Multinomial Identities Arising from the Free Probability Theory
نویسنده
چکیده
PIOTR´SNIADY ABSTRACT. We prove a family of new identities fulfilled by multino-mial coefficients, which were conjectured by Dykema and Haagerup. Our method bases on a study of the, so–called, triangular operator T by the means of the free probability theory.
منابع مشابه
Multinomial identities arising from free probability theory
1.1. Overview. In order to answer some questions in the theory of operator algebras Dykema and Haagerup started investigation of the, so–called, triangular operator T [DH01]. Currently there are many different descriptions of this operator: in terms of random matrices, in terms of free probability theory and a purely combinatorial one (and we will recall them in the following). Dykema and Haage...
متن کاملHigher Order Degenerate Hermite-Bernoulli Polynomials Arising from $p$-Adic Integrals on $mathbb{Z}_p$
Our principal interest in this paper is to study higher order degenerate Hermite-Bernoulli polynomials arising from multivariate $p$-adic invariant integrals on $mathbb{Z}_p$. We give interesting identities and properties of these polynomials that are derived using the generating functions and $p$-adic integral equations. Several familiar and new results are shown to follow as special cases. So...
متن کاملNegative Binomial and Multinomial States: probability distributions and coherent states
Following the relationship between probability distribution and coherent states, for example the well known Poisson distribution and the ordinary coherent states and relatively less known one of the binomial distribution and the su(2) coherent states, we propose interpretation of su(1, 1) and su(r, 1) coherent states in terms of probability theory. They will be called the negative binomial (mul...
متن کاملAsymptotic Normality Through Factorial Cumulants and Partition Identities
In the paper we develop an approach to asymptotic normality through factorial cumulants. Factorial cumulants arise in the same manner from factorial moments as do (ordinary) cumulants from (ordinary) moments. Another tool we exploit is a new identity for 'moments' of partitions of numbers. The general limiting result is then used to (re-)derive asymptotic normality for several models including ...
متن کاملOptimal Selection of the Most Probable Multinomial Alternative
Multinomial selection is concerned with selecting the most probable (best) multinomial alternative. The alternatives compete in a number of independent trials. In each trial, each alternative wins with an unknown probability specific to that alternative. A long-standing research goal has been to find a procedure that minimizes the expected number of trials subject to a lower bound on the probab...
متن کامل