Legacy
Peter Wynn died December 2017. During the last years of his life, he was living in Zacatecas, Mexico. Each year, he came back to the United States for visiting his friends, Manuel Philip Berriozábal and his wife Maria Antonietta, in San Antonio and left them boxes containing unpublished handwritten mathematical documents he did not want to keep with him in Mexico. After the publication, in January 2019, of an issue of the journal Numerical Algorithms (EiC Claude Brezinski), dedicated to Peter Wynn, Claude Brezinski and Michela Redivo-Zaglia were contacted by F. Alexander Norman, a colleague of these friends, in January 2020. He informed them of the existence of these documents. This is how Wynn's legacy came to light.

We think that these manuscripts are valuable additions to the literature on the topics treated, and that they cannot be left unknown since they contain ideas that were never exploited. They can lead to new research and results.
They concern orthogonal polynomials, extrapolation methods, Padé approximation, continued fractions, moment problems, Thiele and rational interpolation, complex analysis, series, software, analysis and abstract algebra.

These documents have been analyzed in the Open Access paper:

C. Brezinski, F.A. Norman, M. Redivo-Zaglia
The Legacy of Peter Wynn
Mathematics, 9 (2021) 1240.
doi:10.3390/math9111240.

The pdf of the documents in this section are in a low-resolution to facilitate their download. A pdf with a better resolution can be obtained by sending an e-mail to one of us (see the section Contacts). The Bürmann series of a function $$f$$ is a generalized form of a Taylor series in which, instead of a series in powers of $$z-z_0$$, we have a series in powers of the analytic function $$\phi(z)-\phi(z_0)$$. It is used in the reversion of series which consists, starting from $$z=f(w)$$, in expressing $$w$$ as $$w=g(z)$$. This paper of 187 pages is concerned with functions of the form $$f(z)=1+\frac{az}{1-az/2+zs(z)}$$ where $$0<a<\infty$$, and $$s(z)=\int_0^\infty \frac{d\psi(t)}{1+z^2t}$$ where $$\psi(t)$$ is a nondecreasing function of bounded variation for $$0 \leq t \leq \infty$$ such that all moments $$c_\nu=\int_0^\infty t^\nu \; d\psi(t)$$ for $$\nu=0,1,\ldots$$ exist. This manuscript has 179 pages with many portions scratched. The problem is to determine a function which satisfies an asymptotic relationship. The proof of the principal theorem is largely based upon results, due to M. Riesz, in the theory of linear functionals. The document is 202 pages long. It dates from December 1976. The first part of the document is in a quite good shape. Results concerning the convergence of forward diagonal sequences of quotients in the Padé table are given. 13 pages. Wynn proposes to give a convergence theorem for the functions $$\varrho_{2r}\{\psi(\mu)\}$$, where, presumably, $$\varrho_{2r}$$ is the $$2r$$th convergent of Thiele's continued fraction. Very well handwritten manuscript has 52 pages. The problem of deriving an extrapolation method from an interpolatory formula and its converse are considered. Related topics are treated. An handwritten complete paper of 90 pages with a bibliography of 27 items. It is about the Thiele-Nörlund interpolation theory. Algorithms are discussed. This document might be the sequel of a report published in 1971 when Wynn was in Montréal. All topics treated are in the noncommutative case. The document begins with a discussion about the application of Euclid's algorithm to rational functions in a ring with a zero and a unit element. Another one on noncommutative continued fractions follows. Recurrence relations for the numerators and the denominators of their successive convergents are given. Vector continued fractions, vector Padé approxiamants, and the ε-algorithm are discussed. This pedagogical document contains 49 pages that correspond to the slides for a talk. It gives an interesting new interpretation of extrapolation by the ε-algorithm. This 120 pages document is clearly a preparatory work for a book. It contains an incredible succession of theorems whose proofs are not given, and comments and explanations between them are missing. It is divided into 23 chapters. This document presents a total of 266 pages. These notes seem to be in a final form. It is about stratified commutative rings. This document of 105 pages probably follows the document (Theory of stratified commutative field). It contains theoretical notes about determinantal identities and LU decomposition over stratified commutative ring. This theoretical document, without title, contains 37 pages. Probably it is also intended to follow the work (Theory of stratified commutative field). This document contains 43 well written pages. It contains formulas between the elements of a nonsingular lower triangular matrix, and those of its inverse. (Numal) was a library of numerical algorithms written in Algol 60 and developed by members of the Mathematical Center in Amsterdam. This document is the presentation of a translation into Fortran of a this library. This is a one page typewritten document ready to be published. It gives necessary and sufficient conditions for $$\alpha$$ and $$\beta$$, belonging to a division algebra of generalized Cayley numbers, to satisfy the relationship $$\alpha \beta=\beta \alpha$$. This is a 8 pages typewritten paper ready to be published. It is on the precise locations for the roots with large modulus of the equation $$[\nu/\mu]_{e^z}=e^z$$. Pages are missing. Conditions for a function to map the left complex plane into the unique disk are given. This document starts from the page numbered 8. The two algorithms that are generalized are processes to compute infinite integrals. An attempt to extract a totally oscillating sequence from the successive convergents of a continued fraction is presented. This is an unfinished document of 60 pages, not numbered but in a good shape. It is probably a part of a monograph. These notes concern interpolation in a field by the use of polynomials and rational functions, the interpolatory argument values being assumed to be discrete. This document of 38 pages (numbered from the 9th one) begins by a first list of 7 topics Wynn wanted to treat. Then, 8 pages that look like two tables of contents with several topics covered follow. It is proposed to study the systems of solutions of equations of the form $$bx =a \quad \mod I$$ where, $$W$$ being a prescribed commutative ring, $$a,b,x \in W$$, and $$I$$ being a system of numbers in $$W$$, the above equation is to be interpreted in the sense that $$u \in I$$ exists for which $$bx-a=u$$. This is a theoretical handwritten draft of 81 pages about what Wynn named (Aggregates). It is quite difficult to understand what it is about since the manuscript is not entirely well written, and seems not to have been finished. This is a long theoretical text of 115 pages with this title, where $$c, E$$ and $$d$$ are mapping systems. Wynn treated the existence of domains of constancy and intersection mapping systems. Then there is a section on invariance of spaces with respect to sequence rearrangement, one on the properties of domains of constancy, another one on properties of prequotient spaces with respect to domains of constancy and intersection systems, a section giving complete factorisation results, and one on spaces of ordered pairs of mapping systems. This document is a typewritten curriculum vitæ by Wynn himself. Lists of documents possessed by Wynn together with rough notes. One of them seems to show that Wynn was trying to express the vectors computed by his vector ε-algorithm as ratios of some kind of (generalized) determinants. This is a list of activities that Wynn has to accomplished. There are 3 pages containing a list of projects. Some of them were probably realized (those preceded by a D) and some of them, with an O, perhaps meaning Ongoing. It is a list of references by various authors, probably a list of Xerox copied papers. This document contains two papers published in the journal Ukrainskii Matematicheskii Zhurnal (Ukrainian Mathematical Journal) in Russian. There are also four handwritten reviews written for zbMATH.
01-Special forms of complementary interpolatory functions of second kind
03-Extensions of determinantal relationships
05-Expression of corresponding continued fraction in closed form
07-The expression of moments as continued fraction integrals
09-Integral transforms of continued fractions an array of functions
11-Numerical experiments in nonassociative algebras
12-Algorithms applied to formal power series
13-The principal and alternating sums of a function defined over a strip in the complex plane
14-Variants of the remainder terms of the Euler-Maclaurin and Boole series
15-The analytic continuation of functions defined by an integral transform