We study hypergeometric functions of nilpotent operators in finite-dimensional settings, motivated by the algebraic structure of exceptional points in non-Hermitian quantum mechanics. Our starting point is the following exact result: if is a nilpotent operator of index in an associative algebra over , then every generalized hypergeometric function evaluated at reduces to a finite polynomial in of degree at most , without any analytic convergence requirement. This “functional collapse” is distinct from the classical parameter-termination mechanism and arises purely from the nilpotent structure of the argument. The main result is a “nilpotent depth criterion” (Theorem 2): if the first non-constant coefficient of a formal series appears in degree , then the nilpotent part has nilpotency index bounded above by . This bound is sharp in generic cases and provides a quantitative measure of how many Jordan levels survive after applying a special function. We apply this criterion to Hamiltonians at exceptional points, where with . Theorem 3 establishes that a function analytic at reduces the Jordan depth of the exceptional point from to at most , where r is the contact order of F at λ (the order of the first nonzero term in the Taylor expansion of around ). As consequences: the time evolution operator preserves the full Jordan depth for all ; a function with a zero of order at annihilates the entire Jordan structure; and the order of the pole of the modified resolvent is reduced from order to at most order Results are illustrated with explicit Jordan block computations for , , and the time evolution operator, confirming sharpness of the bounds.
Ramón Moya (Wed,) studied this question.