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In this paper, we explore a variety of series involving the central binomial coefficients, highlighting their structural properties and connections to other mathematical objects. Specifically, we derive new closed-form representations and examine the convergence properties of infinite series with a repeating alternation pattern of signs involving central binomial coefficients. More concretely, we derive the series $$\sum\limits_{n=0}^{\infty}\frac{(-1)^{ω_n}}{2n+1}\tbinom{2n}{n}x^n,\,\,\, \sum\limits_{n=0}^{\infty}{(-1)^{ω_n}}\tbinom{2n}{n}x^n\,\,\, \text{and} \,\,\, \sum\limits_{n=0}^{\infty}{(-1)^{ω_n}}n\tbinom{2n}{n}x^n,$$ where $ω_n$ represents both $\lfloor\frac{n}{2}\rfloor$ and $\lceil\frac{n}{2}\rceil$. Also, we present novel series involving Fibonacci and Lucas numbers, deriving many interesting identities.
In this paper, we study a Dirichlet series generated by powers of harmonic numbers. As an application of these functions, we derive certain series involving harmonic numbers. We also study the analytic properties of these Dirichlet series such as values negative integers and behavior at poles. In particular, objects similar to the Stieltjes constants are discussed. Asymptotics of the sums involving harmonic numbers are also studied. From these results I showed a connection between its analytic properties and a possible route to showing the irrationality of the Euler-Mascheroni constant.
In this paper, using a generating function approach, we derive several new convolution sum identities involving Fibonacci m-step numbers. As special instances of the results derived herein, we will get many new and known results involving Fibonacci, Tribonacci, Tetranacci and Pentanacci numbers. In addition, we establish some general results providing insights into the inner structure of such convolutions. Finally, some mixed convolutions involving Fibonacci m-step numbers and Jacobsthal and Pell numbers will be stated.
Pausinger recently investigated a special determinant involving prime numbers. In this short note we point out that this type of determinants was already known in linear algebra and its computation is unrelated to prime numbers.
This is an anthology of series involving rational, factorial, and power functions expressed in terms of special functions. New finite expansions involving quotient functions expressed in terms of the Hurwitz-Lerch zeta function are given. These results represent a new form of expressing this special function as a finite series where contour integration is required for derivation. Extended series previously known and derived are extended using differential equations and algebraic methods.
This paper presents a brief survey of the most important and the most remarkable inequalities involving the basic arithmetic functions.
In this article, we study the existence and uniqueness problem for linear Stochastic PDEs involving a bilaplacian operator. Our results on the existence and uniqueness are obtained through an application of a Monotonicity inequality, which we also prove here. As an application of these results, we also obtain a probabilistic representation of the solution for a linear PDE involving the bilaplacian operator.
In this paper, we study Bohr's inequality and refined versions of Bohr-Rogosinski inequalities involving Schwarz functions. Moreover, we establish a version of multidimensional analogue of Bohr inequality and Bohr-Rogosinski inequalities involving Schwarz functions. Finally, we establish a multidimensional analogue of the refined version of Bohr inequalities with the initial coefficient being zero. All the results are proved to be sharp.
The aim of this paper is twofold. In the first part we focus on a functional involving a weighted curvature integral and the quermassintegrals. We prove upper and lower bounds for this functional in the class of convex sets, which provide a stronger form of the classical Aleksandrov-Fenchel inequality involving the $(n-1)$ and $(n-2)$-quermassintegrals, and consequently a stronger form of the classical isoperimetric inequality in the planar case. Moreover, quantitative estimates are proved. In the second part we deal with a shape optimization problem for a functional involving the boundary momentum. It is known that in dimension two the ball is a maximizer among simply connected sets when the perimeter and centroid is fixed. We show that the result still holds in the class of undecomposable sets. In higher dimensions the same result does not hold and we consider a new scaling invariant functional that might be a good candidate to generalize the planar case. For this functional we prove that the ball is a stable maximizer in the class of nearly spherical sets in any dimension.
In this paper, we evaluate in closed form several different series involving the harmonic numbers and skew-harmonic numbers. We consider two classes of series involving these sequences. One class of series involves the product of the $n$th harmonic or skew-harmonic number and a tail. We provide the solution to two open problems concerning these harmonic series with tails from Ovidiu Furdui's book Sharpening Mathematical Analysis Skills. The other class of series is the Hardy series, which involves a logarithm and the Euler-Mascheroni constant being subtracted from the $n$th harmonic number.
