# Addition formulas for q-analogs of trigonometric functions

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?

• 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. Commented Feb 14, 2022 at 16:04
• @PeterTaylor Thanks a lot. Formula (Add) at page 9 looks precisely what I was looking for. Commented Feb 14, 2022 at 17:38
• 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). Commented Feb 15, 2022 at 11:20
• @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. Commented Feb 16, 2022 at 10:27