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14 pages, 491 KiB

Open AccessArticle

A Comparative View of Becker, Lomnitz, and Lambert Linear Viscoelastic Models

by Juan Luis González-Santander and Francesco Mainardi

Mathematics 2024, 12(21), 3426; https://doi.org/10.3390/math12213426 (registering DOI) - 31 Oct 2024

We compare the classical viscoelastic models due to Becker and Lomnitz with respect to a recent viscoelastic model based on the Lambert W function. We take advantage of this comparison to derive new analytical expressions for the relaxation spectrum in the Becker and [...] Read more.

We compare the classical viscoelastic models due to Becker and Lomnitz with respect to a recent viscoelastic model based on the Lambert W function. We take advantage of this comparison to derive new analytical expressions for the relaxation spectrum in the Becker and Lomnitz models, as well as novel integral representations for the retardation and relaxation spectra in the Lambert model. In order to derive these analytical expressions, we have used the analytical properties of the exponential integral and the Lambert W function, as well as the Titchmarsh’s inversion formula of the Stieltjes transform. In addition, we prove some interesting inequalities by comparing the different models considered, as well as the non-negativity of the retardation and relaxation spectral functions. This means that the complete monotonicity of the rate of creep and the relaxation functions is satisfied, as required by the classical theory of linear viscoelasticity. Full article

(This article belongs to the Special Issue Integral Transforms and Special Functions in Applied Mathematics)

17 pages, 316 KiB

Open AccessArticle

A Look at Generalized Trigonometric Functions as Functions of Their Two Parameters and Further New Properties

by Dmitrii Karp and Elena Prilepkina

Mathematics 2024, 12(21), 3383; https://doi.org/10.3390/math12213383 - 29 Oct 2024

Viewed by 328

Abstract

Investigation of the generalized trigonometric and hyperbolic functions containing two parameters has been a very active research area over the last decade. We believe, however, that their monotonicity and convexity properties with respect to parameters have not been thoroughly studied. In this paper, [...] Read more.

Investigation of the generalized trigonometric and hyperbolic functions containing two parameters has been a very active research area over the last decade. We believe, however, that their monotonicity and convexity properties with respect to parameters have not been thoroughly studied. In this paper, we make an attempt to fill this gap. Our results are not complete; for some functions, we manage to establish (log)-convexity/concavity in parameters, while for others, we only managed the prove monotonicity, in which case we present necessary and sufficient conditions for convexity/concavity. In the course of the investigation, we found two hypergeometric representations for the generalized cosine and hyperbolic cosine functions which appear to be new. In the last section of the paper, we present four explicit integral evaluations of combinations of generalized trigonometric/hyperbolic functions in terms of hypergeometric functions. Full article

(This article belongs to the Special Issue Integral Transforms and Special Functions in Applied Mathematics)

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12 pages, 259 KiB

Open AccessArticle

Partial Derivative Equations and Identities for Hermite-Based Peters-Type Simsek Polynomials and Their Applications

by Eda Yuluklu

Mathematics 2024, 12(16), 2505; https://doi.org/10.3390/math12162505 - 14 Aug 2024

Viewed by 448

Abstract

The objective of this paper is to investigate Hermite-based Peters-type Simsek polynomials with generating functions. By using generating function methods, we determine some of the properties of these polynomials. By applying the derivative operator to the generating functions of these polynomials, we also [...] Read more.

The objective of this paper is to investigate Hermite-based Peters-type Simsek polynomials with generating functions. By using generating function methods, we determine some of the properties of these polynomials. By applying the derivative operator to the generating functions of these polynomials, we also determine many of the identities and relations that encompass these polynomials and special numbers and polynomials. Moreover, using integral techniques, we obtain some formulas covering the Cauchy numbers, the Peters-type Simsek numbers and polynomials of the first kind, the two-variable Hermite polynomials, and the Hermite-based Peters-type Simsek polynomials. Full article

(This article belongs to the Special Issue Integral Transforms and Special Functions in Applied Mathematics)

20 pages, 311 KiB

Open AccessArticle

Parameterized Finite Binomial Sums

by Necdet Batır and Junesang Choi

Mathematics 2024, 12(16), 2450; https://doi.org/10.3390/math12162450 - 7 Aug 2024

Viewed by 542

Abstract

We offer intriguing new insights into parameterized finite binomial sums, revealing elegant identities such as [...] Read more.

