1. Introduction This paper has three parts. The first part is a simplified presentation of the basic ideas of the renormalization group and the ε expansion applied to critical phenomena , following roughly a summary exposition given in 1972 1. The second part is an account of the history (as I remember it) of work leading up to the papers in I971-1972 on the renormalization group. Finally, some of the developments since 197 1 will be summarized, and an assessment for the future given.

The renormalization group and critical phenomena

We list the so-called Askey-scheme of hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation and generating functions of all classes of orthogonal polynomials in this scheme. In chapeter 2 we give all limit relation between different classes of orthogonal polynomials listed in the Askey-scheme. 
In chapter 3 we list the q-analogues of the polynomials in the Askey-scheme. We give their definition, orthogonality relation, three term recurrence relation and generating functions. In chapter 4 we give the limit relations between those basic hypergeometric orthogonal polynomials. Finally in chapter 5 we point out how the `classical` hypergeometric orthogonal polynomials of the Askey-scheme can be obtained from their q-analogues.

/pdf/the-askey-scheme-of-hypergeometric-orthogonal-polynomials-46afheftah.pdf

The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue

Nonlinear Realization and Hidden Local Symmetries

Zeros of orthogonal polynomials

This article is devoted to the perturbative renormalization of the abelian Higgs-Kibble model, within the class of renormalizable gauges which are odd under charge conjugation. The Bogoliubov Parasiuk Hepp-Zimmermann renormalization scheme is used throughout, including the renormalized action principle proved by Lowenstein and Lam. The whole study is based on the fulfillment to all orders of perturbation theory of the Slavnov identities which express the invariance of the Lagrangian under a supergauge type family of non-linear transformations involving the Faddeev-Popov ghosts. Direct combinatorial proofs are given of the gauge independence and unitarity of the physicalS operator. Their simplicity relies both on a systematic use of the Slavnov identities as well as suitable normalization conditions which allow to perform all mass renormalizations, including those pertaining to the ghosts, so that the theory can be given a setting within a fixed Fock space. Some simple gauge independent local operators are constructed.

/pdf/renormalization-of-the-abelian-higgs-kibble-model-14fr0vp7oj.pdf

Renormalization of the Abelian Higgs-Kibble Model

For the example of the infinite well potential, we point out some paradoxes which are solved by a careful analysis of what is a truly self-adjoint operator. We then describe the self-adjoint extensions and their spectra for the momentum and the Hamiltonian operators in different settings. Additional physical requirements such as parity, time reversal, and positivity are used to restrict the large class of self-adjoint extensions of the Hamiltonian.

Self-adjoint extensions of operators and the teaching of quantum mechanics

Linear birth and death models and associated Laguerre and Meixner polynomials

We consider two indeterminate moment problems: One corresponding to a birth and death process with quartic rates and the other corresponding to the Al-Salam-Carlitz q-polynomials. Using the Darboux method, we calculate their Nevanlinna matrices and several families of orthogonality measures.

https://www.intlpress.com/site/pub/files/_fulltext/journals/maa/1994/0001/0002/MAA-1994-0001-0002-a003.pdf

The Nevanlinna parametrization for some indeterminate Stieltjes moment problems associated with birth and death processes

On a class of compact and non-compact quasi-Einstein metrics and their renormalizability properties

Quantum integrability of classical integrable systems given by quadratic Killing tensors on curved configuration spaces is investigated. It is proven that, using a “minimal” quantization scheme, quantum integrability is ensured for a large class of classic examples.

Galliano Valent

Papers

Self-adjoint extensions of operators and the teaching of quantum mechanics

Linear birth and death models and associated Laguerre and Meixner polynomials

The Nevanlinna parametrization for some indeterminate Stieltjes moment problems associated with birth and death processes

On a class of compact and non-compact quasi-Einstein metrics and their renormalizability properties

Quantum integrability of quadratic Killing tensors