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We show that there is a full correspondence between the parameters space ofthe degenerate biconfluent Heun connection (BHC) and that of Painlev\'{e} IVthat admits special solutions. The BHC degenerates when either the Stokes' datafor the irregular singularity at $\infty$ degenerates or the regular singularpoint at the origin becomes an apparent singularity. We show that if the BHC iswritten as isomonodromy family of biconfluent Heun equations (BHE), then theBHE degenerates precisely when it admits eigen-solutions of the biconfluentHeun operators, after choosing appropriate accessory parameter, of speciallyconstructed invariant subspaces of finite dimensional solution spaces spannedby parabolic cylinder functions. We have found all eigen-solutions over thisparameter space apart from three exceptional cases after choosing the rightaccessory parameters. These eigen-solutions are expressed as certain finite sumof parabolic cylinder functions. We extend the above sum to new convergentseries expansion in terms of parabolic cylinder functions to the BHE. Theinfinite sum solutions of the BHE terminates precisely when the parameters ofthe BHE assumes the same values as those of the degenerate biconfluent Heunconnection except at three instances after choosing the right accessoryparameter.

Invariant subspaces of biconfluent Heun operators and special solutions of Painlev\'e IV