In the context of \(f(T,{\mathcal {T}})\) gravity, where \(T\) denotes the torsion scalar and \({\mathcal {T}}\) signifies the trace of the energy-momentum tensor, our study delved into holographic inflation cosmology. Our investigation involved incorporating a holographic potential alongside three distinct types of inflationary potentials widely recognized in the scientific community. Our primary aim was to delineate the parameter regime governing the interplay between geometry and matter, crucial for describing scenarios of cosmological inflation. To achieve this, we derived the slow-roll parameters, enabling predictions of fundamental cosmological parameters such as the scalar spectral index \(n_s\) , the tensor-to-scalar ratio \(r\) , and the tensor spectral index \(n_T\) . These predictions were expressed in terms of parameters inherent to the inflationary potential. By comparing our results with those obtained in Teleparallel gravity (equivalent to General Relativity (GR)), we highlighted the significant influence of the parameter linking geometry and matter. This influence was underscored by comprehensive data presented in multiple tables, particularly within the framework of \(f(T,{\mathcal {T}})\) gravity. Our practical findings not only underscore the significance of geometry-matter interactions but also demonstrate strong alignment with both Planck 2018 and WMAP data, thereby bolstering the credibility and importance of our discoveries.
