Exponential Generating Function for Boubaker Polynomials

Theorem
The Boubaker polynomials, defined as:
 * $B_n \left({x}\right) = \begin{cases}

1 & : n = 0 \\ x & : n = 1 \\ x^2+2 & : n = 2 \\ x B_{n-1} \left({x}\right) - B_{n-2} \left({x}\right) & : n > 2 \end{cases}$

have as an exponential generating function:
 * $\displaystyle f_{B_n, \operatorname{EXP}} \left({x, t}\right) = \sum_{n=0}^\infty B_n \left({x}\right) \frac{t^n}{n!} =

4e^{t \frac x 2} \frac {\sin \left({t \sqrt {1 - \left({\frac x 2}\right)^2}}\right)} {\sqrt {1 - \left({{\frac x 2}^2}\right)}} - 2 e^{t \frac x 2} \cos \left({t \sqrt {1 - \left({\frac x 2}\right)^2}}\right) - 3$

Proof
Proof: not available