scholarly journals The Hermite polynomials and the Bessel functions from a general point of view

2003 ◽  
Vol 2003 (57) ◽  
pp. 3633-3642 ◽  
Author(s):  
G. Dattoli ◽  
H. M. Srivastava ◽  
D. Sacchetti
Keyword(s):  
Difference Equations ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Bessel Functions ◽  
Hermite Polynomials ◽  
Point Of View ◽  
General Point ◽  
Ordinary Case

We introduce new families of Hermite polynomials and of Bessel functions from a point of view involving the use of nonexponential generating functions. We study their relevant recurrence relations and show that they satisfy differential-difference equations which are isospectral to those of the ordinary case. We also indicate the usefulness of some of these new families.

scholarly journals Some Properties of the Hermite Polynomials and Their Squares and Generating Functions

2016 ◽  
Author(s):  
Feng Qi ◽  
Bai-Ni Guo
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Hermite Polynomials ◽  
Higher Order ◽  
Explicit Formulas ◽  
Derivatives Of

In the paper, the authors consider the generating functions of the Hermite polynomials and their squares, present explicit formulas for higher order derivatives of the generating functions of the Hermite polynomials and their squares, which can be viewed as ordinary differential equations or derivative polynomials, find differential equations that the generating functions of the Hermite polynomials and their squares satisfy, and derive explicit formulas and recurrence relations for the Hermite polynomials and their squares.

Some properties of the Hermite polynomials

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Feng Qi ◽  
Bai-Ni Guo
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Hermite Polynomials ◽  
Higher Order ◽  
Bell Polynomials ◽  
Explicit Formulas ◽  
Faà Di Bruno Formula ◽  
Derivatives Of

Abstract In this paper, by the Faà di Bruno formula and properties of Bell polynomials of the second kind, the authors reconsider the generating functions of Hermite polynomials and their squares, find an explicit formula for higher-order derivatives of the generating function of Hermite polynomials, and derive explicit formulas and recurrence relations for Hermite polynomials and their squares.

The bessel functions and the hermite polynomials from a unified point of view

2001 ◽  
Vol 80 (3-4) ◽  
pp. 379-384 ◽  
Author(s):  
Giuseppe Dattoli ◽  
Paolo E. Ricci ◽  
Clemente Cesarano
Keyword(s):  
Bessel Functions ◽  
Hermite Polynomials ◽  
Point Of View

New family of Whitney numbers

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 309-320 ◽  
Author(s):  
B.S. El-Desouky ◽  
Nenad Cakic ◽  
F.A. Shiha
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Explicit Formulas ◽  
Basic Properties ◽  
New Family ◽  
Whitney Numbers ◽  
Special Cases

In this paper we give a new family of numbers, called ??-Whitney numbers, which gives generalization of many types of Whitney numbers and Stirling numbers. Some basic properties of these numbers such as recurrence relations, explicit formulas and generating functions are given. Finally many interesting special cases are derived.

scholarly journals Lessons from the COVID-19 Pandemic on the Use of Artificial Intelligence in Digital Radiology: The Submission of a Survey to Investigate the Opinion of Insiders

Healthcare ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 331
Author(s):  
Daniele Giansanti ◽  
Ivano Rossi ◽  
Lisa Monoscalco
Keyword(s):  
Artificial Intelligence ◽  
General Purpose ◽  
Point Of View ◽  
General Point ◽  
Thoracic Radiography ◽  
Digital Radiology ◽  
Electronic Survey ◽  
Radiology Technicians ◽  
Medical Radiology ◽  
Great Development

The development of artificial intelligence (AI) during the COVID-19 pandemic is there for all to see, and has undoubtedly mainly concerned the activities of digital radiology. Nevertheless, the strong perception in the research and clinical application environment is that AI in radiology is like a hammer in search of a nail. Notable developments and opportunities do not seem to be combined, now, in the time of the COVID-19 pandemic, with a stable, effective, and concrete use in clinical routine; the use of AI often seems limited to use in research applications. This study considers the future perceived integration of AI with digital radiology after the COVID-19 pandemic and proposes a methodology that, by means of a wide interaction of the involved actors, allows a positioning exercise for acceptance evaluation using a general purpose electronic survey. The methodology was tested on a first category of professionals, the medical radiology technicians (MRT), and allowed to (i) collect their impressions on the issue in a structured way, and (ii) collect their suggestions and their comments in order to create a specific tool for this professional figure to be used in scientific societies. This study is useful for the stakeholders in the field, and yielded several noteworthy observations, among them (iii) the perception of great development in thoracic radiography and CT, but a loss of opportunity in integration with non-radiological technologies; (iv) the belief that it is appropriate to invest in training and infrastructure dedicated to AI; and (v) the widespread idea that AI can become a strong complementary tool to human activity. From a general point of view, the study is a clear invitation to face the last yard of AI in digital radiology, a last yard that depends a lot on the opinion and the ability to accept these technologies by the operators of digital radiology.

