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On some analytic properties of slice poly‐regular Hermite polynomials
Author(s) -
El Hamyani Amal,
Ghanmi Allal
Publication year - 2018
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5264
Subject(s) - mathematics , hermite polynomials , quaternion , hilbert space , pure mathematics , square integrable function , univariate , gaussian measure , exponential type , laguerre polynomials , mathematical analysis , gaussian , multivariate statistics , statistics , physics , geometry , quantum mechanics
We consider a quaternionic analogue of the univariate complex Hermite polynomials and study some of their analytic properties in some detail. We obtain their integral representation as well as the operational formulas of exponential and Burchnall types they obey. We show that they form an orthogonal basis of the slice Hilbert space L 2 ( L I ; e − | q | 2d λ I ) of all quaternionic‐valued functions defined the whole quaternions space and subject to norm boundedness with respect to the Gaussian measure on a given slice as well as of the full left quaternionic Hilbert space L 2 ( H ; e − | q | 2d λ ) of square integrable functions on quaternions with respect to the Gaussian measure on the whole H ≡ R 4 . We also provide different types of generating functions. Remarkable identities, including quadratic recurrence formulas of Nielsen type, are also derived.