Product of a sequence identities

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Discussion Overview

The discussion centers around the search for product identities of sequences, specifically looking for comprehensive resources rather than specific examples, with a focus on non-trivial identities and their mathematical implications.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant seeks a resource for product identities of sequences, indicating that previous searches yielded only trigonometric identities.
  • Another participant provides links to the Wallis product and a list of formulae involving π, questioning whether the original poster had already explored these resources.
  • A later reply reiterates the need for a comprehensive list of identities, expressing interest in what constitutes a "non-trivial" product identity and discussing the relationship between additive and product identities.
  • One participant shares a link to a resource they found useful, suggesting it meets the original request.

Areas of Agreement / Disagreement

Participants generally agree on the need for a comprehensive list of product identities, but there is no consensus on what constitutes a "non-trivial" identity or the methods for transforming product identities into additive identities.

Contextual Notes

The discussion includes assumptions about the definitions of product and additive identities, as well as the conditions under which transformations between them may or may not be valid.

eddybob123
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HI, does anyone know a decent site where I can find a few product identities? I googled it, but all that came up were trig identities. I am not looking for those; I am specifically looking for product of a sequence identities: ∏
 
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Those are quite interesting, but not what I am looking for. I mean if there are any sites that post a full list of identities, not just specific ones. That would be very helpful.
 
eddybob123 said:
Those are quite interesting, but not what I am looking for. I mean if there are any sites that post a full list of identities, not just specific ones. That would be very helpful.

It's an interesting question what would constitute a "non-trivial" product identity. An additive identity \sum_{i=1}^{\infty} X_i = Y implies a product identity \prod_{i=1}^{\infty} e^{x_i} = e^Y. A product identity that had negative factors could not be converted back into an additive identity by taking logarithms, but there might be some other methods of transformation that would revert it to an additive identity.
 

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