Changes in the Prognosis of Functional Psychoses Since the Days of Kraepelin

1967 ◽  
Vol 113 (501) ◽  
pp. 813-822 ◽  
Author(s):  
Örnulv Ödegård
Keyword(s):  
Special Problem ◽  
Point Of View ◽  
General Point ◽  
Course Of Illness ◽  
Dynamic Reasoning ◽  
Starting Point ◽  
Functional Psychoses ◽  
External Conditions ◽  
Point Of Departure ◽  
The Universe

My choice of Kraepelin as a point of departure for this lecture has definite reasons. If one wants to stay within the field of clinical psychiatry (as opposed to psychiatric history), that is as far back as one can reasonably go. By this no slight is intended upon the pre-Kraepelinian psychiatrists. For our topic Henry Maudsley would indeed have been a most appropriate starting point, and by no means for reasons of courtesy. His general point of view is admirably sound as a basis for the scientific study of prognosis in psychiatry. I quote: “There is no accident in madness. Causality, not casualty, governs its appearance in the universe, and it is very far from being a good and sufficient practice simply to mark its phenomena and straightway to pass on as if they belonged not to an order but to a disorder of events that called for no explanation.” On the special problem of prognosis he shows his clinical acumen by stating that the outlook is poor when the course of illness is insidious, but this only means that these cases develop their psychoses on the basis of mental deviations which go very far back in the patient's life, so that in fact they are generally in a chronic stage at the time of their first admission to hospital. Here he actually corrects a mistake which is still quite often made. He shows his dynamic attitude when he says that prognosis is to a large extent modified by external conditions, in particular by the attitude of friends and relatives. Maudsley's dynamic reasoning was limited by the narrow framework of the degeneration hypothesis of those days. He had a sceptical attitude towards classification, which he regarded as artificial and dangerously pseudo-exact. His own classification was deliberately provisional, with very wide groups. He held that a description of various sub-forms of chronic insanity was useless, as it would mean nothing but a tiresome enumeration of unconnected details.

The solution of an infinite set of differential-difference equations occurring in polymerization and queueing problems

1963 ◽  
Vol 59 (1) ◽  
pp. 117-124 ◽  
Author(s):  
A. Wragg
Keyword(s):  
Difference Equations ◽  
Queueing Theory ◽  
Bessel Functions ◽  
Formal Solution ◽  
Monomer Concentration ◽  
Arrival Rate ◽  
Time Dependent ◽  
Type Integral ◽  
Dependent Parameter ◽  
Infinite Set

AbstractThe time-dependent solutions of an infinite set of differential-difference equations arising from queueing theory and models of ‘living’ polymer are expressed in terms of modified Bessel functions. Explicit solutions are available for constant values of a parameter describing the arrival rate or monomer concentration; for time-dependent parameter a formal solution is obtained in terms of a function which satisfies a Volterra type integral equation of the second kind. These results are used as the basis of a numerical method of solving the infinite set of differential equations when the time-dependent parameter itself satisfies a differential equation.

scholarly journals Recurrence relations for exceptional Hermite polynomials

2016 ◽  
Vol 204 ◽  
pp. 1-16 ◽  
Author(s):  
D. Gómez-Ullate ◽  
A. Kasman ◽  
A.B.J. Kuijlaars ◽  
R. Milson
Keyword(s):  
Recurrence Relations ◽  
Hermite Polynomials

scholarly journals I.—The Metamorphosis of the Lizard Gabbros

1886 ◽  
Vol 3 (11) ◽  
pp. 481-489 ◽  
Author(s):  
J. J. H. Teall
Keyword(s):  
Mechanical Energy ◽  
Earth Pressure ◽  
Careful Consideration ◽  
Point Of View ◽  
Igneous Rocks ◽  
General Point ◽  
Present Moment ◽  
General View ◽  
Original Rock ◽  
The Subject

If we take a general view of the present position of geological science, we are struck by the fact that, although there is substantial agreement amongst geologists on matters relating to the origin of the rocks usually designated as aqueous and igneous, the greatest diversity of opinion prevails with regard to the circumstances under which the so-called metamorphic rocks have been produced. Every fragment of evidence calculated to throw light on the origin of these rocks, therefore, deserves the most careful consideration. Of recent years special attention has been directed to the effects of mechanical energy in modifying the mineralogical and structural characters of rocks originally formed by aqueous and igneous agencies; and a suspicion has been aroused that it is in this direction that we must look for a solution of many of the problems connected with the origin of the crystalline schists. A visit to the Lizard Peninsula of Cornwall during the present summer has convinced me of the immense importance of this view so far as that district is concerned. That portion of the peninsula which lies south of a line drawn from Porthalla on the east to Polurrian Cove on the west is formed.partly of igneous rocks—such as gabbro, greenstone, serpentine, and granite—and partly of crystalline schists. The igneous rocks, in certain places, become foliated and sohistose and sometimes show a definite banding due to a variation in the relative proportions of the different constituents. In other words they present characters which are usually regarded as distinctive of the crystalline schists. There is, moreover, evidence to show that these characters are mainly the result of a yielding to earth-pressure subsequent to the consolidation of the original rock. At the present moment, having just returned from the district, I am unable to treat the subject from a general point of view with any prospect of success; but it has occurred to me that some details with regard to one of the rocks may not be without interest to members of the Association.

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