scholarly journals Some New Recurrence Relations Concerning Jacobi Functions

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Hatem Mejjaoli ◽  
Ahmedou Ould Ahmed Salem
Keyword(s):  
Bessel Function ◽  
Recurrence Relations ◽  
Elementary Functions ◽  
Jacobi Function ◽  
Jacobi Functions ◽  
Asymptotic Relationship

The aim of this paper is to obtain some new recurrence relations for the “modified” Jacobi functionsΦλα,β. Based on an asymptotic relationship between the Jacobi function and the Bessel function, the expression of Bessel function in terms of elementary functions follows as particular cases.

scholarly journals THE GENERALIZED Q-BESSEL MATRIX FUNCTION OF TWO VARIABLES

2020 ◽  
Vol 1 (4) ◽  
Author(s):  
Fadhl S. N. Alsarahi
Keyword(s):  
Difference Equation ◽  
Bessel Function ◽  
Generating Function ◽  
Matrix Function ◽  
Recurrence Relations ◽  
Special Function ◽  
Applied Mathematics ◽  
Function Of Two Variables

The Bessel function is probably the best known special function, within pure and applied mathematics. In this paper, we introduce the generalized q-analogue Bessel matrix function of two variables. Some properties of this function, such as generating function, q-difference equation, and recurrence relations are obtained.

Uniform asymptotic expansions of solutions of linear second-order differential equations for large values of a parameter

1958 ◽  
Vol 250 (984) ◽  
pp. 479-517 ◽  
Keyword(s):  
Differential Equations ◽  
Bessel Function ◽  
Complex Variable ◽  
Asymptotic Expansions ◽  
Recurrence Relations ◽  
Airy Function ◽  
Formal Series ◽  
Complex Parameter ◽  
Second Order Differential Equations ◽  
Uniform Asymptotic Expansions

An investigation is made of the differential equations d 2w 1 da; ■ l, /t2—1 f,>4 d ? = i d ^ + r + V - + / ( w ) r } in which u is a large complex parameter, u a real or complex parameter independent of u , and z is a complex variable whose domain of variation may depend on arg u and u , and need not be bounded. General conditions are obtained under which solutions exist having the formal series w oo A P(Z) 5=0“ + f'(z) u2 V s=0«2* as their asymptotic expansions for large | u|, uniformly valid with respect to z, arg u and u. Here P(z) is respectively an exponential function, Airy function or Bessel function of order u , and the coefficients As and B5 are given by recurrence relations.

scholarly journals A note on monotonicity property of Bessel functions

1997 ◽  
Vol 20 (3) ◽  
pp. 561-566 ◽  
Author(s):  
Stamatis Koumandos
Keyword(s):  
Bessel Function ◽  
Unique Solution ◽  
Bessel Functions ◽  
Positive Root ◽  
Monotonicity Property ◽  
Elementary Functions

A theorem of Lorch, Muldoon and Szegö states that the sequence{∫jα,kjα,k+1t−α|Jα(t)|dt}k=1∞is decreasing forα>−1/2, whereJα(t)the Bessel function of the first kind orderαandjα,kitskth positive root. This monotonicity property implies Szegö's inequality∫0xt−αJα(t)dt≥0, whenα≥α′andα′is the unique solution of∫0jα,2t−αJα(t)dt=0.We give a new and simpler proof of these classical results by expressing the above Bessel function integral as an integral involving elementary functions.

scholarly journals Certain Recurrence Relations of Two Parametric Mittag-Leffler Function and Their Application in Fractional Calculus

2021 ◽  
Vol 5 (4) ◽  
pp. 215
Author(s):  
Dheerandra Shanker Sachan ◽  
Shailesh Jaloree ◽  
Junesang Choi
Keyword(s):  
Fractional Calculus ◽  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Recurrence Relations ◽  
Differential Operators ◽  
Elementary Functions ◽  
Generalized Hypergeometric Functions

The purpose of this paper is to develop some new recurrence relations for the two parametric Mittag-Leffler function. Then, we consider some applications of those recurrence relations. Firstly, we express many of the two parametric Mittag-Leffler functions in terms of elementary functions by combining suitable pairings of certain specific instances of those recurrence relations. Secondly, by applying Riemann–Liouville fractional integral and differential operators to one of those recurrence relations, we establish four new relations among the Fox–Wright functions, certain particular cases of which exhibit four relations among the generalized hypergeometric functions. Finally, we raise several relevant issues for further research.

scholarly journals MATLAB GUI for computing Bessel functions using continued fractions algorithm

