The Generating Function for the Bessel Functions

We construct the generating function for the Bessel functions by appealing to the three term recursion formula for Bessel functions. We derive a differential equation for the generating function and appeal to special values of Bessel functions to find a closed form.The generating function is an essential tool in writing down Fourier expansions of Bessel functions when we insert complex parameters into the expansion. #mikethemathematician, #mikedabkowski, #profdabkowski