## 11B68 Bernoulli and Euler numbers and polynomials

### Refine

#### Keywords

- Algebra (1)
- Bernoulli convolutions (1)
- Bernoulli-Faltungen (1)
- Bialgebra (1)
- Eulerian numbers (1)
- Fock-Raum (1)
- Funktionalanalysis (1)
- GNS-Konstruktion (1)
- Grenzwertsatz (1)
- Hopf-Algebra (1)

We consider Iterated Function Systems (IFS) on the real line and on the complex plane. Every IFS defines a self-similar measure supported on a self-similar set. We study the transfer operator (which acts on the space of continuous functions on the self-similar set) and the Hutchinson operator (which acts on the space of Borel regular measures on the self-similar set). We show that the transfer operator has an infinitely countable set of polynomial eigenfunctions. These eigenfunctions can be regarded as generalized Bernoulli polynomials. The polynomial eigenfuctions define a polynomial approximation of the self-similar measure. We also study the moments of the self-similar measure and give recursions for computing them. Further, we develop a numerical method based on Markov chains to study the spectrum of the Hutchinson and transfer operators. This method provides numerical approximations of the invariant measure for which we give error bounds in terms of the Wasserstein-distance. The standard example in this thesis is the parametric family of Bernoulli convolutions.

This thesis revolves around a new concept of independence of algebras. The independence nicely fits into the framework of universal products, which have been introduced to classify independence relations in quantum probability theory; the associated product is called (r,s)-product and depends on two complex parameters r and s. Based on this product, we develop a theory which works without using involutive algebras or states. The following aspects are considered: 1. Classification: Universal products are defined on the free product of algebras (the coproduct in the category of algebras) and model notions of independence in quantum probability theory. We distinguish universal products according to their behaviour on elements of length two, calling them (r,s)-universal products with complex parameters r and s respectively. In case r and s equal 1, Muraki was able to show that there exist exactly five universal products (Murakiâ€™s five). For r equals s nonzero we get five one parameter families (q-Murakiâ€™s five). We prove that in the case r not equal to s the (r,s)-product, a two parameter deformation of the Boolean product, is the only universal product satisfying our set of axioms. The corresponding independence is called (r,s)-independence. 2. Dual pairs and GNS construction: By use of the GNS construction, one can associate a product of representations with every positive universal product. Since the (r,s)-product does not preserve positivity, we need a substitute for the usual GNS construction for states on involutive algebras. In joint work with M. Gerhold, the product of representations associated with the (r,s)-product was determined, whereby we considered representations on dual pairs instead of Hilbert spaces. This product of representations is - as we could show - essentially different from the Boolean product. 3. Reduction and quantum LÃ©vy processes: U. Franz introduced a category theoretical concept which allows a reduction of the Boolean, monotone and antimonotone independence to the tensor independence. This existing reduction could be modified in order to apply to the (r,s)-independence. Quantum LÃ©vy processes with (r,s)-independent increments can, in analogy with the tensor case, be realized as solutions of quantum stochastic differential equations. To prove this theorem, the previously mentioned reduction principle in the sense of U. Franz and a generalization of M. SchÃ¼rmannâ€™s theory for symmetric Fock spaces over dual pairs are used. As the main result, we obtain the realization of every (r,s)-LÃ©vy process as solution of a quantum stochastic differential equation. When one, more generally, defines LÃ©vy processes in a categorial way using U. Franzâ€™s definition of independence for tensor categories with inclusions, compatibility of the inclusions with the tensor category structure plays an important role. For this thesis such a compatibility condition was formulated and proved to be equivalent to the characterization proposed by M. Gerhold. 4. Limit distributions: We work with so-called dual semigroups in the sense of D. V. Voiculescu (comonoids in the tensor category of algebras with free product). The polynomial algebra with primitive comultiplication is an example for such a dual semigroup. We use a "weakened" reduction which we call reduction of convolution and which essentially consists of a cotensor functor constructed from the symmetric tensor algebra. It turns dual semigroups into commutative bialgebras and also translates the convolution exponentials. This method, which can be nicely described in the categorial language, allows us to formulate central limit theorems for the (r,s)-independence and to calculate the correponding limit distributions (convergence in moments). We calculate the moments appearing in the central limit theorem for the (r,s)-product: The even moments are homogeneous polynomials in r and s with the Eulerian numbers as coefficients; the odd moments vanish. The moment sequence that we get from the central limit theorem for an arbitrary universal product is the moment sequence of a probability measure on the real line if and only if r equals s greater or equal to 1. In this case we present an explicit formula for the probability measure.