  For the overlap matrix element between Hartree-Fock-Bogoliubov states, there
are two analytically different formulae: one with the square root of the
determinant (the Onishi formula) and the other with the Pfaffian (Robledo's
Pfaffian formula). The former formula is two-valued as a complex function,
hence it leaves the sign of the norm overlap undetermined (i.e., the so-called
sign problem of the Onishi formula). On the other hand, the latter formula does
not suffer from the sign problem. The derivations for these two formulae are so
different that the reasons are obscured why the resultant formulae possess
different analytical properties. In this paper, we discuss the reason why the
difference occurs by means of the consistent framework, which is based on the
linked cluster theorem and the product-sum identity for the Pfaffian. Through
this discussion, we elucidate the source of the sign problem in the Onishi
formula. We also point out that different summation methods of series
expansions may result in analytically different formulae.
