  We present a summary of current knowledge about the AGT relations for
conformal blocks with additional insertion of the simplest degenerate operator,
and a special choice of the corresponding intermediate dimension, when the
conformal blocks satisfy hypergeometric-type differential equations in position
of the degenerate operator. A special attention is devoted to representation of
conformal block through the beta-ensemble resolvents and to its asymptotics in
the limit of large dimensions (both external and intermediate) taken
asymmetrically in terms of the deformation epsilon-parameters. The
next-to-leading term in the asymptotics defines the generating differential in
the Bohr-Sommerfeld representation of the one-parameter deformed Seiberg-Witten
prepotentials (whose full two-parameter deformation leads to Nekrasov
functions). This generating differential is also shown to be the one-parameter
version of the single-point resolvent for the corresponding beta-ensemble, and
its periods in the perturbative limit of the gauge theory are expressed through
the ratios of the Harish-Chandra function. The Shr\"odinger/Baxter equations,
considered earlier in this context, directly follow from the differential
equations for the degenerate conformal block. This provides a powerful method
for evaluation of the single-deformed prepotentials, and even for the
Seiberg-Witten prepotentials themselves. We mostly concentrate on the
representative case of the insertion into the four-point block on sphere and
one-point block on torus.
