  The main purpose of this paper is to introduce and investigate the notion of
Jacobi-Jordan conformal algebra. They are a generalization of Jacobi-Jordan
algebras which correspond to the case in which the formal parameter lambda
equals 0. We consider some related structures such as conformal modules,
corresponding representations and O-operators. Therefore, conformal derivations
from Jacobi-Jordan conformal algebras to their conformal modules are used to
describe conformal derivations of Jacobi-Jordan conformal algebras of
semidirect product type. Moreover, we study a class of Jacobi-Jordan conformal
algebras called quadratic Jacobi-Jordan conformal algebras, which are
characterized by mock-Gel'fand Dorfman bialgebras. Finally, the C[delta]-split
extending structures problem for Jacobi-Jordan conformal algebras is studied.
Furthermore, we introduce an unified product of a given Jacobi-Jordan conformal
algebra $J$ and a given C[delta]-module K. This product includes some other
interesting products of Jacobi-Jordan conformal algebras such as twisted
product or crossed product. Using this product, a cohomological type object is
constructed to provide a theoretical answer to the C[delta]-split extending
structures problem.
