New exponential operators connected with a2 + x2: a generalization of Post-Widder and Ismail-May operators

The present study offers a general exponential operator connected with a(2) + x(2), for positive real a. We estimate the asymptotic formula for simultaneous and ordinary approximation of the constructed operator. We also consider the limiting case a -> 0, which provides Post-Widder operators. We graphically present the convergence of the operator to two functions, viz., "x sin(x)" and "- x/2 cos(pi x)". In addition, we analyze each particular case of the defined operator and determine the optimal value of a, that would yield the greatest approximation. This allows us to compare the well-known operators in the literature, especially the Post-Widder operators and the Ismail-May operators. In the end, we conclude that the convergence of the operators improves as the value of a approaches zero from the right side.