In this paper we study for a given azimuthal quantum number k the eigenvalues of
the Chandrasekhar–Page angular equation with respect to the parameters mªam
and nªav, where a is the angular momentum per unit mass of a black hole, m is
the rest mass of the Dirac particle and v is the energy of the particle (as measured
at infinity). For this purpose, a self-adjoint holomorphic operator family Ask ;m ,nd
associated to this eigenvalue problem is considered. At first we prove that for fixed
kPR\ s−1
2 , 1
2 d the spectrum of Ask ;m ,nd is discrete and that its eigenvalues depend
analytically on sm ,ndPC2. Moreover, it will be shown that the eigenvalues satisfy
a first order partial differential equation with respect to m and n, whose characteristic
equations can be reduced to a Painlevé III equation. In addition, we derive a
power series expansion for the eigenvalues in terms of n −m and n +m, and we give
a recurrence relation for their coefficients. Further, it will be proved that for fixed
sm ,ndPC2 the eigenvalues of Ask ;m ,nd are the zeros of a holomorphic function Q
which is defined by a relatively simple limit formula. Finally, we discuss the problem
if there exists a closed expression for the eigenvalues of the Chandrasekhar–
Page angular equation.