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q-Sine


There are several q-analogs of the sine function.

The two natural definitions of the q-sine defined by Koekoek and Swarttouw (1998) are given by

sin_q(z)=sum_(n=0)^(infty)((-1)^nz^(2n+1))/((q;q)_(2n+1))
(1)
=(e_q(iz)-e_q(-iz))/(2i)
(2)
Sin_q(z)=(E_q(iz)-E_q(-iz))/(2i),
(3)

where e_q(z) and E_q(z) are q-exponential functions. The q-cosine and q-sine functions satisfy the relations

sin_q(z)Sin_q(z)+cos_q(z)Cos_q(z)=1
(4)
sin_q(z)Cos_q(z)-Sin_q(z)cos_q(z)=0.
(5)

Another definition of the q-sine considered by Gosper (2001) is given by

sin_q^*(piz)=(q^((z-1/2)^2)(q^(2z);q^2)_infty(q^(2-2z);q^2)_infty)/((q;q^2)_infty^2)
(6)
=iq^(z^2)(theta_1(izlnq))/(theta_4)
(7)
=(theta_1(piz,p))/(theta_1(1/2pi,p)),
(8)

where theta_1(z,p) is a Jacobi theta function and p is defined via

 (lnp)(lnq)=pi^2.
(9)

This is an odd function of unit amplitude and period 2pi with double and triple angle formulas and addition formulas which are analogous to ordinary sine and cosine. For example,

 sin_q^*(2z)=(q^2+1)(pi_q)/(pi_(q^2))cos_(q^2)^*zsin_(q^2)^*z,
(10)

where cos_q^*z is the q-cosine and pi_q is q-pi (Gosper 2001).


See also

q-Cosine, q-Exponential Function, q-Factorial, q-Pi

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References

Gosper, R. W. "Experiments and Discoveries in q-Trigonometry." In Symbolic Computation, Number Theory,Special Functions, Physics and Combinatorics. Proceedings of the Conference Held at the University of Florida, Gainesville, FL, November 11-13, 1999 (Ed. F. G. Garvan and M. E. H. Ismail). Dordrecht, Netherlands: Kluwer, pp. 79-105, 2001.Koekoek, R. and Swarttouw, R. F. The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q-Analogue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98-17, pp. 18-19, 1998.

Referenced on Wolfram|Alpha

q-Sine

Cite this as:

Weisstein, Eric W. "q-Sine." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/q-Sine.html

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