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Reaction rate determination is an essential problem in the theories of cosmological and stellar nucleosynthesis. I investigate the closed-form analytic evaluation of thermonuclear reaction rates, which are necessary for both cosmic and celestial nucleosynthesis. An alternate velocity distribution to the Maxwell-Boltzmann distribution can be considered in nuclear fusion processes occurring in the interior of stars when there is a diversion from hydrostatic equilibrium. The extension of non-resonant reaction rate integrals in standard, cut-off, depletion, and screening instances is examined in this study, utilising the MacDonald distribution. This study is primarily on the use of generalized special functions that represent the extended non-resonant thermonuclear functions in mathematical physics. Moreover, an attempt is made to analyse the reaction probability integrals corresponding to the several forms of the slowly varying cross-section factor S(E).

In contrast to potential-driven warm inflation models, this paper presents a new inflationary scenario driven purely by noncanonical kinetic terms. We derive the evolution equations and the associated slow-roll approximations specific to the kinetic case. The model incorporates a nonminimal derivative coupling that enhances gravitational friction; when combined with thermal damping, this leads to a significantly slower evolution of the pure kinetic inflaton. The resulting slow-roll approximations differ fundamentally from those of potential-driven inflation. The attractor behavior of this warm $k$-inflation with nonminimal derivative coupling is explored, confirming that slow-roll solutions can approach a strict exponential expansion attractor under relaxed slow-roll conditions. We further calculate the density fluctuation equations and obtain analytic expressions for the power spectrum, spectral index, and tensor-to-scalar ratio. Compared to standard inflation in general relativity, the energy scale at horizon crossing is lower, and the tensor-to-scalar ratio is significantly reduced due to the combined effects of thermal damping and nonminimal derivative coupling. The field excursion remains comfortably sub-Planckian. The model's predictions are in excellent agreement with the latest Planck 2018 data, offering a novel and successful extension of the warm inflation paradigm.