Geodesic
Certain Extensions of the Mehler Formula
Canadian mathematical bulletin, Tome 15 (1972) no. 1, pp. 115-117

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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,...). 
Singhal, J. P. Certain Extensions of the Mehler Formula. Canadian mathematical bulletin, Tome 15 (1972) no. 1, pp. 115-117. doi: 10.4153/CMB-1972-020-5
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     title = {Certain {Extensions} of the {Mehler} {Formula}},
     journal = {Canadian mathematical bulletin},
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     doi = {10.4153/CMB-1972-020-5},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1972-020-5/}
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