Reciprocity relations and generalized entropic quantifiers that lack trace-form

In this effort we show that the Legendre reciprocity relations,thermodynamic's essential formal feature, are respected by any entropic functional, even if it is NOT of trace-form nature, as Shannon's is. Further, with reference to the MaXent variational process, we encounter important cases, relevant to physical applications currently discussed in the research literature, in which the associated reciprocity relations exhibit anomalies. We show that these anomalies can be cured by carefully discriminating between apparently equivalent entropic forms.