Recent zbMATH articles in MSC 40https://zbmath.org/atom/cc/402024-09-27T17:47:02.548271ZWerkzeugOn irrationality criteria for the Ramanujan summation of certain serieshttps://zbmath.org/1541.110702024-09-27T17:47:02.548271Z"Khurana, Suraj Singh"https://zbmath.org/authors/?q=ai:khurana.suraj-singhThis paper deals with irrationality criteria for the Ramanujan summation of certain series; see [\textit{B. Candelpergher}, Ramanujan summation of divergent series. Cham: Springer (2017; Zbl 1386.40001)] for the definition and properties of Riemann summation. The main result is a sufficient condition for the Ramanujan summation of a series of the form
\[
\sum_{k\geq 1} \frac{ \prod_{i=1}^{m-1} | k+\alpha_i| ^{2-1/w}}{ \prod_{i=1}^{m} | k+\beta_i| ^{2-1/w}}
\]
to be irrational, with \(1/2 <w < 1\). In a special case, this shows that for \(0<\theta<1\), if the fractional part of \(B_{\theta,n}-A_{\theta,n}\) is greater than \(8^{-n}\) for infinitely many \(n\), then \(\zeta(\theta)\) is irrational; here \(\zeta\) is the Riemann zeta function and \(A_{\theta,n}\), \(B_{\theta,n}\) are finite sums defined explicitly.
These irrationality criteria are related to the one of \textit{J. Sondow} [Proc. Am. Math. Soc. 131, No. 11, 3335--3344 (2003; Zbl 1113.11040)] concerning Euler's constant, which is the Riemann summation of the harmonic series.
Reviewer: Stéphane Fischler (Paris)Extensions of Yamamoto-Nayak's theoremhttps://zbmath.org/1541.150202024-09-27T17:47:02.548271Z"Huang, Huajun"https://zbmath.org/authors/?q=ai:huang.huajun"Tam, Tin-Yau"https://zbmath.org/authors/?q=ai:tam.tin-yauA result of \textit{S. Nayak} [Linear Algebra Appl. 679, 231--245 (2023; Zbl 1529.15007)] asserts that \(\lim_{m\to \infty}|A^m|^{1/m}\) exists for each square complex matrix \(A\), where \(|A| = (A^*A)^{1/2}\). This extends the famous Beruling-Gelfand's spectral radius formula and its generalization by \textit{T. Yamamoto} [J. Math. Soc. Japan 19, 173--178 (1967; Zbl 0152.01404)]. The paper under review extends, with a different proof, the result of S. Nayak [loc. cit.] by proving \(\lim_{m\to \infty}|BA^mC|^{1/m}\) exists for any square complex matrices \(A, B\), and \(C\), where \(B\) and \(C\) are nonsingular. Extensions in the context of real semisimple Lie group are also given.
