Relation between Resonant states and Resonance Resonant states and Resonance can be defined as

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*A resonant state is an eigenstate of the Schrödinger equation with the Siegert condition
given in this paper.

*Resonance is defined as condition, which leads to the peaking of Transmission coefficient to its maxima. This definition is from Wikipedia
I am unable to understand the relation between resonance and a resonant state. Any help will be appreciated
 A: TLDR: The situation in the wiki link you shared is not applicable to resonant states.
Trying to connect various schemes for describing resonance type phenomena can be confusing, because they occur in such a wide array of physical situations, and subtleties are often glossed over in favour of simple, concrete physical examples.
For quantum scattering problems I suggest thinking in terms of the S-matrix.
Resonant states are scattering states consisting of only outgoing waves, e.g. waves moving away from some point (more correctly away from some finite range potential) -  this is the Siegert condition. They are useful when modelling radioactive decay via tunnelling, where the finite range potential corresponds to e.g. some atomic potential barrier. These resonant states clearly require the coefficient of the incoming wave to be 0. The transmission coefficient in the wiki link you shared is therefore poorly defined for such models, because there is strictly speaking no incoming particle flux.
In terms of the S-matrix, resonances in the usual sense of the word correspond to poles. In the simplest case elements of the S-matrix are given by ratios of the scattering coefficients (something like the transmission coefficient in your wikipedia link) . If the coefficient of the incoming wave is 0, these elements diverge, i.e. they are poles of the matrix, and hence resonances.
I think this is a genuinely confusing use of nomenclature that is often glossed over. I hope the above is helpful.
