Well, Bernoulli umbra is an umbra whose moments are the Bernoulli numbers.

But what is it philosophically?

For instance, we can consider imaginary unit $i$ an umbra with moments $\{1,0,−1,0,1,\ldots\}$, hyperbolic unity $j$ as an umbra with moments $\{1,0,1,0,1,0,\dots\}$.

Can we in the same way somehow think about Bernoulli umbra as some kind of a hypercomplex "number", vector or series or whatever other object that can be geometrically represented and imagined, that has algebraic properties besides having those moments?

Is there any geometric or set-theoretic object that represents Bernoulli umbra?

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    $\begingroup$ Where is a definition of this concept? I looked at the Wikipedia page, but it doesn't really say much. $\endgroup$ Commented Jan 28, 2023 at 4:17
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    $\begingroup$ At first glance it looks a like "umbral calculus" is a type of formula mysticism. $\endgroup$ Commented Jan 28, 2023 at 4:22
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    $\begingroup$ I suspect the downvotes are by people who don't know what the umbral calculus is. The question seems perfectly fine to me, though it's true that it would have been good to include some references. For those unfamiliar with the umbral calculus, I recommend Gessel's Applications of the classical umbral calculus as a starting point. $\endgroup$ Commented Jan 28, 2023 at 13:54
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    $\begingroup$ I see now that the OP has already posted several related questions here on MO; e.g., here, here, here and here. $\endgroup$ Commented Jan 28, 2023 at 14:32
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    $\begingroup$ Also here and here. Plus some of these have been cross-posted to math.SE as well. $\endgroup$ Commented Jan 28, 2023 at 14:40

1 Answer 1


Since you used the terms hypercomplex and geometric, I interpret your question as asking whether people have found any connection between Bernoulli umbrae and Clifford algebra. I'm not too familiar with this area of research myself, but the answer seems to be yes. See for example Bernoulli and Euler Polynomials in Clifford Analysis by G.F. Hassan and L. Aloui (Advances in Applied Clifford Algebras 25 (2015), 351–376) or (Discrete) Almansi Type Decompositions: An umbral calculus framework based on $\mathfrak{osp}(1|2)$ symmetries by Nelson Faustino and Guangbin Ren (arXiv:1102.5434) and the references therein.


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