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In this paper, we propose a new method to calculate the relativistic plasma dispersion (RPD) function for wave propagation perpendicular to the magnetic field in a fully relativistic magnetized Maxwellian plasma. Explicit representations for the RPD function are developed in terms of some relatively more familiar special functions and binomial coefficients. As will be seen, the present formulation yields compact closed-form expressions that enable the ready analytical calculation of the RPD function. Extensive numerical tests and comparisons with some already existing methods show that the approach proposed here is highly efficient and competitive. The convergence rate of the series is estimated and discussed.