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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, by Descartes' rule of signs and other techniques, the authors present the negativity, zeros, and extreme values of the single-variable polynomials \(\begin{align*} G(t)&=5 t^{43}-218 t^{30}+720 t^{17}-455 t^{13}-52,\\ H(t)&=5t^{29}\sum_{\ell=0}^{12}(13-\ell)t^\ell-t^{16}\sum_{\ell=0}^{12}(2704-213 \ell)t^{\ell}\\ &\quad-169t^{13}\sum_{\ell=0}^{2} (7+3\ell) t^\ell-52\sum_{\ell=0}^{12}(\ell+1)t^{\ell},\\ J(t)&=43 t^{30}-1308 t^{17}+2448 t^4-1183,\\ K(t)&=43t^{17}\sum_{k=0}^{12}t^k-1265t^4\sum_{k=0}^{12}t^{k} +1183\sum_{k=0}^{3}t^k. \end{align*}\)

By virtue of two approaches, the authors prove that the function \(G(t)=5 t^{43}-218 t^{30}+720 t^{17}-455 t^{13}-52\) is negative on \([0,1)\) and \(G(1)=0\).

In this paper, our aim is to provide as many as one hundred and ninety-two summation formulas for the generalized hypergeometric series F 6 5 − 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 article, the authors present two identities involving products of the Bernoulli numbers, provide four alternative proofs for these two identities, derive two closed-form formulas for the Bernoulli numbers in terms of central factorial numbers of the second kind, and supply simple proofs of series expansions of (hyperbolic) cosecant and cotangent functions.

In the paper, the authors find series expansions and identities for positive integer powers of inverse (hyperbolic) sine and tangent, for composite of incomplete gamma function with inverse hyperbolic sine, in terms of the first kind Stirling numbers, apply a newly established series expansion to derive a closed-form formula for specific partial Bell polynomials and to derive a series representation of generalized logsine function, and deduce combinatorial identities involving the first kind Stirling numbers.

In this paper, the authors study the poly-Bernoulli numbers of the second kind, which are defined by using polyexponential functions introduced by Kims. Also by using unipoly function, we study the unipoly Bernoulli numbers of the second kind, which are attached to an arithmetic function. We derive their explicit expressions and some identities involving poly-Bernoulli numbers of the second kind and unipoly Bernoulli numbers of the second kind.

In this paper, with the aid of the Faà di Bruno formula and by virtue of properties of the Bell polynomials of the second kind, the authors define a kind of notion of degenerate Narumi numbers and polynomials, establish explicit formulas for degenerate Narumi numbers and polynomials, and derive explicit formulas for the Narumi numbers and polynomials and for (degenerate) Cauchy numbers.

In the paper, the authors establish Maclaurin's series expansions and series identities for positive integer powers of the inverse sine function, for positive integer powers of the inverse hyperbolic sine function, for the composite of incomplete gamma functions with the inverse hyperbolic sine function, for positive integer powers of the inverse tangent function, and for positive integer powers of the inverse hyperbolic tangent function, in terms of the first kind Stirling numbers and binomial coefficients, apply the newly established Maclaurin's series expansion for positive integer powers of the inverse sine function to derive a closed-form formula for specific values of partial Bell polynomials and to derive a series representation of the generalized logsine function, and deduce several combinatorial identities involving the first kind Stirling numbers. Some of these results simplify and unify some known ones. All of these newly established Maclaurin's series expansions of positive integer powers of the inverse (hyperbolic) sine and tangent functions can be used to derive infinite series representations of the circular constant Pi and of positive integer powers of Pi.

International audience; In the paper, the authors introduce a notion "multivariate exponential polynomials" which generalize exponential numbers and polynomials, establish explicit formulas, inversion formulas, and recurrence relations for multivariate exponential polynomials in terms of the Stirling numbers of the first and second kinds with the help of the Fàa di Bruno formula, two identities for the Bell polynomials of the second kind, and the inversion theorem for the Stirling numbers of the first and second kinds, construct some determinantal inequalities and product inequalities for multivariate exponential polynomials with the aid of some properties of completely monotonic functions and other known results, derive the logarithmic convexity and logarithmic concavity for multivariate exponential polynomials, and finally find an application of multi-variate exponential polynomials to white noise distribution theory by confirming that multivariate exponential polynomials satisfy conditions for sequences required in white noise distribution theory.

