Matrix Product Ensembles of Hermite Type

       We investigate spectral statistics of the product of GUE and  LUE random matrices, and find an explicit expression of tlie joint  probability density fimction as a bi-orthogonal ensemble. By providing  explicit expressions both for the bi-ortliogoiial fimctions and tlie correlation  kernel, a new double-side kernel is fomid at tlie origin, wliich is slightly  different from tlie well-known Bessel kernel.