Your question should be more specific. First, don't confuse "Bessel functions" and "Modified Bessel functions": they are different. In one of each of whose two sets of functions, they can be of the "first kind" or of the "second kind" : again different sub-sets of functions. And in each one of these sub-sets, they are different Bessel functions of various order.
For examples:
The modified Bessel function of the first kind and order $0$ is $I_0(x)$. One integral definition is : $$I_0(x)=\frac{1}{\pi}\int_0^\pi \exp\left(x\cos(t)\right)dt$$
The modified Bessel function of the second kind and order $0$ is $K_0(x)$. One integral definition is : $$K_0(x)=\int_0^\infty \cos\left(x \sinh(t) \right)dt$$
Series expressions can be found in : < <
and related differential equation : <