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We develop the theory of irregular conformal blocks of the Virasoro algebra.In previous studies, expansions of irregular conformal blocks at regularsingular points were obtained as degeneration limits of regular conformalblocks; however, such expansions at irregular singular points were not clearlyunderstood. This is because precise definitions of irregular vertex operatorshad not been provided previously. In this paper, we present precise definitionsof irregular vertex operators of two types and we prove that one of our vertexoperators exists uniquely. Then, we define irregular conformal blocks with atmost two irregular singular points as expectation values of given irregularvertex operators. Our definitions provide an understanding of expansions ofirregular conformal blocks and enable us to obtain expansions at irregularsingular points.  As an application, we propose conjectural formulas of series expansions ofthe tau functions of the fifth and fourth Painlev\'e equations, usingexpansions of irregular conformal blocks at an irregular singular point.

Irregular conformal blocks, with an application to the fifth and fourth Painlev\'e equations