We offer intriguing new insights into parameterized finite binomial sums, revealing elegant identities such as

\sum_{k = 0, k \neq n}^{m + n} \frac{{(- 1)}^{k}}{n - k} (\binom{m + n}{k}) = {(- 1)}^{n} (\binom{m + n}{n}) (H_{m} - H_{n})

, where

n, m

are non-negative integers and

H_{n}

is the harmonic number. These formulas beautifully capture the intricate relationship between harmonic numbers and binomial coefficients, providing a fresh and captivating perspective on combinatorial sums. Full article

(This article belongs to the Special Issue Integral Transforms and Special Functions in Applied Mathematics)

12 pages, 278 KiB

Open AccessArticle

Bisection Series Approach for Exotic ₃F₂(1)-Series

by Marta Na Chen and Wenchang Chu

Mathematics 2024, 12(12), 1915; https://doi.org/10.3390/math12121915 - 20 Jun 2024

Viewed by 636

Abstract

By employing the bisection series approach, two classes of nonterminating ₃

F_{2} (1)

-series are examined. Several new summation formulae are established in closed form through the summation formulae of Gauss and Kummer for the ₂ [...] Read more.

By employing the bisection series approach, two classes of nonterminating ₃

F_{2} (1)

-series are examined. Several new summation formulae are established in closed form through the summation formulae of Gauss and Kummer for the ₂

F_{1} (\pm 1)

-series. They are expressed in terms of well-known functions such as

π

, Euler–Gamma, and logarithmic functions, which can be used in physics and applied sciences for numerical and theoretical analysis. Full article

(This article belongs to the Special Issue Integral Transforms and Special Functions in Applied Mathematics)

18 pages, 335 KiB

Open AccessArticle

On Some General Tornheim-Type Series

by Kwang-Wu Chen

Mathematics 2024, 12(12), 1867; https://doi.org/10.3390/math12121867 - 14 Jun 2024

Cited by 2 | Viewed by 457

Abstract

In this paper, we solve the open problem posed by Kuba by expressing [...] Read more.

In this paper, we solve the open problem posed by Kuba by expressing

\sum_{j, k \geq 1} \frac{H_{k}^{(u)} H_{j}^{(v)} H_{j + k}^{(w)}}{j^{r} k^{s} {(j + k)}^{t}}

as a linear combination of multiple zeta values. These sums include Tornheim’s double series as a special case. Our approach is based on employing two distinct methods to evaluate the specific integral proposed by Yamamoto, which is associated with the two-poset Hasse diagram. We also provide a new evaluation formula for the general Mordell–Tornheim series and some similar types of double and triple series. Full article

(This article belongs to the Special Issue Integral Transforms and Special Functions in Applied Mathematics)

12 pages, 271 KiB

Open AccessArticle

Gauss’ Second Theorem for

{}_{2}F_{1} (1 / 2)

-Series and Novel Harmonic Series Identities

by Chunli Li and Wenchang Chu

Mathematics 2024, 12(9), 1381; https://doi.org/10.3390/math12091381 - 1 May 2024

Cited by 3 | Viewed by 882

Abstract

Two summation theorems concerning the

F_{1}^{2} (1 / 2)

-series due to Gauss and Bailey will be examined by employing the “coefficient extraction method”. Forty infinite series concerning harmonic numbers and binomial/multinomial coefficients will be evaluated in closed form, [...] Read more.