Reviewer: Minghua Lin (Xi'an)Regularized limit, analytic continuation and finite-part integrationhttps://zbmath.org/1541.300012024-09-27T17:47:02.548271Z"Galapon, Eric A."https://zbmath.org/authors/?q=ai:galapon.eric-aSummary: Finite-part integration is a recent method of evaluating a convergent integral in terms of the finite-parts of divergent integrals deliberately induced from the convergent integral itself [\textit{E. A. Galapon}, Proc. R. Soc. Lond., A, Math. Phys. Eng. Sci. 473, No. 2197, Article ID 20160567, 18 p. (2017; Zbl 1404.26013)]. Within the context of finite-part integration of the Stieltjes transform of functions with logarithmic growths at the origin, the relationship is established between the analytic continuation of the Mellin transform and the finite-part of the resulting divergent integral when the Mellin integral is extended beyond its strip of analyticity. It is settled that the analytic continuation and the finite-part integral coincide at the regular points of the analytic continuation. To establish the connection between the two at the isolated singularities of the analytic continuation, the concept of regularized limit is introduced to replace the usual concept of limit due to Cauchy when the later leads to a division by zero. It is then shown that the regularized limit of the analytic continuation at its isolated singularities equals the finite-part integrals at the singularities themselves. The treatment gives the exact evaluation of the Stieltjes transform in terms of finite-part integrals and yields the dominant asymptotic behavior of the transform for arbitrarily small values of the parameter in the presence of arbitrary logarithmic singularities at the origin.Class of bounds of the generalized Volterra functionshttps://zbmath.org/1541.330122024-09-27T17:47:02.548271Z"Mehrez, Khaled"https://zbmath.org/authors/?q=ai:mehrez.khaled"Brahim, Kamel"https://zbmath.org/authors/?q=ai:brahim.kamel"Sitnik, Sergei M."https://zbmath.org/authors/?q=ai:sitnik.sergei-mihailovichSummary: In the present paper, we prove the monotonicity property of the ratios of the generalized Volterra function. As consequences, new and interesting monotonicity concerning ratios of the exponential integral function, as well as it yields some new functional inequalities including Turán-type inequalities. Moreover, two-side bounding inequalities are then obtained for the generalized Volterra function. The main mathematical tools are some integral inequalities. As applications, a few of upper and lower bound inequalities for the exponential integral function are derived. The various results, which are established in this paper, are presumably new, and their importance is illustrated by several interesting consequences and examples accompanied by graphical representations to substantiate the accuracy of the obtained results. Some potential directions for analogous further research on the subject of the present investigation are indicated in the concluding section.On the summability and convergence of formal solutions of linear \(q\)-difference-differential equations with constant coefficientshttps://zbmath.org/1541.351282024-09-27T17:47:02.548271Z"Ichinobe, Kunio"https://zbmath.org/authors/?q=ai:ichinobe.kunio"Michalik, Sławomir"https://zbmath.org/authors/?q=ai:michalik.slawomirIn this article, the authors are interested in the Cauchy problem for homogeneous linear \(q\)-difference-differential equations of the form
\[
\begin{cases}
P(D_{q,t},\partial_z)u=0\\
D_{q,t}^ju(0,z)=\varphi_j(z)\text{ for }j=0,\dots,p-1
\end{cases}
\]
where \(P(D_{q,t},\partial_z)\) is a general linear \(q\)-difference-differential operator with constant coefficients of order \(p\) with respect to the \(q\)-difference operator \(D_{q,t}\) defined by \[D_{q,t}u(t,z)=\dfrac{u(qt,z)-u(t,z)}{qt-t},\quad q\in[0,1[,\] and where the initial data \(\varphi_j(z)\) are holomorphic functions in a neighborhood of the origin \(0\in\mathbb{C}\) for all \(j=0,\dots,p-1\).
They characterize convergent, \(k\)-summable and multisummable formal power series solutions in terms of analytic continuation properties and growth estimates of the initial data \(\varphi_j(z)\). They also introduce and characterize sequences preserving summability, which make a very useful tool, especially in the context of moment differential equations.
Reviewer: Pascal Remy (Carrières-sur-Seine)Retracted article: ``On weighted generator of triple sequences and its Tauberian conditions''https://zbmath.org/1541.400012024-09-27T17:47:02.548271Z"Jan, Asif Hussain"https://zbmath.org/authors/?q=ai:jan.asif-hussain"Jalal, Tanweer"https://zbmath.org/authors/?q=ai:jalal.tanweerEditors' remark.
The Editors have retracted this Article because of concerns regarding the originality and novelty of this work.
