Appell polynomials and their relatives
By Michael Anshelevich. 2003
65 citations recorded by the source corpus.
This paper summarizes some known results about Appell polynomials and investigates their various analogs. The primary of these are the free Appell polynomials. In the multivariate case, they can be considered as natural analogs of the Appell polynomials among polynomials in noncommuting variables. They also fit well into the framework of free probability. For the free Appell polynomials, a number of combinatorial and “diagram” formulas are proven, such as the formulas for their linearization coefficients. An explicit formula for their generating function is obtained. These polynomials are also martingales for free Levy processes. For more general free Sheffer families, a necessary condition for pseudo-orthogonality is given. Another family investigated is that of the Kailath-Segall polynomials. These are multivariate polynomials, which share with the Appell polynomials nice combinatorial properties, but are always orthogonal. Their origins lie in the Fock space representations, or in the theory of multiple stochastic integrals. Diagram formulas are proven for these polynomials as well, even in the q-deformed case.
Subfields: Algebra and Number Theory, Applied Mathematics
View Paper: https://www.semanticscholar.org/paper/722a22edde77f289ba7f46a5e5a4f002d4d1a3f3
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In Algebra and Number Theory
In Applied Mathematics
Citations in the corpus
- Conditional moments of q-Meixner processes (2004, background): “Non-com m utative processes with free increm ents can be thought as a special casecorresponding to thevalueq= 0 ofthem oregeneralclassofq-L evy processes [3],[6].”
- Second order freeness and fluctuations of random matrices: I. Gaussian and Wishart matrices and cyclic Fock spaces (2004, background): “Again one knows (see [ Ans , Theorem 4.11]) that the linear Wick polynomials Wn(d) := W(d ⊗n ) are”
- Askey--Wilson polynomials, quadratic harnesses and martingales (2008, background): “Orthogonal martingale polynomials play also a prominent role in non-commutative probability [1, 2], and can serve as a connection to the so called “classical versions” of non-commutative processes.”