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Certain Extensions of the Mehler Formula

For the Hermite polynomials H n (z) defined by 1 it is easy to see from the Rodrigues formula that 2 where, as usual, D z =d/dz. In recent papers ([1], [2]) Carlitz has proved the following formulae: 3 4 where, on the right-hand side of (4), the range of each summation is from i= 1 to i=k(k=1,2,…).