Mehler's formulas for the univariate complex Hermite polynomials and applications |
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| Authors: | Allal Ghanmi |
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| Affiliation: | Center of Mathematical Research and Applications of Rabat (CeReMAR), Analysis and Spectral Geometry (A.G.S.), Laboratory of Mathematical Analysis and Applications (L.A.M.A.), Department of Mathematics, Faculty of Sciences, Mohammed V University, Rabat, Morocco |
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| Abstract: | We give 2 widest Mehler's formulas for the univariate complex Hermite polynomials  , by performing double summations involving the products  and  . They can be seen as the complex analogues of the classical Mehler's formula for the real Hermite polynomials. The proof of the first one is based on a generating function giving rise to the reproducing kernel of the generalized Bargmann space of level m. The second Mehler's formula generalizes the one appearing as a particular case of the so‐called Kibble‐Slepian formula. The proofs we present here are direct and more simpler. Moreover, direct applications are given and remarkable identities are derived. |
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| Keywords: | complex hermite polynomials heat kernel integral representation magnetic Laplacian Mehler's formula |
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, by performing double summations involving the products 
and 
. They can be seen as the complex analogues of the classical Mehler's formula for the real Hermite polynomials. The proof of the first one is based on a generating function giving rise to the reproducing kernel of the generalized Bargmann space of level m. The second Mehler's formula generalizes the one appearing as a particular case of the so‐called Kibble‐Slepian formula. The proofs we present here are direct and more simpler. Moreover, direct applications are given and remarkable identities are derived.