scholarly journals Determinantal Expressions and Recursive Relations for the Bessel Zeta Function and for a Sequence Originating from a Series Expansion of the Power of Modified Bessel Function of the First Kind

2021 ◽  
Vol 127 (3) ◽  
pp. 1-15
Author(s):  
Yan Hong ◽  
Bai-Ni Guo ◽  
Feng Qi
Keyword(s):  
Zeta Function ◽  
Bessel Function ◽  
Series Expansion ◽  
Modified Bessel Function ◽  
Recursive Relations

scholarly journals On polynomials connected to powers of Bessel functions

2014 ◽  
Vol 10 (05) ◽  
pp. 1245-1257 ◽  
Author(s):  
Victor H. Moll ◽  
Christophe Vignat
Keyword(s):  
Zeta Function ◽  
Bessel Function ◽  
Series Expansion ◽  
Bessel Functions ◽  
Bell Polynomials ◽  
Modified Bessel Function ◽  
Probabilistic Version

The series expansion of a power of the modified Bessel function of the first kind is studied. This expansion involves a family of polynomials introduced by C. Bender et al. New results on these polynomials established here include recurrences in terms of Bell polynomials evaluated at values of the Bessel zeta function. A probabilistic version of an identity of Euler yields additional recurrences. Connections to the umbral formalism on Bessel functions introduced by Cholewinski are established.

scholarly journals Bounds and algorithms for the -Bessel function of imaginary order

2013 ◽  
Vol 16 ◽  
pp. 78-108 ◽  
Author(s):  
Andrew R. Booker ◽  
Andreas Strömbergsson ◽  
Holger Then
Keyword(s):  
Bessel Function ◽  
Convergent Series ◽  
Steepest Descent ◽  
Modified Bessel Function ◽  
Software Library ◽  
Asymptotic Bounds ◽  
Mixed Derivatives ◽  
Rigorous Computation ◽  
Working Out ◽  
Precise Asymptotic

AbstractUsing the paths of steepest descent, we prove precise bounds with numerical implied constants for the modified Bessel function${K}_{ir} (x)$of imaginary order and its first two derivatives with respect to the order. We also prove precise asymptotic bounds on more general (mixed) derivatives without working out numerical implied constants. Moreover, we present an absolutely and rapidly convergent series for the computation of${K}_{ir} (x)$and its derivatives, as well as a formula based on Fourier interpolation for computing with many values of$r$. Finally, we have implemented a subset of these features in a software library for fast and rigorous computation of${K}_{ir} (x)$.

scholarly journals An Integral involving a Modified Bessel Function of the Second Kind and an E-function

1953 ◽  
Vol 1 (3) ◽  
pp. 119-120 ◽  
Author(s):  
Fouad M. Ragab
Keyword(s):  
Positive Integer ◽  
Bessel Function ◽  
Modified Bessel Function ◽  
Image Position

§ 1. Introductory. The formula to be established iswhere m is a positive integer,and the constants are such that the integral converges.

scholarly journals The Distribution of Larvae of Randomly Moving Insects

1961 ◽  
Vol 14 (4) ◽  
pp. 598 ◽  
Author(s):  
EJ Williams
Keyword(s):  
Normal Distribution ◽  
Bessel Function ◽  
Theoretical Distribution ◽  
Frequency Function ◽  
Published Data ◽  
Modified Bessel Function ◽  
Theoretical Conclusion ◽  
Constant Rate ◽  
Spatially Distributed ◽  
Good Agreement

Though randomly moving insects released from a central point in a uniform environment are often found to be distributed according to a circular normal distribution, their larvae will not conform to this distribution. When such insects lay at a constant rate and are subject to constant mortality, their larvae are found to be spatially distributed according to a highly peaked frequency function, depending on the modified Bessel function of the second kind. This theoretical conclusion is in good agreement with published data. Some of the properties of the theoretical distribution are discussed.

On the Square of the First Zero of the Bessel Function Jv(z)

1999 ◽  
Vol 42 (1) ◽  
pp. 56-67 ◽  
Author(s):  
Árpád Elbert ◽  
Panayiotis D. Siafarikas
Keyword(s):  
Convex Function ◽  
Bessel Function ◽  
Series Expansion ◽  
Positive Zero ◽  
Monotonicity Properties

AbstractLet Jv,1 be the smallest (first) positive zero of the Bessel function Jv(z), v > −1, which becomes zero when v approaches −1. Then can be continued analytically to −2 < v < −1, where it takes on negative values. We show that is a convex function of v in the interval −2 < v ≤ 0, as an addition to an old result [Á. Elbert and A. Laforgia, SIAM J. Math. Anal. 15(1984), 206–212], stating this convexity for v > 0. Also the monotonicity properties of the functions are determined. Our approach is based on the series expansion of Bessel function Jv(z) and it turned out to be effective, especially when −2 < v < −1.

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