Several characterizations of Bessel functions and their applications

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Tabinda Nahid ◽  
Mahvish Ali
Keyword(s):  
Imaginary Part ◽  
Generating Function ◽  
Bessel Functions ◽  
Special Functions ◽  
Complex Form ◽  
Hybrid Form ◽  
The Real ◽  
Mathematical Investigation ◽  
Main Motive ◽  
Replacement Technique

Abstract The present work deals with the mathematical investigation of some generalizations of Bessel functions. The main motive of this paper is to show that the generating function can be employed efficiently to obtain certain results for special functions. The complex form of Bessel functions is introduced by means of the generating function. Certain enthralling properties for complex Bessel functions are investigated using the generating function method. By considering separately the real and the imaginary part of complex Bessel functions, we get respectively cosine-Bessel functions and sine-Bessel functions for which several novel identities and Jacobi–Anger expansions are established. Also, the generating function of degenerate Bessel functions is investigated and certain novel identities are obtained for them. A hybrid form of degenerate Bessel functions, namely, of degenerate Fubini–Bessel functions, is constructed using the replacement technique. Finally, the explicit forms of the real and the imaginary part of complex Bessel functions are established by a hypergeometric approach.

scholarly journals On the complex q-Appell polynomials

2020 ◽  
Vol 74 (1) ◽  
pp. 31
Author(s):  
Thomas Ernst
Keyword(s):  
Imaginary Part ◽  
Generating Function ◽  
Complex Number ◽  
Analytic Functions ◽  
Complex Case ◽  
Euler Polynomials ◽  
Appell Polynomials ◽  
The Real ◽  
Cauchy Riemann

The purpose of this article is to generalize the ring of \(q\)-Appell polynomials to the complex case. The formulas for \(q\)-Appell polynomials thus appear again, with similar names, in a purely symmetric way. Since these complex \(q\)-Appell polynomials are also \(q\)-complex analytic functions, we are able to give a first example of the \(q\)-Cauchy-Riemann equations. Similarly, in the spirit of Kim and Ryoo, we can define \(q\)-complex Bernoulli and Euler polynomials. Previously, in order to obtain the \(q\)-Appell polynomial, we would make a \(q\)-addition of the corresponding \(q\)-Appell number with \(x\). This is now replaced by a \(q\)-addition of the corresponding \(q\)-Appell number with two infinite function sequences \(C_{\nu,q}(x,y)\) and \(S_{\nu,q}(x,y)\) for the real and imaginary part of a new so-called \(q\)-complex number appearing in the generating function. Finally, we can prove \(q\)-analogues of the Cauchy-Riemann equations.

scholarly journals Appell-Type Functions and Chebyshev Polynomials

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 679 ◽  
Author(s):  
Pierpaolo Natalini ◽  
Paolo Emilio Ricci
Keyword(s):  
Imaginary Part ◽  
Chebyshev Polynomials ◽  
Recent Article ◽  
Bessel Functions ◽  
Appell Polynomials ◽  
The Real

In a recent article we noted that the first and second kind Cebyshev polynomials can be used to separate the real from the imaginary part of the Appell polynomials. The purpose of this article is to show that the same classic polynomials can also be used to separate the even part from the odd part of the Appell polynomials and of the Appell–Bessel functions.

scholarly journals Real Representations of Finite Symplectic Groups over Fields of Characteristic Two

2018 ◽  
Vol 2020 (5) ◽  
pp. 1281-1299 ◽  
Author(s):  
C Ryan Vinroot
Keyword(s):  
Irreducible Representation ◽  
Generating Function ◽  
Prime Power ◽  
Symplectic Groups ◽  
Real Numbers ◽  
The Real ◽  
Characteristic Two ◽  
Finite Symplectic Groups

Abstract We prove that when q is a power of 2 every complex irreducible representation of $\textrm{Sp}\big (2n, \mathbb{F}_{q}\big )$ may be defined over the real numbers, that is, all Frobenius–Schur indicators are 1. We also obtain a generating function for the sum of the degrees of the unipotent characters of $\textrm{Sp}\big(2n, \mathbb{F}_{q}\big )$, or of $\textrm{SO}\big(2n+1,\mathbb{F}_{q}\big )$, for any prime power q.

scholarly journals Complex q-Rung Orthopair Fuzzy Aggregation Operators and Their Applications in Multi-Attribute Group Decision Making

