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A recently formulated conjecture of Gamayun, Iorgov and Lisovyy gives anasymptotic expansion of the Jimbo--Miwa--Ueno isomonodromic $\tau$-function forcertain Painlev\'e transcendents. The coefficients in this expansion are givenin terms of conformal blocks of a two-dimensional conformal field theory, whichcan be written as infinite sums over pairs of partitions. In this note adiscrete matrix model is proposed on a lattice whose partition function can beused to obtain a multiple integral representation for the length restrictedpartial sums of the Painlev\'e conformal blocks. This leads to expressions ofthe partial sums involving H\"ankel determinants associated to the discretemeasure of the matrix model, or equivalently, Wronskians of the correspondingmoment generating function which is shown to be of the generalizedhypergeometric type.

Discrete matrix models for partial sums of conformal blocks associated to Painlev\'e transcendents