An investigation conducted after its publication confirmed that it contains material that substantially overlaps with [\textit{Z.~Önder} et al., Adv. Oper. Theory 8, No.~3, Paper No.~38, 23~p. (2023; Zbl 1530.40002)]. Furthermore, the authors failed to even cite [loc. cit.] in their text. These are clear and egregious violations of the Journal ethics policy.Duelling idiots and Abel sumshttps://zbmath.org/1541.400022024-09-27T17:47:02.548271Z"Matis, Anton"https://zbmath.org/authors/?q=ai:matis.anton"Slavík, Antonín"https://zbmath.org/authors/?q=ai:slavik.antoninSummary: We investigate a puzzle involving the winning probabilities in a duel of two players. The problem of calculating limiting probabilities leads to the summation of a divergent infinite series. The solution admits a generalization that applies to a wide class of duels.Series expansion, asymptotic behavior and computation of the values of the Schwarzschild-Milne integrals arising in a radiative transferhttps://zbmath.org/1541.400032024-09-27T17:47:02.548271Z"Aygar, Y."https://zbmath.org/authors/?q=ai:aygar.yelda"Bairamov, E."https://zbmath.org/authors/?q=ai:bairamov.elgizSummary: In this paper, we investigate the uniform convergence and series expansion of the Schwarzschild-Milne (SM) integrals arising in the study of radiative transfer in a two-dimensional planar-medium. Using the uniform convergence, we study asymptotic behavior of the SM integrals. Hence, knowing the interval of uniform convergency is important for an effective computation, we also give an accurate and efficient algorithm for computing the values of the SM integrals.Some Knopp's core type theorems via idealshttps://zbmath.org/1541.400042024-09-27T17:47:02.548271Z"Edely, O. H."https://zbmath.org/authors/?q=ai:edely.osama-h-h"Mursaleen, M."https://zbmath.org/authors/?q=ai:mursaleen.mohammadSummary: In this paper, we characterize the matrix class \((\mathcal{I}_c \cap l_\infty, \mathcal{I}_c \cap l_\infty)_{reg}\), where \(\mathcal{I}_c\) is the space of all ideal convergent sequences and \(l_\infty\) denotes the space of all bounded sequences. We use this class to establish some core theorems analogous to Knopp's core theorem [\textit{K. Knopp}, Math. Z. 31, 97--127 (1929; JFM 55.0730.07)].On asymptotically lacunary statistical equivalent of order \(\widetilde{\alpha}\) of difference double sequenceshttps://zbmath.org/1541.400052024-09-27T17:47:02.548271Z"Et, Mikail"https://zbmath.org/authors/?q=ai:et.mikail"Kandemir, Hacer Şengül"https://zbmath.org/authors/?q=ai:kandemir.hacer-sengul"Çinar, Muhammed"https://zbmath.org/authors/?q=ai:cinar.muhammedSummary: In this article, we introduce and investigate the concepts of \(\Delta_\theta^m\)-asymptotically statistical equivalent of order \(\widetilde{\alpha}\) and strong \(\Delta_\theta^m\)-asymptotically equivalent of order \(\widetilde{\alpha}\) of double sequences. Also, we give some relationships related to these concepts.
{{\copyright} 2021 John Wiley \& Sons, Ltd.}Rates of asymptotically statistical equivalents of measurable functionshttps://zbmath.org/1541.400062024-09-27T17:47:02.548271Z"Savaş, Rabia"https://zbmath.org/authors/?q=ai:savas.rabiaSummary: The primary goal of this article is to present the concepts of asymptotically equivalent function and asymptotic regular function transformations. Moreover, by using these definitions, we examine the bivariate function transformation of asymptotically statistical equivalent measurable functions.
{{\copyright} 2021 John Wiley \& Sons, Ltd.}On asymptotically statistical equivalent functions on time scaleshttps://zbmath.org/1541.400072024-09-27T17:47:02.548271Z"Sözbir, Bayram"https://zbmath.org/authors/?q=ai:sozbir.bayram"Altundağ, Selma"https://zbmath.org/authors/?q=ai:altundag.selmaSummary: In this paper, we introduce the concepts of asymptotically $f$-statistical equivalence, asymptotically $f$-lacunary statistical equivalence, and strong asymptotically $f$-lacunary equivalence for non-negative two delta measurable real-valued functions defined on time scales with the aid of modulus function $f$. Furthermore, the relationships between these new concepts are investigated. We also present some inclusion theorems.