The Bessel function of the first kind $J_{N}\left(kx\right)$ is expanded in a Fourier-Legendre series, as is the modified Bessel functions of the first kind $I_{N}\left(kx\right)$. The purpose of these expansions in Legendre polynomials was not an attempt to rival established \emph{numerical methods} for calculating Bessel functions, but to provide a form for $J_{N}\left(kx\right)$ useful for \emph{analytical} work in the area of strong laser fields, where analytical integration over scattering angles is essential. Despite their primary purpose, we can easily truncate the series at 21 terms to provide 33-digit accuracy that matches IEEE extended precision in some compilers. The analytical theme is furthered by showing that infinite series of like-powered contributors (involving $\,_{2}F_{3}$ hypergeometric functions) extracted from the Fourier-Legendre series may be summed, having values that are inverse powers of the eight primes $1/\left(2^{i}3^{j}5^{k}7^{l}11^{m}13^{n}17^{o}19^{p}\right)$ multiplying powers of the coefficient $k$.
In this note we prove an inequality involving primes and the product of consecutive primes.
Since the study by Jacobi and Hecke, Hecke-type series have received extensive attention. Especially, Hecke-type series involving infinite products have attracted broad interest among many mathematicians including Kac, Peterson, Andrews, Bressoud and Liu. Motivated by the works of these people, we study Hecke-type series involving infinite products. In particular, we establish some Hecke-type series involving infinite products and then obtain the truncated versions of these series as well as some other known series of the same type. As consequences, three families of inequalities for certain partition functions are also presented. Our proofs heavily rely on a formula from the work of Zhi-Guo Liu.
A class of functions involving the divided differences of the psi function and the polygamma functions and originating from Kershaw's double inequality are proved to be completely monotonic. As applications of these results, the monotonicity and convexity of a function involving ratio of two gamma functions and originating from establishment of the best upper and lower bounds in Kershaw's double inequality are derived, two sharp double inequalities involving ratios of double factorials are recovered, the probability integral or error function is estimated, a double inequality for ratio of the volumes of the unit balls in $\mathbb{R}^{n-1}$ and $\mathbb{R}^n$ respectively is deduced, and a symmetrical upper and lower bounds for the gamma function in terms of the psi function is generalized.
We introduce a full solution to a problem considered by Wang and Chu concerning series involving the squares of finite sums of the form $1 + \frac{1}{3}+ \cdots + \frac{1}{2n-1}$. Our proof involves techniques from the theory of colored multiple zeta values.
We prove the analogue of an identity of Huard, Ou, Spearman and Williams and apply it to evaluate a variety of sums involving divisor functions in two variables. It turns out that these sums count representations of positive integers involving radicals.
In the paper, necessary and sufficient conditions are presented for a function involving a ratio of gamma functions to be logarithmically completely monotonic. This extends and generalizes the main result in [\emph{Inequalities and monotonicity for the ratio of gamma functions}, Taiwanese J. Math. \textbf{7} (2003), no.~2, 239 obreakdash--247.] and others. As applications, several inequalities involving the volume of the unit ball in $\mathbb{R}^n$ are derived, which refine, generalize and extend some known inequalities.
We offer a generalization of a formula of Popov involving the Von Mangoldt function. Some commentary on its relation to other results in analytic number theory is mentioned as well as an analogue involving the m$\ddot{o}$bius function.
In the paper, by establishing the monotonicity of some functions involving the sine and cosine functions, the authors provide concise proofs of some known inequalities and find some new sharp inequalities involving the Seiffert, contra-harmonic, centroidal, arithmetic, geometric, harmonic, and root-square means of two positive real numbers $a$ and $b$ with $a e b$.
This article fits in many studies of multifractal analysis of measure. We took as a starting point the work of F. Ben Nasr in " Calculs de dimension de packing " to give a new inequality involving $Dim(\bar{X}^α)$ which would be, in certain cases, finer than the inequality established by L. Olsen in " A multifractal formalism " . Besides we elaborated an application of our result which gives a better inequality involving $Dim(\bar{X}^α)$.