Padé approximants are often better than Taylor series at representing a function. Given a Taylor series, one can use Wynn's epsilon algorithm to easily produce the Padé approximants to it.


[Volterra series](https://en.wikipedia.org/wiki/Volterra_series) are a generalization of Taylor series that can also model "memory" phenomena. Does there exist a similar generalization of Padé approximants that can model these phenomena, or an algorithm like Wynn's to compute them from the Volterra series?


I would be at least happy to know the answer for the discrete Volterra series, which (I think) would be equivalent to something like a multivariate Padé approximant.


[Originally asked at MSE](https://math.stackexchange.com/questions/3297364/taylor-series-is-to-volterra-series-as-pad%C3%A9-approximant-is-to), but seems too advanced for that site.



“Taylor series” is to “Volterra series” as “Padé approximant” is to _________?

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> "I would be at least happy to know the answer for the discrete Volterra series, which (I think) would be equivalent to something like a multivariate Padé approximant."
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> 
> 


Multi-variate versions of Padé approximants got attention in the 1970s, for example a generalization to functions of two variables was published by J.S.R. Chisolm in 1973 (see ["**Rational Approximates Defined from
Double Power Series**"](https://www.ams.org/journals/mcom/1973-27-124/S0025-5718-1973-0382928-6/)) and a generalization to functions of ***countably*** more variables was studied at around the same time by P.R. Graves-Morris and D.E. Roberts who were both at the University of Kent in Canterbury, leading to this generalization being called the "Canterbury approximant" (see ["**Calculation of Canterbury approximants**"](https://www.sciencedirect.com/science/article/abs/pii/0010465575900685?via%3Dihub)).


The original papers about Canterbury approximants never seemed to get many citations, so it's easy to look through the entire list of papers that cited them to find ***further*** generalizations (for example, to functions of an uncountably infinite number of variables). Some of the most significant post-Canterbury papers on multivariate versions of Padé approximants that can be found in this list are:


* [Annie Cuyt "**How well can the concept of Padé approximant be generalized to the multivariate case?**" (May 1999)](https://www.sciencedirect.com/science/article/pii/S037704279900028X)
* [Philippe Guillaume and Alain Huard. "**Multivariate Padé approximation**" (September 2000)](https://www.sciencedirect.com/science/article/pii/S037704270000337X)


Finally, one of the original authors of the Canterbury approximant co-wrote a book with George Baker Jr. called "[**Padé Approximants**](https://books.google.ca/books?hl=en&lr=&id=Vkk4JNLKbLoC&oi=fnd&pg=PP1&ots=kUWgJtTqM0&sig=rFjYf82DbW2YwlNPUzs4pdfpKUQ#v=onepage&q&f=false)" (first edition: 1982, second edition: 1996) which provides you with any developments which they considered significant between the aforementioned papers from the 1970s and about the time of the aforementioned papers from 1999-2000.


Finally, some steps towards a generalization to functions of ***complex*** variables can be found in my analogous question: [“Taylor series” is to “Volterra series” as “Laurent series” is to \_\_\_\_\_\_\_\_\_?](https://mathoverflow.net/q/413016/66334)


