Search for Articles:
Title / Keyword
Author / Affiliation / Email
Journal
Article Type
 
 
Advanced Search
 
Section
Special Issue
Volume
Issue
Number
Page
 

Saved Queries

Sign in to use this feature.

Search Filter Reset All

Years

Between: -

Subjects

remove_circle_outline
remove_circle_outline
remove_circle_outline
remove_circle_outline
remove_circle_outline
remove_circle_outline
Add Subjects Reset
Select Subjects

Journals

remove_circle_outline
remove_circle_outline
remove_circle_outline
remove_circle_outline
Add Journals Reset
Select Journals

Article Types

remove_circle_outline
remove_circle_outline
Add Article Types Reset
Select Article Types

Countries / Regions

remove_circle_outline
remove_circle_outline
remove_circle_outline
Add Countries / Regions Reset
Select Countries / Regions
Update Search
share announcement

Search Results (11)

Search Parameters:
Keywords = maclaurin power series expansion

Order results
Result details
Results per page
Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:
52 pages, 869 KiB  
Review
Series and Connections Among Central Factorial Numbers, Stirling Numbers, Inverse of Vandermonde Matrix, and Normalized Remainders of Maclaurin Series Expansions †
by Feng Qi
Mathematics 2025, 13(2), 223; https://doi.org/10.3390/math13020223 - 10 Jan 2025
Cited by 5 | Viewed by 898
Abstract
This paper presents an extensive investigation into several interrelated topics in mathematical analysis and number theory. The author revisits and builds upon known results regarding the Maclaurin power series expansions for a variety of functions and their normalized remainders, explores connections among central [...] Read more.
This paper presents an extensive investigation into several interrelated topics in mathematical analysis and number theory. The author revisits and builds upon known results regarding the Maclaurin power series expansions for a variety of functions and their normalized remainders, explores connections among central factorial numbers, the Stirling numbers, and specific matrix inverses, and derives several closed-form formulas and inequalities. Additionally, this paper reveals new insights into the properties of these mathematical objects, including logarithmic convexity, explicit expressions for certain quantities, and identities involving the Bell polynomials of the second kind. Full article
(This article belongs to the Section C1: Difference and Differential Equations)
►▼ Show Figures

Figure 1

11 pages, 271 KiB  
Article
On Qi’s Normalized Remainder of Maclaurin Power Series Expansion of Logarithm of Secant Function
by Hong-Chao Zhang, Bai-Ni Guo and Wei-Shih Du
Axioms 2024, 13(12), 860; https://doi.org/10.3390/axioms13120860 - 8 Dec 2024
Cited by 6 | Viewed by 977
Abstract
In the study, the authors introduce Qi’s normalized remainder of the Maclaurin power series expansion of the function lnsecx=−lncosx; in view of a monotonicity rule for the ratio of two Maclaurin power series and by [...] Read more.
In the study, the authors introduce Qi’s normalized remainder of the Maclaurin power series expansion of the function lnsecx=−lncosx; in view of a monotonicity rule for the ratio of two Maclaurin power series and by virtue of the logarithmic convexity of the function (2x−1)ζ(x) on (1,∞), they prove the logarithmic convexity of Qi’s normalized remainder; with the aid of a monotonicity rule for the ratio of two Maclaurin power series, the authors present the monotonic property of the ratio between two Qi’s normalized remainders. Full article
(This article belongs to the Special Issue New Theory and Applications of Nonlinear Analysis, Fractional Calculus and Optimization, 2nd Edition)
11 pages, 260 KiB  
Article
Absolute Monotonicity of Normalized Tail of Power Series Expansion of Exponential Function
by Feng Qi
Mathematics 2024, 12(18), 2859; https://doi.org/10.3390/math12182859 - 14 Sep 2024
Cited by 10 | Viewed by 1428
Abstract
In this work, the author reviews the origination of normalized tails of the Maclaurin power series expansions of infinitely differentiable functions, presents that the ratio between two normalized tails of the Maclaurin power series expansion of the exponential function is decreasing on the [...] Read more.
In this work, the author reviews the origination of normalized tails of the Maclaurin power series expansions of infinitely differentiable functions, presents that the ratio between two normalized tails of the Maclaurin power series expansion of the exponential function is decreasing on the positive axis, and proves that the normalized tail of the Maclaurin power series expansion of the exponential function is absolutely monotonic on the whole real axis. Full article
(This article belongs to the Special Issue Polynomials: Theory and Applications, 2nd Edition)
15 pages, 300 KiB  
Article
Some Properties on Normalized Tails of Maclaurin Power Series Expansion of Exponential Function
by Zhi-Hua Bao, Ravi Prakash Agarwal, Feng Qi and Wei-Shih Du
Symmetry 2024, 16(8), 989; https://doi.org/10.3390/sym16080989 - 5 Aug 2024
Cited by 10 | Viewed by 1710
Abstract
In the paper, (1) in view of a general formula for any derivative of the quotient of two differentiable functions, (2) with the aid of a monotonicity rule for the quotient of two power series, (3) in light of the logarithmic convexity of [...] Read more.
In the paper, (1) in view of a general formula for any derivative of the quotient of two differentiable functions, (2) with the aid of a monotonicity rule for the quotient of two power series, (3) in light of the logarithmic convexity of an elementary function involving the exponential function, (4) with the help of an integral representation for the tail of the power series expansion of the exponential function, and (5) on account of Čebyšev’s integral inequality, the authors (i) expand the logarithm of the normalized tail of the power series expansion of the exponential function into a power series whose coefficients are expressed in terms of specific Hessenberg determinants whose elements are quotients of combinatorial numbers, (ii) prove the logarithmic convexity of the normalized tail of the power series expansion of the exponential function, (iii) derive a new determinantal expression of the Bernoulli numbers, deduce a determinantal expression for Howard’s numbers, (iv) confirm the increasing monotonicity of a function related to the logarithm of the normalized tail of the power series expansion of the exponential function, (v) present an inequality among three power series whose coefficients are reciprocals of combinatorial numbers, and (vi) generalize the logarithmic convexity of an extensively applied function involving the exponential function. Full article
(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)
20 pages, 368 KiB  
Article
Monotonicity Results of Ratios between Normalized Tails of Maclaurin Power Series Expansions of Sine and Cosine
by Da-Wei Niu and Feng Qi
Mathematics 2024, 12(12), 1781; https://doi.org/10.3390/math12121781 - 7 Jun 2024
Cited by 10 | Viewed by 1293
Abstract
In the paper, the authors establish the monotonicity results of the ratios between normalized tails of the Maclaurin power series expansions of the sine and cosine functions and restate them in terms of the generalized hypergeometric functions. Full article
(This article belongs to the Section E: Applied Mathematics)
►▼ Show Figures

