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Lie algebra \({\mathcal {G}}(0, 1)\) and 3-variable 2-parameter Hermite polynomials

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Abstract

This article is an attempt to stress the usefulness of multi-variable special functions by expressing them in terms of the corresponding Lie algebra or Lie group. The problem of framing the 3-variable 2-parameter Hermite polynomials (3V2PHP) into the context of the irreducible representation \( \downarrow _{0,1}\) of \({\mathcal {G}}(0,1)\) is considered. Certain relations involving 3V2PHP \(H_{m}(x,y,z;\nu _{1},\nu _{2})\) are obtained using the approach adopted by Miller. Certain examples involving other forms of Hermite polynomials are derived as special cases. Further, some Volterra integral equations involving the 3V2PHP \(H_{m}(x,y,z;\nu _{1},\nu _{2})\) are also obtained.

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Acknowledgements

The authors are thankful to the reviewers for several useful comments and suggestions towards the improvement of this paper.

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  1. Department of Applied Mathematics, Aligarh Muslim University, Aligarh, India

    Ghazala Yasmin & Abdulghani Muhyi

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  1. Ghazala Yasmin
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  2. Abdulghani Muhyi
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Correspondence to Ghazala Yasmin.

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Yasmin, G., Muhyi, A. Lie algebra \({\mathcal {G}}(0, 1)\) and 3-variable 2-parameter Hermite polynomials. Afr. Mat. 30, 231–246 (2019). https://doi.org/10.1007/s13370-018-0639-4

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