Abstract
In the present paper, we treat an infinite series (<TEX>$\alpha$</TEX>, <TEX>$\beta$</TEX>)-metric L =<TEX>$\beta$</TEX><TEX>$^2$</TEX>/(<TEX>$\beta$</TEX>-<TEX>$\alpha$</TEX>). First, we find the conditions that a Finsler metric F<TEX>$^{n}$</TEX> with the metric above be a Berwald space, a Douglas space, and a projectively flat Finsler space, respectively. Next, we investigate the condition that a two-dimensional Finsler space with the metric above be a Landsbeg space. Then the differential equations of the geodesics are also discussed.
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