I am reading Proofs and Confirmations by David Bressoud. On page $150$ is a long excerpt by Richard Askey, from "How can mathematicians and mathematical historians help each other?" There is mentioned

R. Narasimhan told me that he found a definition of "comfortable" series in one of the late volumes of Euler's collected works. For Euler, a comfortable series is a power series whose term ratio is a rational function of $n$. When I asked Narasimhan to give me a specific reference, he was unable to find it again. I will be very pleased to pay $\\\$50$ U.S. for this reference.

When I did a Google Search for this using Euler "comfortable series", only a handful of matches were found. Has this Euler reference been recovered yet?

  • $\begingroup$ For context: Notices of the American Mathematical Society, Vol. 31, No. 2, February 1984, p. 164: "Queries 296. Richard Askey [...] R. Narasimhan found a passage in Euler where Euler referred to "comfortable series" as series whose term ratio is a rational function. Thus Euler isolated the essential property of hypergeometric series. Narasimhan thinks this is in one of the recently published volumes of Euler's collected works, but he has forgotten where. Can someone find this again?" $\endgroup$
    – njuffa
    Dec 12, 2022 at 21:11
  • $\begingroup$ The volume of Euler's Opera Omnia published by Birkhauser/Springer in closest temporal proximity to Askey's query appears to be Correspondance de Leonhard Euler avec A. C. Clairaut, J. d'Alembert et J. L. Lagrange (1980), but I have yet to find a comprehensive list (with date of publication) of the books in the series. $\endgroup$
    – njuffa
    Dec 12, 2022 at 21:43
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    $\begingroup$ How sure are you that the term in English should be comfortable series? That term sounds so awkward. I wonder if the original Latin term should be translated as "convenient series" or "suitable series". Reason #1: Euler did write about "numeri idonei", which is often rendered in English as idoneal numbers, but really means convenient/suitable numbers. It is easy to find accounts of his work on such numbers. Reason #2: both "comfortable" and "convenient" can be translated into Russian as the same word (удобный), and Russians learning English often confuse them. Is it also so with Latin? $\endgroup$
    – KCd
    Dec 13, 2022 at 5:58
  • $\begingroup$ If the "comfortable series" (as quoted by Askey in his 1984 query; see comment above) appeared in the volume of Opera Omnia published closest to his query, it would have been part of Euler's French correspondence. How solid was Euler's French? I do not know. For now I am assuming we are looking for instances of the French word "confortable" in Euler's writing. I don't have access to Opera Omnia to check. $\endgroup$
    – njuffa
    Dec 13, 2022 at 6:51
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    $\begingroup$ @MBN Euler's "comfortable series" is what is today called hypergeometric series. "Удобные числа Эйлера", i.e. the idoneal numbers, are something else. $\endgroup$
    – Conifold
    Dec 17, 2022 at 9:24


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