Boole-Dunkl polynomials and generalizations

Appell sequences of polynomials can be extended to the Dunkl context replacing the ordinary derivative by the Dunkl operator on the real line, and the exponential function by the Dunkl kernel. In a similar way, discrete Appell sequences can be extended to the Dunkl context; here, the role of the ord...

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Bibliographic Details
Published in:Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas Físicas y Naturales. Serie A, Matemáticas, 2024, Vol.118 (1), Article 16
Main Authors: Gil Asensi, Alejandro, Labarga, Edgar, Mínguez Ceniceros, Judit, Varona, Juan Luis
Format: Article
Language:English
Subjects:
Applications of Mathematics
Context
Exponential functions
Factorials
Mathematical analysis
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Operators (mathematics)
Original Paper
Polynomials
Sequences
Theoretical
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Summary:Appell sequences of polynomials can be extended to the Dunkl context replacing the ordinary derivative by the Dunkl operator on the real line, and the exponential function by the Dunkl kernel. In a similar way, discrete Appell sequences can be extended to the Dunkl context; here, the role of the ordinary translation is played by the Dunkl translation, which is a much more intricate operator. Some sequences as the falling factorials or the Bernoulli polynomials of the second kind have already been extended and investigated in the mathematical literature. In this paper, we study the discrete Appell version of the Euler polynomials, usually known as Euler polynomials of the second kind or Boole polynomials. We show how to define the Dunkl extension of these polynomials (and some of their generalizations), and prove some relevant properties and relations with other polynomials and with Stirling-Dunkl numbers.
ISSN:1578-7303
1579-1505
DOI:10.1007/s13398-023-01518-3