scholarly journals Some Properties and Generating Functions of Generalized Harmonic Numbers

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 577 ◽  
Author(s):  
Giuseppe Dattoli ◽  
Silvia Licciardi ◽  
Elio Sabia ◽  
Hari M. Srivastava
Keyword(s):  
Generating Functions ◽  
Integral Transforms ◽  
Higher Order ◽  
Harmonic Numbers ◽  
Efficient Tool ◽  
Type Method ◽  
The Subject ◽  
Generalized Harmonic Numbers

In this paper, we introduce higher-order harmonic numbers and derive their relevant properties and generating functions by using an umbral-type method. We discuss the link with recent works on the subject, and show that the combinations of umbral and other techniques (such as the Laplace and other types of integral transforms) yield a very efficient tool to explore the properties of these numbers.

HANKEL DETERMINANTS OF FACTORIAL FRACTIONS

2021 ◽  
pp. 1-12
Author(s):  
WENCHANG CHU
Keyword(s):  
Integral Transforms ◽  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Hankel Determinant ◽  
Harmonic Numbers ◽  
Expansion Formula ◽  
Hankel Determinants ◽  
Laplace Expansion ◽  
Generalized Harmonic Numbers

Abstract By making use of the Cauchy double alternant and the Laplace expansion formula, we establish two closed formulae for the determinants of factorial fractions that are then utilised to evaluate several determinants of binomial coefficients and Catalan numbers, including those obtained recently by Chammam [‘Generalized harmonic numbers, Jacobi numbers and a Hankel determinant evaluation’, Integral Transforms Spec. Funct.30(7) (2019), 581–593].

On 3-adic Valuations of Generalized Harmonic Numbers

Integers ◽  
2012 ◽  
Vol 12 (2) ◽  
Author(s):  
Ken Kamano
Keyword(s):  
Harmonic Numbers ◽  
Generalized Harmonic Numbers

Abstract.We investigate 3-adic valuations of generalized harmonic numbers

scholarly journals Convolution and deconvolution: two mathematical tools to help performing tests in research and industry

2021 ◽  
Vol 12 ◽  
pp. 6
Author(s):  
Jean-Pierre Fanton
Keyword(s):  
Experimental Data ◽  
Inverse Problem ◽  
Electrical Energy ◽  
Integral Transforms ◽  
Physical Measurement ◽  
Photographic Images ◽  
Test Standards ◽  
Energy Networks ◽  
The Third ◽  
The Subject

The concepts of convolution and deconvolution are well known in the field of physical measurement. In particular, they are of interest in the field of metrology, since they can positively influence the performance of the measurement. Numerous mathematical models and computer developments dedicated to convolution and deconvolution have emerged, enabling a more efficient use of experimental data; this in sectors as different as biology, astronomy, manufacturing and energy industries. The subject finds today a new topicality because it has been made accessible to a large public for applications such as processing photographic images. The purpose of this paper is to take into account some recent evolutions such as the introduction of convolution methods in international test standards. Thus, its first part delivers a few reminders of some associated definitions. They concern linear systems properties, and integral transforms. If convolution, in most cases, does not create major calculation problems, deconvolution on the contrary is an inverse problem, and as such needs more attention. The principles of some of the methods available today are exposed. In the third part, illustrations are given on recent examples of applications, belonging to the domain of electrical energy networks and photographic enhancement.

Adverbs

2018 ◽  
Author(s):  
James Higginbotham
Keyword(s):  
Thirteenth Century ◽  
Logical Form ◽  
Semantic Theory ◽  
Higher Order ◽  
Order Logic ◽  
Philosophical Logic ◽  
Language Understanding ◽  
Higher Order Logic ◽  
The Subject

