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On new exact conformal blocks and Nekrasov functions

Recently, an intriguing family of the one-point toric conformal blocks AGT related to the $\mathcal{N}=2^*\,\, SU(2)$ Nekrasov functions was discovered by M. Beccaria and G. Macorini. Members of the family are distinguished by having only finite amount of poles as functions of the intermediate dimension/v.e.v. in gauge theory. Another remarkable property is that these conformal blocks/Nekrasov functions can be found in closed form to all orders in the coupling expansion. In the present paper we use Zamolodchikov's recurrence equation to systematically account for these exceptional conformal blocks. We conjecture that the family is infinite-dimensional and describe the corresponding parameter set. We further apply the developed technique to demonstrate that the four-point spheric conformal blocks feature analogous exact expressions. We also study the modular transformations of the finite-pole blocks.