  The equivalent representations for the hadronic vacuum polarization
contributions to the muon anomalous magnetic moment $a^{\text{HVP}}_{\mu}$ in
the presence of the quark flavour thresholds are studied. Specifically, the
explicit relations between the contributions given by the integration over a
finite kinematic interval to $a^{\text{HVP}}_{\mu}$ expressed in terms of the
hadronic vacuum polarization function, Adler function, and the $R$-ratio of
electron-positron annihilation into hadrons are derived. It is shown that the
quark flavour thresholds of the hadronic vacuum polarization function generate
additional contributions to $a^{\text{HVP}}_{\mu}$ expressed in terms of the
Adler function and the $R$-ratio and the explicit expressions for such
contributions are obtained. The commonly employed dispersion relations, which
bind together hadronic vacuum polarization function, Adler function, and
$R$-ratio, are extended to account for the effects due to the quark flavour
thresholds. The examined additional contributions due to the heavy quark
thresholds to $a^{\text{HVP}}_{\mu}$ expressed in terms of the $R$-ratio appear
to be quite sizable, that can be of a particular relevance for the data-driven
method of assessment of the hadronic part of the muon anomalous magnetic
moment.
