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Often times, we consult resources, like Abramowitz and Stegun's Handbook of Mathematical Functions https://www.math.ubc.ca/~cbm/aands/, NIST's database on special functions https://www.nist.gov/programs-projects/special-functions, or Mathematica to find identities which aid us with some kind of computation.

However, what if we want to know if we have found a new identity, want to systematically check against the above resources, and want to add to the library in the case the identity is new? Also, are there journals which, even today, still consider mathematical effort toward discovering identities of classical functions?

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    $\begingroup$ Sum[Binomial[2j,j](Cos[x]/2)^(2j),{j,0,\[Infinity]}] in Mathematica gives 1/Sqrt[Sin[x]^2] $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Sep 1, 2021 at 20:19
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    $\begingroup$ @მამუკაჯიბლაძე Thanks! I've removed this part of the question so that more important (soft) question remains. $\endgroup$
    – garserdt216
    Commented Sep 1, 2021 at 20:38
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    $\begingroup$ If you really discovered a new identity, just contact the editors of NIST, and ask for inclusion. $\endgroup$
    – Alexandre Eremenko
    Commented Sep 1, 2021 at 21:35
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    $\begingroup$ By the way, $\sum_{j=0}^\infty {2j \choose j} x^j = (1-4x)^{-1/2}$ is well known and your result follows. $\endgroup$
    – Somos
    Commented Sep 2, 2021 at 23:54

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Q: Are there journals which would publish identities of classical functions?

A: Elsevier's Applied Mathematics and Computation has published quite a number of papers in that category, see this search listing.

It is ranked as a Q1 journal, open access is optional.

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    $\begingroup$ In case anyone is wondering, "Q1 journal" appears to means this has a top quartile rank, in somebody's ranking. $\endgroup$
    – Ryan Budney
    Commented Sep 2, 2021 at 4:22
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    $\begingroup$ some of us face the complication when choosing a publication venue that only Q1 journals count for things like promotion or tenure, so this might be relevant data; the full list in mathematics is here $\endgroup$
    – Carlo Beenakker
    Commented Sep 2, 2021 at 6:19

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