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 The Modified Bessel Function Kn(z)

1919 ◽  
Vol 38 ◽  
pp. 10-19
Author(s):  
T. M. MacRobert
Keyword(s):  
Asymptotic Expansion ◽  
Bessel Function ◽  
Bessel Functions ◽  
Modified Bessel Function ◽  
Definition Of ◽  
Image Position

Gray and Mathews, in their treatise on Bessel Functions, define the function Kn(z) to beWe shall denote this function by Vn(z). This definition only holds when z is real, and R(n)≧0. The asymptotic expansion of the function is also given; but the proof, which is said to be troublesome and not very satisfactory, is omitted. Basset (Proc. Camb. Phil. Soc., Vol. 6) gives a similar definition of the function.

scholarly journals Bounds for the generalized Marcum function of the second kind

2021 ◽  
Author(s):  
Árpád Baricz ◽  
Nitin Bisht ◽  
Sanjeev Singh ◽  
V. Antony Vijesh
Keyword(s):  
Bessel Function ◽  
Bessel Functions ◽  
Modified Bessel Functions ◽  
Modified Bessel Function ◽  
Monotonicity Properties

AbstractIn this paper, we focus on the generalized Marcum function of the second kind of order $$\nu >0$$ ν > 0 , defined by $$\begin{aligned} R_{\nu }(a,b)=\frac{c_{a,\nu }}{a^{\nu -1}} \int _b ^ {\infty } t^{\nu } e^{-\frac{t^2+a^2}{2}}K_{\nu -1}(at)\mathrm{d}t, \end{aligned}$$ R ν ( a , b ) = c a , ν a ν - 1 ∫ b ∞ t ν e - t 2 + a 2 2 K ν - 1 ( a t ) d t , where $$a>0, b\ge 0,$$ a > 0 , b ≥ 0 , $$K_{\nu }$$ K ν stands for the modified Bessel function of the second kind, and $$c_{a,\nu }$$ c a , ν is a constant depending on a and $$\nu $$ ν such that $$R_{\nu }(a,0)=1.$$ R ν ( a , 0 ) = 1 . Our aim is to find some new tight bounds for the generalized Marcum function of the second kind and compare them with the existing bounds. In order to deduce these bounds, we include the monotonicity properties of various functions containing modified Bessel functions of the second kind as our main tools. Moreover, we demonstrate that our bounds in some sense are the best possible ones.

II.—Some Formulæ for the Associated Legendre Functions of the Second Kind; with corresponding Formulæ for the Bessel Functions

1938 ◽  
Vol 57 ◽  
pp. 19-25 ◽  
Author(s):  
T. M. MacRobert
Keyword(s):  
Bessel Function ◽  
Bessel Functions ◽  
Legendre Functions ◽  
Modified Bessel Function ◽  
Associated Legendre Functions ◽  
Analogous Formula

In two former papers (Proc. Roy. Soc. Edin., vol. liv, 1934, pp. 135–144; vol. lv, 1935, pp. 85–90) a number of integrals and series, involving Associated Legendre Functions of the First Kind regarded as functions of their degrees, were evaluated. In this paper similar integrals and series for the Associated Legendre Functions of the Second Kind, and also for Bessel Functions, are discussed. The latter are deduced from Bessel's Integral in its generalised form and from the corresponding integral for the Modified Bessel Function of the First Kind; the former from an analogous formula for the Associated Legendre Functions of the Second Kind.

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${K}_{ir} (x)$.

Higher monotonicity properties of normalized Bessel functions

2014 ◽  
Vol 12 (05) ◽  
pp. 1461007
Author(s):  
Yongping Liu
Keyword(s):  
Bessel Function ◽  
Bessel Functions ◽  
Positive Zero ◽  
Local Maxima ◽  
Monotonicity Properties

Denote by Jν the Bessel function of the first kind of order ν and μν,k is its kth positive zero. For ν > ½, a theorem of Lorch, Muldoon and Szegö states that the sequence [Formula: see text] is decreasing, another theorem of theirs states that the sequence [Formula: see text] has higher monotonicity properties. In the present paper, we proved that when ν > ½ the sequence [Formula: see text] has higher monotonicity properties and the properties imply those of the sequence of the local maxima of the function x-ν+1|Jν-1(x)|, x ∈ (0, ∞), i.e. the sequence [Formula: see text] has higher monotonicity properties.

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.

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