The following seemingly arbitrary polynomial has popped up in my research on period polynomials of modular forms, and I would like to try to understand it a little better. I hope it is OK to ask about it here, and I would welcome any other related comments!
Let $N\equiv 3\pmod{4}$ be a positive integer and let $M:=(N-1)/2$. Consider the following homogeneous polynomial of degree $N-1$:
$$T_N:=\sum_{r=0}^{M-1}\frac{2 i^r X^r Y^{N-1-r}}{r!(N-1-r)!}+\frac{i^M X^M Y^M}{(M!)^2}\in\mathbb{Q}(i)[X,Y].$$
Question: Is there a simpler expression for $T_N$ that does not involve summation notation, and depends only on $N$?
Of course, if the upper summation index in the definition of $T_N$ changed from $M-1$ to $N-1$ (call this modified version $T_N'$), we could use the binomial formula to write
$$T_N' = \frac{2}{(N-1)!}(i X+Y)^{N-1} + \frac{i^M X^M Y^M}{(M!)^2}.$$
However in $T_N$ the summation index runs only up to $M-1$.