{{\copyright} 2021 John Wiley \& Sons, Ltd.}Some interesting properties of the matrix classes \((\ell,c)\) and \((\ell,c;P')\)https://zbmath.org/1541.400082024-09-27T17:47:02.548271Z"Natarajan, P. N."https://zbmath.org/authors/?q=ai:natarajan.pinnangudi-narayanasubramanianSummary: In this paper, entries of sequences and infinite matrices are real or complex numbers. We prove some nice properties of the infinite matrix classes \((\ell,c)\) and \((\ell,c;P')\).Characterization of the matrix class \((l_\alpha,l_\beta)\), \(0<\alpha\le\beta\), over complete ultrametric fieldshttps://zbmath.org/1541.400092024-09-27T17:47:02.548271Z"Natarajan, P. N."https://zbmath.org/authors/?q=ai:natarajan.pinnangudi-narayanasubramanianSummary: Throughout the present paper, \(K\) denotes a complete, non-trivially valued, ultrametric (or non-Archimedean) field. Entries of sequences, infinite series and infinite matrices are in \(K\). In this paper, we characterize the matrix class \((\ell_\alpha,\ell_\beta)\), \(0<\alpha\le\beta\).A novel approach to evaluating improper integralshttps://zbmath.org/1541.440052024-09-27T17:47:02.548271Z"Gordon, Russell A."https://zbmath.org/authors/?q=ai:gordon.russell-a"Stewart, Séan M."https://zbmath.org/authors/?q=ai:stewart.sean-markSummary: We explain and apply a recently developed method for evaluating improper integrals of the form \(\int^\infty_0 f(t) dt\) using Laplace transforms. A number of examples are provided to illustrate the method, along with some results that streamline the computations. We show how the method can be used to readily determine values for entire classes of certain integrals which, using other more familiar methods, are difficult to find. We also indicate how the method can determine the values of integrals for which other methods fail.Erratum to: ``Triple series evaluated in \(\pi\) and \(\ln 2\) as well as Catalan's constant \(G\)''https://zbmath.org/1541.810432024-09-27T17:47:02.548271Z"Li, Chunli"https://zbmath.org/authors/?q=ai:li.chunli"Chu, Wenchang"https://zbmath.org/authors/?q=ai:chu.wenchangErratum to the authors' paper [ibid. 63, No. 11, 2005--2023 (2023; Zbl 1536.81037)] .Row-column duality and combinatorial topological stringshttps://zbmath.org/1541.811392024-09-27T17:47:02.548271Z"Padellaro, Adrian"https://zbmath.org/authors/?q=ai:padellaro.adrian"Radhakrishnan, Rajath"https://zbmath.org/authors/?q=ai:radhakrishnan.rajath"Ramgoolam, Sanjaye"https://zbmath.org/authors/?q=ai:ramgoolam.sanjayeSummary: Integrality properties of partial sums over irreducible representations, along columns of character tables of finite groups, were recently derived using combinatorial topological string theories (CTST). These CTST were based on Dijkgraaf-Witten theories of flat \(G\)-bundles for finite groups \(G\) in two dimensions, denoted \(G\)-TQFTs. We define analogous combinatorial topological strings related to two dimensional topological field theories (TQFTs) based on fusion coefficients of finite groups. These TQFTs are denoted as \(R(G)\)-TQFTs and allow analogous integrality results to be derived for partial row sums of characters over conjugacy classes along fixed rows. This relation between the \(G\)-TQFTs and \(R(G)\)-TQFTs defines a row-column duality for character tables, which provides a physical framework for exploring the mathematical analogies between rows and columns of character tables. These constructive proofs of integrality are complemented with the proof of similar and complementary results using the more traditional Galois theoretic framework for integrality properties of character tables. The partial row and column sums are used to define generalised partitions of the integer row and column sums, which are of interest in combinatorial representation theory.
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