Is there an equation like Jacobi-Anger expansion for the Bessel function of the second kind

Question

The Jacobi-Anger expansion is represented as $$ e^{ix \cos\theta} = \sum_{n=-\infty}^{\infty} i^{n}J_{n}(x)e^{in \theta} $$ then is there an expression for the Bessel function of the second kind $Y_n$ without summing $$\sum_{n=-\infty}^{\infty} i^{n}Y_{n}(x)e^{in \theta} = ?$$

Answer 1

The Neumann functions at integer order cannot have a Jacobi-Anger type expansion due to their asymptotic behavior at large order $n$. They grow at a super-exponential rate $\propto n^n$ (also see here) which implies that multiplying them by powers of some number $a$ cannot possibly describe a convergent infinite sum.

The best one can do is define the exponential generating function

$$G(a,z)=\sum_{n=0}^\infty \frac{a^nY_n(z)}{n!}$$

since the factorial in the denominator cancels out the bulk of the asymptotic behavior and hence the series should have at least a bounded region of convergence in the $z-$plane. Then by using the integral representation 10.9.7 and after a few manipulations (I can provide more details upon request) one can write down the following integral expression for the generating function:

$$G(a,z)=\frac{1}{\pi}\int_0^\pi e^{a\cos\theta}\sin[(z-a)\sin\theta]d\theta-\frac{1}{\pi}\int_0^\infty e^{-(z-a)\sinh t}\cosh(a\cosh t)dt ,\\ \text{Re}(z)>\max(0,2\text{Re}(a))$$

It doesn't look like there is a simple analytic form for this integral representation.