<p>In the present paper, we study the approximation properties of the compositions of Jain operators and Szász–Mirakjan operators. First, we estimate the moment-generating function and moments of the new operators in terms of the Lambert <i>W</i> function and then we establish some convergence results and their quantitative estimates for the difference of the operators, while emphasizing on the preservation of the modulus of continuity. We further discuss Korovkin-type theorem using a Chebyshev system of exponential functions and Voronovskaja-type asymptotic formulae for these operators. In the last section, we provide a comparative study of the rates of convergence of these operators using graphs and numerical tables.</p>