Q-Sin
From specialfunctionswiki
The function $\mathrm{Sin}_q$ is defined by $$\mathrm{Sin}_q(z)=\dfrac{E_q(iz)-E_q(-iz)}{2i},$$ where $E_q$ denotes the $q$-exponential $E_q$.
Properties
Theorem: The following formula holds: $$D_q \mathrm{Sin}_q(bz) = b \mathrm{Cos}_q(bz),$$ where $D_q$ is the q-difference operator, $\mathrm{Sin}_q$ is the $q$-Sine function, and $\mathrm{Cos}_q$ is the $q$-cosine function.
Proof: █
Theorem: The general solution of the $q$-difference equation $D_q^2 y(x) + k^2 y(x) = 0$ is $y(x)=c_1 \mathrm{Cos}_q(kz) + c_2 \mathrm{Sin}_q(kz).$
Proof: █
External links
References
- 2012: Thomas Ernst: A Comprehensive Treatment of q-Calculus ... (previous) ... (next): (6.168)
