  In this paper, we aim to explore the properties and applications on the
operators consisting of the Hodge star operator together with the exterior
derivative, whose action on an arbitrary $p$-form field in $n$-dimensional
spacetimes makes its form degree remain invariant. Such operations are able to
generate a variety of $p$-forms with the even-order derivatives of the
$p$-form. To do this, we first investigate the properties of the operators,
such as the Laplace-de Rham operator, the codifferential and their
combinations, as well as the applications of the operators in the construction
of conserved currents. On basis of two general p-forms, then we construct a
general n-form with higher-order derivatives. Finally, we propose that such an
n-form could be applied to define a generalized Lagrangian with respect to a
p-form field according to the fact that it incudes the ordinary Lagrangians for
the $p$-form and scalar fields as special cases.
