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Sine and Cosine functions possess notable formulas for addition of angles $$ \sin(a+b) = \sin(a)\cos(b) + \cos(a)\sin(b) \qquad \text{or} \qquad \cos(a+b) = \cos(a)\cos(b) - \sin(a)\sin(b). $$ One can come up with several $q$ analogs of trigonometric functions. For instance, from the relation $$ \sin(\pi z) = \frac{\pi}{\Gamma(z)\Gamma(1-z)}, $$ one can define the $q$-sine function as $$ \sin_q(\pi z) = \frac{\pi}{\Gamma_q(z)\Gamma_q(1-z)}. $$

My question is: Are there notable $q$ deformations of trigonometric functions satisfying addition formulas for angles?

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    $\begingroup$ There's Gosper's q-trigonometric functions but I believe that the addition formulae (and, indeed, most of the identities in the paper) are conjectural, and while it's easy to find a paper which proves the double-angle formulae I haven't seen one which proves the addition formulae. $\endgroup$
    – Peter Taylor
    Commented Feb 14, 2022 at 16:04
  • $\begingroup$ @PeterTaylor Thanks a lot. Formula (Add) at page 9 looks precisely what I was looking for. $\endgroup$
    – Matteo
    Commented Feb 14, 2022 at 17:38
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    $\begingroup$ Ernst, T. A method for q-calculus, J. Nonlinear Math. Phys. 10.4 (2003) pp 487-525 is also of interest, but I must be misunderstanding the notation because I can't see the relevance of Hahn addition in (3.22) and (3.23) or Ward-AlSalam addition in (4.35) to (4.42). $\endgroup$
    – Peter Taylor
    Commented Feb 15, 2022 at 11:20
  • $\begingroup$ @PeterTaylor thanks again for the additional reference. I found that some of the identities by Gosper were proven by El Bachraoui and some co-authors; see for instance link. $\endgroup$
    – Matteo
    Commented Feb 16, 2022 at 10:27

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