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A new expression of the generalised Marcum Q-function is obtained in terms of incomplete cylindrical function. Based on the new representation, new lower and upper bounds of the first-order Marcum Q-function are formulated. The bounds are represented in terms of the error and modified Bessel functions. Unlike the existing bounds, the tightness of the new bounds is maintained over the entire range of arguments of the Marcum Q-function, which is numerically and theoretically demonstrated. To show the usefulness of the formulated bound, the upper and lower of a generic integral is formulated in the performance analysis of an antenna diversity system under a Nakagami fading channel. The uniform tightness of the bound of the integral is also observed. Then, the authors consider the average bit error rate (ABER) probability of the equal gain combining diversity system in the fading channel, which show that the proposed bound can estimate very tight bound of the ABER probability.