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In the article, by virtue of Maclaurin's expansions of the arcsine function and its square and cubic, the authors give a short proof of a sum formula of a Maclaurin's series with coefficients containing reciprocals of the Catalan numbers; establish four sum formulas for finite sums containing the ratio or product of two central binomial coefficients or the Catalan numbers. The instant proof simplifies discussions in the journal papers: College Math. J. 43 (2012), no. 2, 141–146; Amer. Math. Monthly 121 (2014), no. 3, 267–267; Amer. Math. Monthly 123 (2016), no. 4, 405–406; Elem. Math. 71 (2016), no. 3, 109–121; and Mathematics 5 (2017), no. 3, Article 40, 31 pages.

In this paper, the authors derive several recursive and closed-form formulas for some specific values of partial Bell polynomials.

In 1979, Apéry proved the irrationality of ζ(2) and ζ(3). Since then, there has been much research interest in investigating the Apéry-like series for values of Riemann zeta function, Ramanujan-like series for π and other infinite series involving central binomial coefficients. The purpose of this work is to present the first 20 results related to the Apéry-like series in the form of 4 lemmas, each containing 5 results. The Sherman’s results are applied to attain this. Thereafter, these 20 results are further used to establish up to 104 results pertaining to the Apéry-like series in the form of 4 theorems, with 26 results each. These findings are finally been described in terms of the generalized hypergeometric functions. Symmetry occurs naturally in the generalized hypergeometric functions.

In the paper, the authors 1. establish general expressions of series expansions of $ (\arcsin x)^\ell $ for $ \ell\in\mathbb{N} $; 2. find closed-form formulas for the sequence $ \begin{equation*} {\rm{B}}_{2n,k}\biggl(0,\frac{1}{3},0,\frac{9}{5},0,\frac{225}{7},\dotsc, \frac{1+(-1)^{k+1}}{2}\frac{[(2n-k)!!]^2}{2n-k+2}\biggr), \end{equation*} $ where $ {\rm{B}}_{n, k} $ denotes the second kind Bell polynomials; 3. derive series representations of generalized logsine functions. The series expansions of the powers $ (\arcsin x)^\ell $ were related with series representations for generalized logsine functions by Andrei I. Davydychev, Mikhail Yu. Kalmykov, and Alexey Sheplyakov. The above sequence represented by special values of the second kind Bell polynomials appeared in the study of Grothendieck's inequality and completely correlation-preserving functions by Frank Oertel.

Using a hypergeometric series approach, a general combinatorial identity is found in this note, and among its special cases are well-known and classical combinatorial identities due to Gould and Touchard.

Abstract In the paper, the authors consider a ratio of finite many gamma functions and find its monotonicity properties such as complete monotonicity, the Bernstein function property, and logarithmically complete monotonicity.

In this paper, our aim is to provide as many as one hundred and ninety-two summation formulas for the generalized hypergeometric series F65−1 in terms of gamma functions. For this, we have established eight theorems containing general results. This is achieved by means of applying generalized Watson’s and Dixon’s summation formulas obtained earlier by Lavoie et al. into a known transformation formula available in the literature. Results obtained earlier by Zhao follows special cases of our main findings.

In this paper, the author considers the twisted Catalan numbers and the twisted Catalan-Daehee numbers, which are arisen from p-adic fermionic integrals and p-adic invariant integrals, respectively. We give some explicit identities and properties for those twisted numbers and polynomials by using p-adic integrals or generating functions.

Abstract In the paper, the authors establish some inequalities for generalized eigenvalues of perturbation problems on Hermitian matrices and modify shortcomings of some known inequalities for generalized eigenvalues in the related literature.

In this paper, the author considers twisted q-analogues of Catalan-Daehee numbers and polynomials by using p-adic q-integral on Zp. We derive some explicit identities for those twisted numbers and polynomials related to various special numbers and polynomials.

In this paper, we consider a new type of degenerate derangement polynomial and number, which shall be called the degenerate derangement polynomials and numbers of the second kind. These concepts are motivated by Kim et al.’s work on degenerate derangement polynomials and numbers. We investigate some properties of these new degenerate derangement polynomials and numbers and explore their connections with the degenerate gamma distributions for the case λ∈(−1,0). In more detail, we derive their explicit expressions, recurrence relations, and some identities involving our degenerate derangement polynomials and numbers and other special polynomials and numbers, which include the fully degenerate Bell polynomials, the degenerate Fubini polynomials, and the degenerate Stirling numbers of the first and the second kinds. We also show that those polynomials and numbers are connected with the moments of some variants of the degenerate gamma distributions. Moreover, we compare the degenerate derangement polynomials and numbers of the second kind to those of Kim et al.

Abstract In the paper, the author considers the Fourier series related to higher-order Daehee and Changhee functions and establishes some new identities for higher-order Daehee and Changhee functions.

In the paper, by virtue of the p-adic invariant integral on Z p , the authors consider a type 2 w-Daehee polynomials and present some properties and identities of these polynomials related with well-known special polynomials. In addition, we present some symmetric identities involving the higher order type 2 w-Daehee polynomials. These identities extend and generalize some known results.

In the paper, the authors present some symmetric identities involving the Stirling polynomials and higher order Bernoulli polynomials under all permutations in the finite symmetric group of degree n. These identities extend and generalize some known results.

In recent decades, many researchers have investigated the ecological models with three and more species to understand complex dynamical behaviors of ecological systems in nature. However, when they studied the models with three species, they have just considered the functional responses between prey and mid-predator and between mid-predator and top predator as the same type. However, in the paper, in order to describe more realistic ecological world, a three-species food chain system with two types of functional response, Holling type and Beddington-DeAngelis type, is considered. It is shown that this system is dissipative. Also, the local and global stability of equilibrium points of the system is established. In addition, conditions for the persistence of the system are found according to the existence of limit cycles. Some numerical examples are given to substantiate our theoretical results. Moreover, we provide numerical evidence of the existence of chaotic phenomena by illustrating bifurcation diagrams of system and by calculating the largest Lyapunov exponent.

In the paper, with the aid of the series expansions of the square or cubic of the arcsine function, the authors establish several possibly new combinatorial identities containing the ratio of two central binomial coefficients which are related to the Catalan numbers in combinatorial number theory.