**Description** The *Bessel functions* are solutions to certain DEs that emerge in various applications involving wave-like phenomena. They satisfy the [[Recurrence Relation]] $J_{k+1}(x)=\frac{2k}{x} J_{k} (x) - J_{k-1}(x), \quad k=1,2,3,\dots$Here, $x \in(0,\infty)$ is a parameter, so this is a difference equation for whole functions. **Bessel function of order zero** Consider the [[Ordinary Differential Equation|DE]] $xy'' + y ' +xy = 0$Suppose there were a solution given a [[Power Series|power series]] $y=\sum_{n=0}^{\infty} a_{n} x^{n}$Then computing its first and second [[Power Series is Termwise Differentiable within Radius of Convergence|derivatives]] gives $xy'' = \sum_{n=2}^{\infty} n(n-1)a_{n} x^{n-1} \quad \text{ and} \quad y' = \sum_{n=1}^{\infty} na_{n} x^{n-1} $while $xy=\sum_{n=2}^{\infty} a_{n-2} x^{n-1}$So $a_{1} + \sum_{n=2}^{\infty} (n^{2} a_{n} + a_{n-2}) x^{n-1} = 0$ Consider the coefficients of each power $a_{1}, \; a_{0}+4a_{2}, \; a_{1}+9a_{3}, \; a_{2}+16a_{4}, \dots$We can make all these zero by choosing all the odd indexed $a_{i}$ to be zero and take $a_{0}=1,\; a_{2}=-\frac{1}{4}, \; a_{4} = \frac{1}{64},\dots$More generally $a_{2m}= \frac{(-1)^{m}}{4^{m}(m!)^{2}}$If we define a function $J_{0}$ by $J_{0}(x)=\sum_{0}^{\infty} \frac{(-1)^{m}}{4^{m}(m!)^{2}}x^{2m}$ then the series has infinite radius of convergence and its derivatives satisfy $xy''+y'+xy=0$ This function is called the Bessel function of order zero.