Adverbs are so named from their role in modifying verbs and other non-nominal expressions. For example, in ‘John ran slowly’, the adverb ‘slowly’ modifies ‘ran’ by characterizing the manner of John’s running. The debate on the semantic contribution of adverbs centres on two approaches. On the first approach, adverbs are understood as predicate operators: for example, in ‘John ran slowly’, ‘ran’ would be taken to be a predicate and ‘slowly’ an operator affecting its meaning. Working this out in detail requires the resources of higher-order logic. On the second approach, adverbs are understood as predicates of ‘objects’ such as events and states, reference to which is revealed in logical form. For example, ‘John ran slowly’ would be construed along the lines of ‘there was a running by John and it was slow’, in which the adverb ‘slowly’ has become a predicate ‘slow’ applied to the event that was John’s running. Since adverbs are exclusively modifiers, they are classed among the syncategorematic words of terminist logic, the investigation of which carried the subject forward from Aristotle in the thirteenth century. (The contrasting ‘categoremata’ – grammatical subjects and predicates – are those words which have meaning independently.) They are of contemporary interest for philosophical logic and semantic theory, because particular accounts of them carry implications for the nature of combinatorial semantics and language understanding, and for ontology.

scholarly journals On a Reduced Cost Higher Order Traub-Steffensen-Like Method for Nonlinear Systems

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 891 ◽  
Author(s):  
Janak Raj Sharma ◽  
Deepak Kumar ◽  
Lorentz Jäntschi
Keyword(s):  
Order Of Convergence ◽  
Higher Order ◽  
New Method ◽  
Systems Of Nonlinear Equations ◽  
Third Order ◽  
Type Method ◽  
Cpu Time ◽  
Chebyshev’S Method ◽  
Derivative Free ◽  
Fifth Order

We propose a derivative-free iterative method with fifth order of convergence for solving systems of nonlinear equations. The scheme is composed of three steps, of which the first two steps are that of third order Traub-Steffensen-type method and the last is derivative-free modification of Chebyshev’s method. Computational efficiency is examined and comparison between the efficiencies of presented technique with existing techniques is performed. It is proved that, in general, the new method is more efficient. Numerical problems, including those resulting from practical problems viz. integral equations and boundary value problems, are considered to compare the performance of the proposed method with existing methods. Calculation of computational order of convergence shows that the order of convergence of the new method is preserved in all the numerical examples, which is not so in the case of some of the existing higher order methods. Moreover, the numerical results, including the CPU-time consumed in the execution of program, confirm the accurate and efficient behavior of the new technique.

scholarly journals On some combinatorial identities and harmonic sums

2017 ◽  
Vol 13 (07) ◽  
pp. 1695-1709 ◽  
Author(s):  
Necdet Batir
Keyword(s):  
Integral Representation ◽  
Closed Form ◽  
Generating Function ◽  
Combinatorial Identities ◽  
Harmonic Numbers ◽  
Generalized Harmonic Numbers

For any [Formula: see text] we first give new proofs for the following well-known combinatorial identities [Formula: see text] and [Formula: see text] and then we produce the generating function and an integral representation for [Formula: see text]. Using them we evaluate many interesting finite and infinite harmonic sums in closed form. For example, we show that [Formula: see text] and [Formula: see text] where [Formula: see text] are generalized harmonic numbers defined below.

scholarly journals Quasi-likelihood-based higher-order spectral estimation of random fields with possible long-range dependence

2004 ◽  
Vol 41 (A) ◽  
pp. 35-53 ◽  
Author(s):  
V. V. Anh ◽  
N. N. Leonenko ◽  
L. M. Sakhno
Keyword(s):  
Long Range ◽  
Random Fields ◽  
Spectral Estimation ◽  
Kernel Estimation ◽  
Mathematical Finance ◽  
Higher Order ◽  
Long Range Dependence ◽  
Type Method ◽  
Minimum Contrast ◽  
Consistency And Asymptotic Normality

This paper provides a quasi-likelihood or minimum-contrast-type method for the parameter estimation of random fields in the frequency domain based on higher-order information. The estimation technique uses the spectral density of the general kth order and allows for possible long-range dependence in the random fields. To avoid bias due to edge effects, data tapering is incorporated into the method. The suggested minimum contrast functional is linear with respect to the periodogram of kth order, hence kernel estimation for the spectral densities is not needed. Furthermore, discretization is not required in the estimation of continuously observed random fields. The consistency and asymptotic normality of the resulting estimators are established. Illustrative applications of the method to some problems in mathematical finance and signal detection are given.

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