scholarly journals Spherical curves and quadratic relationships for special functions

1985 ◽  
Vol 27 (1) ◽  
pp. 111-124
Author(s):  
Even Mehlum ◽  
Jet Wimp
Keyword(s):  
Differential Equation ◽  
Hypergeometric Functions ◽  
Bessel Functions ◽  
Special Functions ◽  
Position Vector ◽  
Third Order ◽  
Generalized Hypergeometric Functions ◽  
Transcendental Functions ◽  
Horn Functions ◽  
Vector Differential Equation

AbstractWe show that the position vector of any 3-space curve lying on a sphere satisfies a third-order linear (vector) differential equation whose coefficients involve a single arbitrary function A(s). By making various identifications of A(s), we are led to nonlinear identities for a number of higher transcendental functions: Bessel functions, Horn functions, generalized hypergeometric functions, etc. These can be considered natural geometrical generalizations of sin2t + cos2t = 1. We conclude with some applications to the theory of splines.

scholarly journals Position Vectors of Curves Generalizing General Helices and Slant Helices in Euclidean 3-Space

2021 ◽  
Vol 52 ◽  
Author(s):  
Malika Izid ◽  
Abderrazak El Haimi ◽  
Amina Ouazzani Chahdi
Keyword(s):  
Differential Equation ◽  
Parametric Representation ◽  
Position Vector ◽  
Third Order ◽  
The Third ◽  
Slant Helix ◽  
Vector Differential Equation ◽  
Standard Frame

Inthispaper,wegiveanewcharacterizationofak-slanthelixwhichisageneral- ization of general helix and slant helix. Thereafter, we construct a vector differential equation of the third order to determine the parametric representation of a k-slant helix according to standard frame in Euclidean 3-space. Finally, we apply this method to find the position vector of some examples of 2-slant helix by means of intrinsic equations.

scholarly journals A Guide to Special Functions in Fractional Calculus

Mathematics ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 106
Author(s):  
Virginia Kiryakova
Keyword(s):  
Applied Sciences ◽  
Fractional Calculus ◽  
Hypergeometric Functions ◽  
Special Functions ◽  
Generalized Hypergeometric Functions ◽  
Social Events ◽  
Transcendental Functions ◽  
Meijer G Function ◽  
Wright Generalized Hypergeometric Functions

Dedicated to the memory of Professor Richard Askey (1933–2019) and to pay tribute to the Bateman Project. Harry Bateman planned his “shoe-boxes” project (accomplished after his death as Higher Transcendental Functions, Vols. 1–3, 1953–1955, under the editorship by A. Erdélyi) as a “Guide to the Functions”. This inspired the author to use the modified title of the present survey. Most of the standard (classical) Special Functions are representable in terms of the Meijer G-function and, specially, of the generalized hypergeometric functions pFq. These appeared as solutions of differential equations in mathematical physics and other applied sciences that are of integer order, usually of second order. However, recently, mathematical models of fractional order are preferred because they reflect more adequately the nature and various social events, and these needs attracted attention to “new” classes of special functions as their solutions, the so-called Special Functions of Fractional Calculus (SF of FC). Generally, under this notion, we have in mind the Fox H-functions, their most widely used cases of the Wright generalized hypergeometric functions pΨq and, in particular, the Mittag–Leffler type functions, among them the “Queen function of fractional calculus”, the Mittag–Leffler function. These fractional indices/parameters extensions of the classical special functions became an unavoidable tool when fractalized models of phenomena and events are treated. Here, we try to review some of the basic results on the theory of the SF of FC, obtained in the author’s works for more than 30 years, and support the wide spreading and important role of these functions by several examples.

scholarly journals Distributions of the Ratio and Product of Two Independent Weibull and Lindley Random Variables

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
N. J. Hassan ◽  
A. Hawad Nasar ◽  
J. Mahdi Hadad
Keyword(s):  
Hypergeometric Functions ◽  
Special Functions ◽  
Random Variables ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Parabolic Cylinder ◽  
Generalized Hypergeometric Functions ◽  
Confluent Hypergeometric Functions ◽  
Parabolic Cylinder Functions ◽  
The Moment

In this paper, we derive the cumulative distribution functions (CDF) and probability density functions (PDF) of the ratio and product of two independent Weibull and Lindley random variables. The moment generating functions (MGF) and the k-moment are driven from the ratio and product cases. In these derivations, we use some special functions, for instance, generalized hypergeometric functions, confluent hypergeometric functions, and the parabolic cylinder functions. Finally, we draw the PDF and CDF in many values of the parameters.

