investigating parametric elliptic functions and their properties in mathematical physics

In this paper, we have tried to study how the parametric (spectral) elliptic functions are defined andinvestigated in mathematical physics. in this regard, we firstly introduce a very key formula (eq. [11]), and thenprove that jacobi elliptic functions are part of this parametric framework. by using the (quasi)periodicproperties and modular transformations of jacobi theta functions, we can obtain all jacobi elliptic functions,and even generating some new elliptic functions would be useful in conformal quantum field theory. afterwards,we study the derivation of new elliptic functions to achieve new identities. finally, we introduce new identitiesof jacobi elliptic functions and new elliptic functions.