Information ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 5 ◽  
Author(s):  
Liu ◽  
Mahmood ◽  
Ali
Keyword(s):  
Decision Making ◽  
Fuzzy Sets ◽  
Imaginary Part ◽  
Group Decision Making ◽  
Group Decision ◽  
Aggregation Operators ◽  
Uncertain Information ◽  
Membership Degree ◽  
The Real ◽  
Complex Valued

In this manuscript, the notions of q-rung orthopair fuzzy sets (q-ROFSs) and complex fuzzy sets (CFSs) are combined is to propose the complex q-rung orthopair fuzzy sets (Cq-ROFSs) and their fundamental laws. The Cq-ROFSs are an important way to express uncertain information, and they are superior to the complex intuitionistic fuzzy sets and the complex Pythagorean fuzzy sets. Their eminent characteristic is that the sum of the qth power of the real part (similarly for imaginary part) of complex-valued membership degree and the qth power of the real part (similarly for imaginary part) of complex-valued non‐membership degree is equal to or less than 1, so the space of uncertain information they can describe is broader. Under these environments, we develop the score function, accuracy function and comparison method for two Cq-ROFNs. Based on Cq-ROFSs, some new aggregation operators are called complex q-rung orthopair fuzzy weighted averaging (Cq-ROFWA) and complex q-rung orthopair fuzzy weighted geometric (Cq-ROFWG) operators are investigated, and their properties are described. Further, based on proposed operators, we present a new method to deal with the multi‐attribute group decision making (MAGDM) problems under the environment of fuzzy set theory. Finally, we use some practical examples to illustrate the validity and superiority of the proposed method by comparing with other existing methods.

scholarly journals The Asymptotic Expansions of the Spherical Harmonics

1922 ◽  
Vol 41 ◽  
pp. 82-93
Author(s):  
T. M. MacRobert
Keyword(s):  
Real Axis ◽  
Spherical Harmonics ◽  
Asymptotic Expansions ◽  
Bessel Functions ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
The Real ◽  
Image Position

Associated Legendre Functions as Integrals involving Bessel Functions. Let,where C denotes a contour which begins at −∞ on the real axis, passes positively round the origin, and returns to −∞, amp λ=−π initially, and R(z)>0, z being finite and ≠1. [If R(z)>0 and z is finite, then R(z±)>0.] Then if I−m (λ) be expanded in ascending powers of λ, and if the resulting expression be integrated term by term, it is found that

One Method to Construct the S-Box Based on Improved Chaos Map

2014 ◽  
Vol 651-653 ◽  
pp. 2164-2167
Author(s):  
Hang Zhang ◽  
Xiao Jun Tong
Keyword(s):  
Imaginary Part ◽  
Logistic Map ◽  
Block Ciphers ◽  
Mandelbrot Set ◽  
New Method ◽  
Hénon Map ◽  
Henon Map ◽  
The Real

Many methods of constructing S-box often adopt the classical chaotic equations. Yet study found that some of the chaotic equations exists drawbacks. Based on that, this paper proposed a new method to generate S-Box by improving the Logistic map and Henon map, and combining the real and imaginary part of complex produced by the Mandelbrot set. By comparing with several other S-boxes proposed previously, the results show the S-box here has better cryptographic properties. So it has a good application prospect in block ciphers.

scholarly journals A family of Finch and Skea relativistic stars

2017 ◽  
Vol 26 (03) ◽  
pp. 1750014 ◽  
Author(s):  
S. D. Maharaj ◽  
D. Kileba Matondo ◽  
P. Mafa Takisa
Keyword(s):  
Bessel Functions ◽  
Special Functions ◽  
Graphical Analysis ◽  
System Of Differential Equations ◽  
Elementary Functions ◽  
Relativistic Stars ◽  
Spacetime Geometry ◽  
Special Cases ◽  
Barotropic Equation ◽  
Charged Model

Several new families of exact solution to the Einstein–Maxwell system of differential equations are found for anisotropic charged matter. The spacetime geometry is that of Finch and Skea which satisfies all criteria for physical acceptability. The exact solutions can be expressed in terms of elementary functions, Bessel functions and modified Bessel functions. When a parameter is restricted to be an integer then the special functions reduce to simple elementary functions. The uncharged model of Finch and Skea [R. Finch and J. E. F. Skea, Class. Quantum Grav. 6 (1989) 467.] and the charged model of Hansraj and Maharaj [S. Hansraj and S. D. Maharaj, Int. J. Mod. Phys. D 15 (2006) 1311.] are regained as special cases. The solutions found admit a barotropic equation of state. A graphical analysis indicates that the matter and electric quantities are well behaved.

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