Explicit forms of \(q\) -deformed Lévy-Meixner polynomials and their generating functions. (English) Zbl 1149.05008
Summary: The classical Lévy-Meixner polynomials are distinguished through the special forms of their generating functions. In fact, they are completely determined by 4 parameters: \(c _{1}, c _{2}, \gamma \) and \(\beta \). In this paper, for \(-1 < q < 1\), we obtain a unified explicit form of \(q\)-deformed Lévy-Meixner polynomials and their generating functions in term of \(c _{1}, c _{2}, \gamma \) and \(\beta \), which is shown to be a reasonable interpolation between classical case \((q = 1)\) and fermionic case (\(q = - 1\)). In particular, when \(q = 0\) it’s also compatible with the free case.
MSC:
| 05A30 | \(q\)-calculus and related topics |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 46L54 | Free probability and free operator algebras |
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