2011 ◽  
Vol 33 (1) ◽  
Author(s):  
E. Hernández ◽  
K. Commeford ◽  
M.J. Pérez-Quiles
Keyword(s):  
User Interface ◽  
Bessel Function ◽  
Continued Fractions ◽  
Recurrence Relations ◽  
Engineering Students ◽  
Bessel Functions ◽  
Higher Order ◽  
Graphic User Interface ◽  
Miller’S Algorithm ◽  
The Stability

Higher order Bessel functions are prevalent in physics and engineering and there exist different methods to evaluate them quickly and efficiently. Two of these methods are Miller's algorithm and the continued fractions algorithm. Miller's algorithm uses arbitrary starting values and normalization constants to evaluate Bessel functions. The continued fractions algorithm directly computes each value, keeping the error as small as possible. Both methods respect the stability of the Bessel function recurrence relations. Here we outline both methods and explain why the continued fractions algorithm is more efficient. The goal of this paper is both (1) to introduce the continued fractions algorithm to physics and engineering students and (2) to present a MATLAB GUI (Graphic User Interface) where this method has been used for computing the Semi-integer Bessel Functions and their zeros.

scholarly journals Generalised Bernoulli polynomials and series

2000 ◽  
Vol 61 (2) ◽  
pp. 289-304 ◽  
Author(s):  
Clément Frappier
Keyword(s):  
Bessel Function ◽  
Recurrence Relations ◽  
Bernoulli Polynomials ◽  
Summation Formula

We present several results related to the recently introduced generalised Bernoulli polynomials. Some recurrence relations are given, which permit us to compute efficiently the polynomials in question. The sums , where jk = jk (α) are the zeros of the Bessel function of the first kind of order α, are evaluated in terms of these polynomials. We also study a generalisation of the series appearing in the Euler-MacLaurin summation formula.

scholarly journals On new approximations for the modified Bessel function of the second kind \(K_0(x)\)

2021 ◽  
Vol 5 (1) ◽  
pp. 11-17
Author(s):  
Francisco Caruso ◽  
◽  
Felipe Silveira ◽  
Keyword(s):  
Bessel Function ◽  
Series Representation ◽  
Modified Bessel Function ◽  
Elementary Functions ◽  
Polynomial Approximations ◽  
Kummer’S Function

A new series representation of the modified Bessel function of the second kind \(K_0(x)\) in terms of simple elementary functions (Kummer's function) is obtained. The accuracy of different orders in this expansion is analysed and has been shown not to be so good as those of different approximations found in the literature. In the sequel, new polynomial approximations for \(K_0(x)\), in the limits \(0< x\leq 2\) and \(2\leq x < \infty\), are obtained. They are shown to be much more accurate than the two best classical approximations given by the Abramowitz and Stegun's Handbook, for those intervals.

scholarly journals Quasi-Rational Analytic Approximation for the Modified Bessel Function I1(x) with High Accuracy

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 741
Author(s):  
Pablo Martin ◽  
Eduardo Rojas ◽  
Jorge Olivares ◽  
Adrián Sotomayor
Keyword(s):  
Asymptotic Expansion ◽  
Bessel Function ◽  
Relative Error ◽  
Asymptotic Series ◽  
Approximation Technique ◽  
Maximum Relative Error ◽  
Analytic Approximation ◽  
Modified Bessel Function ◽  
Elementary Functions ◽  
Order Of Magnitude

A new simple and accurate expression to approximate the modified Bessel function of the first kind I1(x) is presented in this work. This new approximation is obtained as an improvement of the multi-point quasi-rational approximation technique, MPQA. This method uses the power series of the Bessel function, its asymptotic expansion, and a process of optimization to fit the parameters of a fitting function. The fitting expression is formed by elementary functions combined with rational ones. In the present work, a sum of hyperbolic functions was selected as elementary functions to capture the first two terms of the asymptotic expansion of I1(x), which represents an important improvement with respect to previous research, where just the leading term of the asymptotic series was captured. The new approximation function presents a remarkable agreement with the analytical solution I1(x), decreasing the maximum relative error in more than one order of magnitude with respect to previous similar expressions. Concretely, the relative error was reduced from 10−2 to 4×10−4, opening the possibility of applying the new improved method to other Bessel functions. It is also remarkable that the new approximation is valid for all positive and negative values of the argument.

Existence of a pair of new recurrence relations for the Meixner-Pollaczek polynomials

2018 ◽  
Vol 11 (3) ◽  
pp. 29-39
Author(s):  
E. I. Jafarov ◽  
A. M. Jafarova ◽  
S. M. Nagiyev
Keyword(s):  
Recurrence Relations ◽  
Pollaczek Polynomials
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