On the Computation of Spectra in Free Probability

Lehner, Franz

doi:10.1006/jfan.2001.3748

Cited by 9 publications

(9 citation statements)

References 18 publications

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“…Thus, Z N is of degree 3N + 2, Z N +1 ≡ Z N (mod x 3N +3 ). After reaching enough terms, plug this into (7). This agrees perfectly with (1), which was calculated by a direct count.…”

Section: The Identity In Proposition 4 Can Be Written Assupporting

confidence: 79%

“…The equalities ( 6) and (7) give a polynomial-time method to calculate coefficients t(n). Indeed, let us start from Z 1 (x) = x 5 (the first primitive a−word is SU 3 S), and let us define polynomials Z N (x) ∈ Z[x] recurrently by…”

Section: Letmentioning

confidence: 99%

“…Franz Lehner has pointed out that the question in consideration is a special case of a "free convolution" and can be obtained via Voiculescu-Woess transform [7,10,11,12]. This technique is implemented as a package and it is part of the library of FriCAS.…”

Section: We Also Introducementioning

confidence: 99%

“…This is the sequence A265434 in [13]. The second sequence t(n), n ≥ 0, starts from 1, 0, 1, 1, 0, 3, 0, 5,3,7,16,12,50,44,123,195,301,718,928,2244, . .…”

mentioning

confidence: 99%

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The modular group and words in its two generators*

Alkauskas

2017

Lith Math J

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Consider the full modular group PSL 2 (Z) with presentation U, S|U 3 , S 2 . Motivated by our investigations on quasi-modular forms and the Minkowski question mark function (so that this paper might be considered as a necessary appendix), we are lead to the following natural question. Some words in the alphabet {U, S} are equal to the unity; for example, U SU 3 SU 2 is such a word of length 8, and U SU 3 SU SU 3 S 3 U is such a word of length 15. We consider the following integer sequence. For each n ∈ N 0 , let t(n) be the number of words in alphabet {U, S} that equal the identity in the group. This is the new entry A265434 into the Online Encyclopedia of Integer Sequences. We investigate the generating function of this sequence and prove that it is an algebraic function over Q(x) of degree 3. As an interesting generalization, we formulate the problem of describing all algebraic functions with a Fermat property.

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“…Thus, Z N is of degree 3N + 2, Z N +1 ≡ Z N (mod x 3N +3 ). After reaching enough terms, plug this into (7). This agrees perfectly with (1), which was calculated by a direct count.…”

Section: The Identity In Proposition 4 Can Be Written Assupporting

confidence: 79%

Section: Letmentioning

confidence: 99%

Section: We Also Introducementioning

confidence: 99%

“…This is the sequence A265434 in [13]. The second sequence t(n), n ≥ 0, starts from 1, 0, 1, 1, 0, 3, 0, 5,3,7,16,12,50,44,123,195,301,718,928,2244, . .…”

mentioning

confidence: 99%

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The modular group and words in its two generators*

Alkauskas

2017

Lith Math J

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“…We prove equation (32); the proof of ( 30) is similar, and also implied by equation (34) below. In fact, we will prove a more general statement, that (33) R A (X u 1 ) , . .…”

Section: S(k) Note That (30) Is the Linearization Coefficient For Th...mentioning

confidence: 91%

Appell polynomials and their relatives II. Boolean theory

Anshelevich¹

2009

Indiana Univ. Math. J.

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ABSTRACT. The Appell-type polynomial family corresponding to the simplest non-commutative derivative operator turns out to be connected with the Boolean probability theory, the simplest of the three universal non-commutative probability theories (the other two being free and tensor/classical probability). The basic properties of the Boolean Appell polynomials are described. In particular, their generating function turns out to have a resolvent-type form, just like the generating function for the free Sheffer polynomials. It follows that the Meixner (that is, Sheffer plus orthogonal) polynomial classes, in the Boolean and free theory, coincide. This is true even in the multivariate case. A number of applications of this fact are described, to the Belinschi-Nica and Bercovici-Pata maps, conditional freeness, and the Laha-Lukacs type characterization.A number of properties which hold for the Meixner class in the free and classical cases turn out to hold in general in the Boolean theory. Examples include the behavior of the Jacobi coefficients under convolution, the relationship between the Jacobi coefficients and cumulants, and an operator model for cumulants. Along the way, we obtain a multivariate version of the Stieltjes continued fraction expansion for the moment generating function of an arbitrary state with monic orthogonal polynomials.

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k-divisible random variables in free probability

Arizmendi

2018

Advances in Applied Mathematics

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We introduce and study the notion of k-divisible elements in a non-commutative probability space. A k-divisible element is a (non-commutative) random variable whose n-th moment vanishes whenever n is not a multiple of k.First, we consider the combinatorial convolution * in the lattices N C of noncrossing partitions and N C k of k-divisible non-crossing partitions and show that convolving k times with the zeta-function in N C is equivalent to convolving once with the zeta-function in N C k . Furthermore, when x is k-divisible, we derive a formula for the free cumulants of x k in terms of the free cumulants of x, involving k-divisible non-crossing partitions.Second, we prove that if a and s are free and s is k-divisible then sps and a are free, where p is any polynomial (on a and s) of degree k − 2 on s. Moreover, we define a notion of R-diagonal k-tuples and prove similar results.Next, we show that free multiplicative convolution between a measure concentrated in the positive real line and a probability measure with k-symmetry is well defined. Analytic tools to calculate this convolution are developed.Finally, we concentrate on free additive powers of k-symmetric distributions and prove that µ t is a well defined probability measure, for all t > 1. We derive central limit theorems and Poisson type ones. More generally, we consider freely infinitely divisible measures and prove that free infinite divisibility is maintained under the mapping µ → µ k . We conclude by focusing on (k-symmetric) free stable distributions, for which we prove a reproducing property generalizing the ones known for one sided and real symmetric free stable laws.

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On the Computation of Spectra in Free Probability

Cited by 9 publications

References 18 publications

The modular group and words in its two generators*

The modular group and words in its two generators*

Appell polynomials and their relatives II. Boolean theory

k-divisible random variables in free probability

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