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q-Cosine


There are several q-analogs of the cosine function.

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

cos_q(z)=sum_(n=0)^(infty)((-1)^nz^(2n))/((q;q)_(2n))
(1)
=(e_q(iz)+e_q(-iz))/2
(2)
Cos_q(z)=(E_q(iz)+E_q(-iz))/2,
(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-cosine considered by Gosper (2001) is given by

cos_q^*(piz)=sin_q^*(pi(1/2-z))
(6)
=(q^(z^2)(q^(1-2z);q^2)_infty(q^(2z+1);q^2)_infty)/((q;q^2)_infty^2)
(7)
=q^(z^2)(theta_4(izlnq))/(theta_4)
(8)
=(theta_2(piz,p))/(theta_2(p)),
(9)

where theta_2(z,p) is a Jacobi theta function and p is defined via

 (lnp)(lnq)=pi^2.
(10)

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

cos_q^*(2z)=(cos_(q^2)^*z)^2-(sin_(q^2)^*z)^2
(11)
=(cos_q^*z)^4-(sin_q^*z)^4,
(12)

where sin_qz is the q-sine, and pi_q is q-pi (Gosper 2001). The q-cosine also satisfies

 cos_q^*(pia)=(sum_(n=-infty)^(infty)(-1)^nq^((n+a)^2))/(sum_(n=-infty)^(infty)(-1)^nq^(n^2)).
(13)

See also

q-Exponential Function, q-Factorial, q-Pi, q-Sine

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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.

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q-Cosine

Cite this as:

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

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