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fock space
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Appell polynomials and their relatives
Appell polynomials and their relatives,10.1155/S107379280413345X,Michael Anshelevich
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Appell polynomials and their relatives
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Citations: 17
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Michael Anshelevich
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 pseudoorthogonality is given. Another family investigated are the KailathSegall 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 qdeformed case.
Published in 2003.
DOI:
10.1155/S107379280413345X
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Citation Context
(13)
...Orthogonal martingale polynomials play also a prominent role in noncommutative probability [1,
2
], and can serve as a connection to the so called “classical versions” of noncommutative processes...
Włodek Bryc
,
et al.
Askey–Wilson polynomials, quadratic harnesses and martingales
...A deep study of free Meixner polynomials of d (d ∈ N) noncommutative variables has been carried out by Anshelevich in [
3
, 5, 6, 7]...
...(Note that, in [
3
, 4], elements of Gn were called extended partitions, with classes labeled +1 called “classes open on the left”.)...
...where c({) ∈ R (compare with [20, Section 4] and [
3, Section 3
])...
Marek Bozejko
,
et al.
Meixner Class of NonCommutative Generalized Stochastic Processes with...
...(In fact, such polynomials had already occurred in many places in the literature even before [2, 19], see [9, p. 62] and [4, p. 864] for bibliographical references.) A deep study of multivariate free Meixner polynomials of noncommutative variables has been carried out by Anshelevich, see [
3
, 4, 5] and the references therein...
...We also refer to [2,
3
, 10] for qinterpolation of the classical and free Meixner classes...
EUGENE LYTVYNOV
,
et al.
LOWERING AND RAISING OPERATORS FOR THE FREE MEIXNER CLASS OF ORTHOGONA...
...The natural qdeformation that interpolates the forementioned families for arbitrary q 1 was defined and studied in [
3
] and is up to ane transformations the socalled AlSalam and Chihara family of orthogonal polynomials ([1])...
Nizar Demni
.
ULTRASPHERICAL TYPE GENERATING FUNCTIONS FOR
...Here the adjective “free” refers to their rela tion to free probability [20], see [3,
4
] for more details...
...The parallel between propositions 1 and 2 can be explained by noticing that they are both particular cases of a more general theorem involving the generating function of a specific basic hypergeometric form, see [1] or Theorem 4.8 of [
4
]...
...So for the rest of the paper, we take as the definition of free cumulants the following impl icit functional relation, see Section 13 of [16] or Proposition 3.1 of [
4
]:...
...Proposition 5. [
4, Theorem 3.21
] Suppose that a family of free Sheffer polyn omials is pseudoorthogonal...
...The following result was already used in the proof of Theorem 3.21 of [
4
]; here we formulate it as a lemma...
Michael Anshelevich
.
Orthogonal polynomials with a resolventtype generating function
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Citations
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(
Citations: 3
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(
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ULTRASPHERICAL TYPE GENERATING FUNCTIONS FOR
Nizar Demni
Published in 2009.