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While reading about the Rician distribution, I stumbled upon "modified Bessel function of the zeroth order of the first kind", i.e. $I_o$.

First thing, I did not know what Bessel function is, but I have learned it (Hurry to Wikipedia). Second, don't know what modified form is! and then the zeroth order and of the first kind...!!! Could not understand what it mean.

I am an electronic engineer who is trying to understand a mathematical paper, can someone help me what does this mean (AND YES I HAVE SEEN WIKIPEDIA AND GOOGLED THE STUFF).

https://en.wikipedia.org/wiki/Rician_fading

  1. What is Zeroth order here?
  2. What is First Kind here?
  3. If I want to have the series expansion for $I_o$, how do I get it?
SJa
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1 Answers1

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The modified Bessel functions of the first and second kinds and order $n$, $I_n(x)$ and $K_n(x)$ respectively, are both solutions of the modified Bessel differential equation

$$ x^2 y'' + x y' - (n^2 + x^2) y = 0 $$

while the corresponding (unmodified) Bessel functions $J_n(x)$ and $Y_n(x)$ are solutions of the Bessel differential equation

$$ x^2 y'' + x y' + (-n^2 + x^2) y = 0 $$

For nonnegative integers $n$, $I_n$ is a solution that is bounded as $x \to 0$, while $K_n$ is a solution that is bounded as $x \to \infty$.

$I_n(x)$ has a series representation

$$I_n(x) = \sum_{k=0}^\infty \frac{(x/2)^{n+2k}}{(n+k)!\; k!} $$

Robert Israel
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