Analytic structure of the associated Legendre functions of the second kind

We consider the complex nu plane structure of the associated Legendre functions of the second kind Q(nu)(-1/2-K)(cosh rho). We find that for any noninteger value of K the Q(nu)(-1/2-K)(cosh rho) have an infinite number of poles in the complex nu plane, but for any negative integer K there are no poles at all. For K = 0 or any positive integer K there is only a finite number of poles, with there only being one single pole (at nu = 0) when K = 0. This pattern is analogous to the pattern of exceptional points that appear in a wide variety of physical contexts. However, while theories with exceptional points usually lose a finite number of degrees of freedom at the exceptional points, the Q(nu)(-1/2-K)(cosh rho) lose an infinite number of poles whenever K is integer. Moreover, while theories with exceptional points usually have a finite number of such exceptional points, the Q(nu)-(1/2-K)(cosh rho) possess an infinite number of points (all integer K) at which they lose degrees of freedom. Other than in the PT-symmetry Jordan-block case, exceptional points usually occur at complex values of parameters. While not being Jordan-block exceptional points themselves, the poles associated with the Q(nu)(-1/2-K)(cosh rho) nonetheless occur at real values of K.