Figure 1

24 pages, 398 KiB  
Article
A New Closed-Form Formula of the Gauss Hypergeometric Function at Specific Arguments
by Yue-Wu Li and Feng Qi
Axioms 2024, 13(5), 317; https://doi.org/10.3390/axioms13050317 - 10 May 2024
Cited by 14 | Viewed by 2992
Abstract
In this paper, the authors briefly review some closed-form formulas of the Gauss hypergeometric function at specific arguments, alternatively prove four of these formulas, newly extend a closed-form formula of the Gauss hypergeometric function at some specific arguments, successfully apply a special case [...] Read more.
In this paper, the authors briefly review some closed-form formulas of the Gauss hypergeometric function at specific arguments, alternatively prove four of these formulas, newly extend a closed-form formula of the Gauss hypergeometric function at some specific arguments, successfully apply a special case of the newly extended closed-form formula to derive an alternative form for the Maclaurin power series expansion of the Wilf function, and discover two novel increasing rational approximations to a quarter of the circular constant. Full article
17 pages, 339 KiB  
Article
Some Properties of Normalized Tails of Maclaurin Power Series Expansions of Sine and Cosine
by Tao Zhang, Zhen-Hang Yang, Feng Qi and Wei-Shih Du
Fractal Fract. 2024, 8(5), 257; https://doi.org/10.3390/fractalfract8050257 - 26 Apr 2024
Cited by 12 | Viewed by 2191
Abstract
In the paper, the authors introduce two notions, the normalized remainders, or say, the normalized tails, of the Maclaurin power series expansions of the sine and cosine functions, derive two integral representations of the normalized tails, prove the nonnegativity, positivity, decreasing property, and [...] Read more.
In the paper, the authors introduce two notions, the normalized remainders, or say, the normalized tails, of the Maclaurin power series expansions of the sine and cosine functions, derive two integral representations of the normalized tails, prove the nonnegativity, positivity, decreasing property, and concavity of the normalized tails, compute several special values of the Young function, the Lommel function, and a generalized hypergeometric function, recover two inequalities for the tails of the Maclaurin power series expansions of the sine and cosine functions, propose three open problems about the nonnegativity, positivity, decreasing property, and concavity of a newly introduced function which is a generalization of the normalized tails of the Maclaurin power series expansions of the sine and cosine functions. These results are related to the Riemann–Liouville fractional integrals. Full article
(This article belongs to the Section General Mathematics, Analysis)
18 pages, 341 KiB  
Article
Two Forms for Maclaurin Power Series Expansion of Logarithmic Expression Involving Tangent Function
by Yue-Wu Li, Feng Qi and Wei-Shih Du
Symmetry 2023, 15(9), 1686; https://doi.org/10.3390/sym15091686 - 1 Sep 2023
Cited by 15 | Viewed by 2239
Abstract
In view of a general formula for higher order derivatives of the ratio of two differentiable functions, the authors establish the first form for the Maclaurin power series expansion of a logarithmic expression in term of determinants of special Hessenberg matrices whose elements [...] Read more.
In view of a general formula for higher order derivatives of the ratio of two differentiable functions, the authors establish the first form for the Maclaurin power series expansion of a logarithmic expression in term of determinants of special Hessenberg matrices whose elements involve the Bernoulli numbers. On the other hand, for comparison, the authors recite and revise the second form for the Maclaurin power series expansion of the logarithmic expression in terms of the Bessel zeta functions and the Bernoulli numbers. Full article
(This article belongs to the Special Issue Symmetry and Asymmetry in Nonlinear Analysis, Optimization and Related Topics)
45 pages, 13262 KiB  
Article
An Exact In-Plane Equilibrium Equation for Transversely Loaded Large Deflection Membranes and Its Application to the Föppl-Hencky Membrane Problem
by Jun-Yi Sun, Ji Wu, Xue Li and Xiao-Ting He
Mathematics 2023, 11(15), 3329; https://doi.org/10.3390/math11153329 - 28 Jul 2023
Cited by 5 | Viewed by 1554
Abstract
In the existing literature, there are only two in-plane equilibrium equations for membrane problems; one does not take into account the contribution of deflection to in-plane equilibrium at all, and the other only partly takes it into account. In this paper, a new [...] Read more.
In the existing literature, there are only two in-plane equilibrium equations for membrane problems; one does not take into account the contribution of deflection to in-plane equilibrium at all, and the other only partly takes it into account. In this paper, a new and exact in-plane equilibrium equation is established by fully taking into account the contribution of deflection to in-plane equilibrium, and it is used for the analytical solution to the well-known Föppl-Hencky membrane problem. The power series solutions of the problem are given, but in the form of the Taylor series, so as to overcome the difficulty in convergence. The superiority of using Taylor series expansion over using Maclaurin series expansion is numerically demonstrated. Under the same conditions, the newly established in-plane equilibrium equation is compared numerically with the existing two in-plane equilibrium equations, showing that the new in-plane equilibrium equation has obvious superiority over the existing two. A new finding is obtained from this study, namely, that the power series method of using Taylor series expansion is essentially different from that of using Maclaurin series expansion; therefore, the recurrence formulas for power series coefficients of using Maclaurin series expansion cannot be derived directly from that of using Taylor series expansion. Full article
(This article belongs to the Special Issue Mathematical Modeling and Numerical Simulation in Engineering, 2nd Edition)
►▼ Show Figures

