Rigorous proof of the antisymmetric Bessel orthogonality identities

Construct a mathematically rigorous proof of the antisymmetric semi-infinite-interval Bessel-function orthogonality identity and the resulting mixed J_p–J_{-p} identity given in equations (B.3b) and (B.4).

Background

The two-time autocorrelator calculation relies on distributional orthogonality relations for Bessel functions J_p and J_{-p} on the positive half-line. The appendix derives the relevant identities using formal asymptotic expansions and distributional arguments, but explicitly acknowledges that these arguments do not constitute a mathematically rigorous proof. The unresolved task is therefore to establish the identities rigorously.