Two summation theorems concerning the

F_{1}^{2} (1 / 2)

-series due to Gauss and Bailey will be examined by employing the “coefficient extraction method”. Forty infinite series concerning harmonic numbers and binomial/multinomial coefficients will be evaluated in closed form, including eight conjectured ones made by Z.-W. Sun. The presented comprehensive coverage for the harmonic series of convergence rate “

1 / 2

” may serve as a reference source for readers. Full article

(This article belongs to the Special Issue Integral Transforms and Special Functions in Applied Mathematics)

20 pages, 315 KiB

Open AccessArticle

Novel Formulas for B-Splines, Bernstein Basis Functions, and Special Numbers: Approach to Derivative and Functional Equations of Generating Functions

by Yilmaz Simsek

Mathematics 2024, 12(1), 65; https://doi.org/10.3390/math12010065 - 24 Dec 2023

Cited by 2 | Viewed by 962

Abstract

The purpose of this article is to give relations among the uniform B-splines, the Bernstein basis functions, and certain families of special numbers and polynomials with the aid of the generating functions method. We derive a relation between generating functions for the [...] Read more.

The purpose of this article is to give relations among the uniform B-splines, the Bernstein basis functions, and certain families of special numbers and polynomials with the aid of the generating functions method. We derive a relation between generating functions for the uniform B-splines and generating functions for the Bernstein basis functions. We derive some functional equations for these generating functions. Using the higher-order partial derivative equations of these generating functions, we derive both the generalized de Boor recursion relation and the higher-order derivative formula of uniform B-splines in terms of Bernstein basis functions. Using the functional equations of these generating functions, we derive the relations among the Bernstein basis functions, the uniform B-splines, the Apostol-Bernoulli numbers and polynomials, the Aposto–Euler numbers and polynomials, the Eulerian numbers and polynomials, and the Stirling numbers. Applying the p-adic integrals to these polynomials, we derive many novel formulas. Furthermore, by applying the Laplace transformation to these generating functions, we derive infinite series representations for the uniform B-splines and the Bernstein basis functions. Full article

(This article belongs to the Special Issue Integral Transforms and Special Functions in Applied Mathematics)

17 pages, 319 KiB

Open AccessArticle

Bohr’s Phenomenon for the Solution of Second-Order Differential Equations

by Saiful R. Mondal

Mathematics 2024, 12(1), 39; https://doi.org/10.3390/math12010039 - 22 Dec 2023

Viewed by 1300

Abstract

The aim of this work is to establish a connection between Bohr’s radius and the analytic and normalized solutions of two differential second-order differential equations, namely [...] Read more.

The aim of this work is to establish a connection between Bohr’s radius and the analytic and normalized solutions of two differential second-order differential equations, namely

y^{″} (z) + a (z) y^{'} (z) + b (z) y (z) = 0

and

z^{2} y^{″} (z) + a (z) y^{'} (z) + b (z) y (z) = d (z)

. Using differential subordination, we find the upper bound of the Bohr and Rogosinski radii of the normalized solution

F (z)

of the above differential equations. We construct several examples by judicious choice of

a (z)

,

b (z)

and

d (z)

. The examples include several special functions like Airy functions, classical and generalized Bessel functions, error functions, confluent hypergeometric functions and associate Laguerre polynomials. Full article

(This article belongs to the Special Issue Integral Transforms and Special Functions in Applied Mathematics)

10 pages, 271 KiB

Open AccessFeature PaperArticle

Integral Representations of a Generalized Linear Hermite Functional

by Roberto S. Costas-Santos

Mathematics 2023, 11(14), 3227; https://doi.org/10.3390/math11143227 - 22 Jul 2023

Viewed by 758

Abstract

In this paper, we find new integral representations for the generalized Hermite linear functional in the real line and the complex plane. As an application, new integral representations for the Euler Gamma function are given. Full article

(This article belongs to the Special Issue Integral Transforms and Special Functions in Applied Mathematics)

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14 pages, 560 KiB

Open AccessFeature PaperArticle

Asymptotic Expansions for Moench’s Integral Transform of Hydrology

by José L. López, Pedro Pagola and Ester Pérez Sinusía

Mathematics 2023, 11(14), 3053; https://doi.org/10.3390/math11143053 - 10 Jul 2023

Cited by 1 | Viewed by 1663

Abstract

Theis’ theory (1935), later improved by Hantush & Jacob (1955) and Moench (1971), is a technique designed to study the water level in aquifers. The key formula in this theory is a certain integral transform [...] Read more.

