What is $1 + \sum_{k=1}^{\infty} \frac{(it)^k}{k!}a^{2k+1}$?

Question

I want to express $$1 + \sum_{k=1}^{\infty} \frac{(it)^k}{k!}a^{2k+1}$$ in terms of standard functions (exp, cos, sin, etc.), but I just don't see what this function is. Does anybody here have an idea, what this series could represent?

Answer 1

Hint: $\sum\frac{(it)^k}{k!} a^{2k+1}=a\sum \frac{(ita^2)^k}{k!}$. What is $\sum \frac{x^k}{k!}$ (note summation starts from $k=1$, not $k=0$)? Then plug $x=ita^2$.

Answer 2

$$1 + \sum_{k=1}^{\infty} \frac{(it)^k}{k!}a^{2k+1} = 1 - a + a\sum_{k=0}^{\infty} \frac{(it)^k}{k!}a^{2k} \\ = 1 - a + a\exp (ita^2) $$