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Long-range correlation plays an important role in analyses of pionicBose-Einstein correlations (BECs). In many cases, such correlations arephenomenologically introduced. In this investigation, we propose an analyticform. By making use of the form, we analyze the OPAL BEC and the L3 BEC at$Z^0$-pole and the CMS BEC at 0.9 and 7 TeV using our formulas and the$\tau$-model. The parameters estimated by both approaches are found to beconsistent. Utilizing the Fourier transform in four-dimensional Euclideanspace, a number of pion-pair density distributions are also studied.

On long-range pionic Bose-Einstein correlations -- Including analyses of OPAL, L3 and CMS BECs