We derive two explicit formulas for two sequences of special values of the Bell polynomials of the second kind in terms of associate Stirling numbers of the second kind, give an explicit formula for associate Stirling numbers of the second kind in terms of the Stirling numbers of the second kind, and, consequently, present two explicit formulas for two sequences of special values of the Bell polynomials of the second kind in terms of the Stirling numbers of the second kind.

Araci et al. introduced a $p$-adic $(\rho,q)$-analogue of the Haar distribution. By means of the distribution, they constructed the $p$-adic $(\rho,q)$-Volkenborn integral. In this paper, by virtue of the Mahler expansion of continuous functions, the author gives the Radon-Nikodym theorem with respect to the $p$-adic $(\rho,q)$-distribution on $\Bbb Z_p$.

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.

본 사례 연구의 목적은 변화율을 포함하는 상황을 그래프로 나타내는 과정에서 드러나는 두 중학생 간의 ‘변화율의 일정함’과 ‘변화율의 일정한 변화’에 대한 이해의 차이를 비교 및 분석하는 것이다. 이를 위해 학교수학에서 일차함수를 학습한 경험이 없는 중학생 2명을 대상으로 2018년 4월부터 6월까지 총 6회(12차시) 수업을 실시하였다. 수업을 분석하는 과정에서 변화율이 음수이거나 변화율이 일정한 비율로 변하는 상황에서 학생 S만이 상황에 적절한 그래프를 나타내고 해석하였다. 이에 변화율을 포함하는 과제의 해결에서 나타나는 두 학생 간의 차이와 그 원인을 분석하였고, 그 결과는 다음과 같다. 1) 직선 모양과 곡선 모양인 그래프를 ‘변화율의 일정함’의 여부로 설명하더라도 학생들이 주목하는 양들 사이의 관계와 그 변화에 대한 인식은 서로 다를 수 있으며, 2) 이러한 차이는 일정한 변화율이 음수인 상황뿐 아니라 변화율이 일정한 비율로 변하는 상황을 이해하는 과정에서도 유사하게 드러남을 확인하였다.

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.

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., Comment: 9 pages

International audience; In the paper, by the aid of mathematical induction and some properties of determinants, the authors present several explicit and recurrent formulas of evaluations for determinants of general tridiagonal matrices in terms of finite generalized continued fractions and apply these newly-established formulas to evaluations for determinants of the Sylvester matrix and two Sylvester type matrices.

International audience; In the paper, the authors extend a function arising from the Bernoulli trials in probability and involving the gamma function to its largest ranges, find logarithmically complete monotonicity of these extended functions, and, in light of logarithmically complete monotonicity of these extended functions, derive some inequalities for multinomial coefficients and multivariate beta functions. These results recover, extend, and generalize some known conclusions.

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.

Motivated by Carlitz’s q-Bernoulli polynomials, properties and identities are given for a new type of polynomial, Carlitz type q-Daehee polynomials. Higher order Carlitz type q-Deahee polynomials are given in terms of p-adic q-integrals and symmetric identities from the symmetric group of degree 3 are given.”

In the paper, by virtue of induction and properties of determinants, the authors discover explicit and recurrent formulas of evaluations for determinants of general tridiagonal matrices in terms of finite generalized continued fractions and apply these formulas to evaluations for determinants of the Sylvester matrix and two Sylvester type matrices.; In the paper, by virtue of induction and properties of determinants, the authors discover explicit and recurrent formulas of evaluations for determinants of general tridiagonal matrices in terms of finite generalized continued fractions and apply these formulas to evaluations for determinants of the Sylvester matrix and two Sylvester type matrices.

In the paper, the authors survey integral representations (including the Levy–Khintchine representations) and applications of some bivariate means (including the logarithmic mean, the identric mean, Stolarsky’s mean, the harmonic mean, the (weighted) geometric means and their reciprocals, and the Toader–Qi mean) and the multivariate (weighted) geometric means and their reciprocals, derive integral representations of bivariate complex geometric mean and its reciprocal, and apply these newly-derived integral representations to establish integral representations of Heronian mean of power 2 and its reciprocal.