Dirichlet series depending on two fixed parameters. In the paper, we consider the approximation of analytic functions by discrete shifts of the Lerch zeta function, and we prove that, for arbitrary parameters and a step [...] Read more.

The Lerch zeta function is defined by a Dirichlet series depending on two fixed parameters. In the paper, we consider the approximation of analytic functions by discrete shifts of the Lerch zeta function, and we prove that, for arbitrary parameters and a step of arithmetic progression, there is a closed non-empty subset of the space of analytic functions defined in the critical strip such that its functions can be approximated by discrete shifts of the Lerch zeta function. The set of those shifts is infinite, and it has a positive density. For the proof, the weak convergence of probability measures in the space of analytic functions is applied. Full article

9 pages, 292 KB

Open AccessArticle

A Family of Analytic and Bi-Univalent Functions Associated with Srivastava-Attiya Operator

by Adel A. Attiya and Mansour F. Yassen

Symmetry 2022, 14(10), 2006; https://doi.org/10.3390/sym14102006 - 25 Sep 2022

Cited by 2 | Viewed by 1539

Abstract

In this paper, we investigate a new family of normalized analytic functions and bi-univalent functions associated with the Srivastava–Attiya operator. We use the Faber polynomial expansion to estimate the bounds for the general coefficients

| a_{n} |

of this family. The bounds [...] Read more.

In this paper, we investigate a new family of normalized analytic functions and bi-univalent functions associated with the Srivastava–Attiya operator. We use the Faber polynomial expansion to estimate the bounds for the general coefficients

| a_{n} |

of this family. The bounds values for the initial Taylor–Maclaurin coefficients of the functions in this family are also established. Full article

(This article belongs to the Special Issue Symmetry in Geometric Function Theory)

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