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In this paper, we aim to explore the properties and applications on theoperators consisting of the Hodge star operator together with the exteriorderivative, whose action on an arbitrary $p$-form field in $n$-dimensionalspacetimes makes its form degree remain invariant. Such operations are able togenerate 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 theircombinations, as well as the applications of the operators in the constructionof conserved currents. On basis of two general p-forms, then we construct ageneral n-form with higher-order derivatives. Finally, we propose that such ann-form could be applied to define a generalized Lagrangian with respect to ap-form field according to the fact that it incudes the ordinary Lagrangians forthe $p$-form and scalar fields as special cases.

Constructing $p,n$-forms from $p$-forms via the Hodge star operator and the exterior derivative