Figure 1

12 pages, 285 KiB  
Article
A Series Expansion of a Logarithmic Expression and a Decreasing Property of the Ratio of Two Logarithmic Expressions Containing Sine
by Xin-Le Liu, Hai-Xia Long and Feng Qi
Mathematics 2023, 11(14), 3107; https://doi.org/10.3390/math11143107 - 14 Jul 2023
Cited by 21 | Viewed by 1813
Abstract
In the paper, by virtue of a derivative formula for the ratio of two differentiable functions and with the help of a monotonicity rule, the authors expand a logarithmic expression involving the sine function into the Maclaurin power series in terms of specific [...] Read more.
In the paper, by virtue of a derivative formula for the ratio of two differentiable functions and with the help of a monotonicity rule, the authors expand a logarithmic expression involving the sine function into the Maclaurin power series in terms of specific determinants and prove a decreasing property of the ratio of two logarithmic expressions containing the sine function. These results are interesting purely in pure mathematics. Full article
(This article belongs to the Section C1: Difference and Differential Equations)
14 pages, 291 KiB  
Article
Determinantal Expressions, Identities, Concavity, Maclaurin Power Series Expansions for van der Pol Numbers, Bernoulli Numbers, and Cotangent
by Zhen-Ying Sun, Bai-Ni Guo and Feng Qi
Axioms 2023, 12(7), 665; https://doi.org/10.3390/axioms12070665 - 5 Jul 2023
Cited by 4 | Viewed by 1933
Abstract
In this paper, basing on the generating function for the van der Pol numbers, utilizing the Maclaurin power series expansion and two power series expressions of a function involving the cotangent function, and by virtue of the Wronski formula and a derivative formula [...] Read more.
In this paper, basing on the generating function for the van der Pol numbers, utilizing the Maclaurin power series expansion and two power series expressions of a function involving the cotangent function, and by virtue of the Wronski formula and a derivative formula for the ratio of two differentiable functions, the authors derive four determinantal expressions for the van der Pol numbers, discover two identities for the Bernoulli numbers and the van der Pol numbers, prove the increasing property and concavity of a function involving the cotangent function, and establish two alternative Maclaurin power series expansions of a function involving the cotangent function. The coefficients of the Maclaurin power series expansions are expressed in terms of specific Hessenberg determinants whose elements contain the Bernoulli numbers and binomial coefficients. Full article
(This article belongs to the Special Issue New Theory and Applications of Nonlinear Analysis, Fractional Calculus and Optimization)
Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:
 
Displaying article 1-50 on page 1 of 1.
Back to TopTop