Theis’ theory (1935), later improved by Hantush & Jacob (1955) and Moench (1971), is a technique designed to study the water level in aquifers. The key formula in this theory is a certain integral transform

H [g] (r, t)

of the pumping function g that depends on the time t and the relative position r to the pumping point as well as on other physical parameters. Several analytic approximations of

H [g] (r, t)

have been investigated in the literature that are valid and accurate in certain regions of r, t and the mentioned physical parameters. In this paper, the analysis of possible analytic approximations of

H [g] (r, t)

is completed by investigating asymptotic expansions of

H [g] (r, t)

in a region of the parameters that is of interest in practical situations, but that has not yet been investigated. Explicit and/or recursive algorithms for the computation of the coefficients of the expansions and estimates for the remainders are provided. Some numerical examples based on an actual physical experiment conducted by Layne-Western Company in 1953 illustrate the accuracy of the approximations. Full article

(This article belongs to the Special Issue Integral Transforms and Special Functions in Applied Mathematics)

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Review

Jump to: Research

39 pages, 570 KiB

Open AccessEditor’s ChoiceReview

Going Next after “A Guide to Special Functions in Fractional Calculus”: A Discussion Survey

by Virginia Kiryakova and Jordanka Paneva-Konovska

Mathematics 2024, 12(2), 319; https://doi.org/10.3390/math12020319 - 18 Jan 2024

Cited by 5 | Viewed by 1072

Abstract

In the survey Kiryakova: “A Guide to Special Functions in Fractional Calculus” (published in this same journal in 2021) we proposed an overview of this huge class of special functions, including the Fox H-functions, the Fox–Wright generalized hypergeometric functions

_{p} Ψ_{q}

[...] Read more.

In the survey Kiryakova: “A Guide to Special Functions in Fractional Calculus” (published in this same journal in 2021) we proposed an overview of this huge class of special functions, including the Fox H-functions, the Fox–Wright generalized hypergeometric functions

_{p} Ψ_{q}

and a large number of their representatives. Among these, the Mittag-Leffler-type functions are the most popular and frequently used in fractional calculus. Naturally, these also include all “Classical Special Functions” of the class of the Meijer’s G- and

_{p} F_{q}

-functions, orthogonal polynomials and many elementary functions. However, it so happened that almost simultaneously with the appearance of the Mittag-Leffler function, another “fractionalized” variant of the exponential function was introduced by Le Roy, and in recent years, several authors have extended this special function and mentioned its applications. Then, we introduced a general class of so-called (multi-index) Le Roy-type functions, and observed that they fall in an “Extended Class of SF of FC”. This includes the I-functions of Rathie and, in particular, the

\bar{H}

-functions of Inayat-Hussain, studied also by Buschman and Srivastava and by other authors. These functions initially arose in the theory of the Feynman integrals in statistical physics, but also include some important special functions that are well known in math, like the polylogarithms, Riemann Zeta functions, some famous polynomials and number sequences, etc. The I- and

\bar{H}

-functions are introduced by Mellin–Barnes-type integral representations involving multi-valued fractional order powers of

Γ

-functions with a lot of singularities that are branch points. Here, we present briefly some preliminaries on the theory of these functions, and then our ideas and results as to how the considered Le Roy-type functions can be presented in their terms. Next, we also introduce Gelfond–Leontiev generalized operators of differentiation and integration for which the Le Roy-type functions are eigenfunctions. As shown, these “generalized integrations” can be extended as kinds of generalized operators of fractional integration, and are also compositions of “Le Roy type” Erdélyi–Kober integrals. A close analogy appears with the Generalized Fractional Calculus with H- and G-kernel functions, thus leading the way to its further development. Since the theory of the I- and

\bar{H}

-functions still needs clarification of some details, we consider this work as a “Discussion Survey” and also provide a list of open problems. Full article

(This article belongs to the Special Issue Integral Transforms and Special Functions in Applied Mathematics)

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