A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order

1951 ◽  
Vol 47 (4) ◽  
pp. 699-712 ◽  
Author(s):  
F. W. J. Olver
Keyword(s):  
Differential Equation ◽  
Bessel Functions ◽  
Numerical Technique ◽  
Third Order ◽  
New Approach ◽  
The Third ◽  
Zeros Of Bessel Functions ◽  
Zeros Of Functions ◽  
Linear Differential ◽  
Image Position

In a recent paper (1) I described a method for the numerical evaluation of zeros of the Bessel functions Jn(z) and Yn(z), which was independent of computed values of these functions. The essence of the method was to regard the zeros ρ of the cylinder functionas a function of t and to solve numerically the third-order non-linear differential equation satisfied by ρ(t). It has since been successfully used to compute ten-decimal values of jn, s, yn, s, the sth positive zeros* of Jn(z), Yn(z) respectively, in the ranges n = 10 (1) 20, s = 1(1) 20. During the course of this work it was realized that the least satisfactory feature of the new method was the time taken for the evaluation of the first three or four zeros in comparison with that required for the higher zeros; the direct numerical technique for integrating the differential equation satisfied by ρ(t) becomes unwieldy for the small zeros and a different technique (described in the same paper) must be employed. It was also apparent that no mere refinement of the existing methods would remove this defect and that a new approach was required if it was to be eliminated. The outcome has been the development of the method to which the first part (§§ 2–6) of this paper is devoted.

Multi-parametric mittag-leffler functions and their extension

2013 ◽  
Vol 16 (2) ◽  
Author(s):  
Anatoly Kilbas ◽  
Anna Koroleva ◽  
Sergei Rogosin
Keyword(s):  
Fractional Calculus ◽  
Historical Development ◽  
Hypergeometric Functions ◽  
Special Functions ◽  
Integral Representations ◽  
Generalized Hypergeometric Functions ◽  
Late Professor ◽  
Wright Generalized Hypergeometric Functions

AbstractThis paper surveys one of the last contributions by the late Professor Anatoly Kilbas (1948–2010) and research made under his advisorship. We briefly describe the historical development of the theory of the discussed multi-parametric Mittag-Leffler functions as a class of the Wright generalized hypergeometric functions. The method of the Mellin-Barnes integral representations allows us to extend the considered functions to the case of arbitrary values of parameters. Thus, the extended Mittag-Leffler-type functions appear. The properties of these special functions and their relations to the fractional calculus are considered. Our results are based mainly on the properties of the Fox H-functions, as one of the widest class of special functions.

The Stokes phenomenon and generalized contiguous transformations of some generalized hypergeometric functions

1974 ◽  
Vol 76 (2) ◽  
pp. 423-442 ◽  
Author(s):  
J. Heading
Keyword(s):  
Differential Equation ◽  
Lower Order ◽  
Hypergeometric Functions ◽  
Common Factors ◽  
Stokes Phenomenon ◽  
Generalized Hypergeometric Functions ◽  
Complete Cycle ◽  
Independent Variable ◽  
Basic Equations ◽  
Periodic Cycles

AbstractPrevious investigations by the author into the Stokes phenomenon pertaining to solutions of the differential equation dnu/dzn = (–1)nzmu are extended in order to find when different equations have the same set of Stokes multipliers, with perhaps a series of zeros being additionally allowed. The reason for periodic cycles to exist (with n fixed and m varying), with the same Stokes multipliers regained after a complete cycle, is traced to certain transformation properties of the equations. Within the first cycle (with n fixed and m varying) further remarkable identities exist between the Stokes multipliers, and this also is traced to special transformations between the equations. Relations are found toexist when values of m are chosen so that the highest common factors of the two integers n and n/(n + m) are identical. Finally, a transformation of the independent variable is deduced whereby the set of Stokes multipliers for an equation of order n is identical (apart from the additional zeros) to that for an equation of lower order. A hierarchy of equations is thrown up, whereby certain basic equations are transformed to yield more advanced equations of higher order and different m.

scholarly journals Differential representation of the Lorentzian spherical timelike curves by using bishop frame

2019 ◽  
Vol 23 (Suppl. 6) ◽  
pp. 2037-2043
Author(s):  
Okullu Balki ◽  
Huseyin Kocayigit
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Timelike Curve ◽  
Position Vector ◽  
Order Linear Differential Equation ◽  
Third Order ◽  
Order Linear ◽  
Bishop Frame ◽  
Lorentzian Sphere ◽  
Linear Differential

In this study, we will give the differential representation of the Lorentzian spherical timelike curves according to Bishop frame and we obtain a third-order linear differential equation which represents the position vector of a timelike curve lying on a Lorentzian sphere.

scholarly journals Convergence of Solutions to a Certain Vector Differential Equation of Third Order

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Cemil Tunç ◽  
Melek Gözen
Keyword(s):  
Differential Equation ◽  
Theoretical Analysis ◽  
Sufficient Conditions ◽  
Third Order ◽  
Convergence Of Solutions ◽  
Nonlinear Vector ◽  
Vector Differential Equation ◽  
Made In

We give some sufficient conditions to guarantee convergence of solutions to a nonlinear vector differential equation of third order. We prove a new result on the convergence of solutions. An example is given to illustrate the theoretical analysis made in this paper. Our result improves and generalizes some earlier results in the literature.

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