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There is no reason to think there is any simple expression for solution $T$ of $$T(x + 1) - T(x) = \tan(x)$$ What we CAN find a simple solution to is this: $$T(x + \pi) - T(x) = \tan(x)$$

I add more details for the solution in the distinguished answer due to Anixx. First, we need the digamma function http://en.wikipedia.org/wiki/Digamma_function which we will call $\Psi(x)$. Importan …

half-discrete analytic First do the formal calculation, then discuss its validity. $$\begin{align} &\sum_{m = 0}^{\infty} \binom{x}{m} \sum_{k = 0}^{m} \binom{m}{k} (-1)^{(m - k)} \operatorname{sin} …

weak discrete-analytic Closed form here involves some ${}_2F_1$ functions, so I cannot provide proofs. Numerically, though, it seems that $\sin(ax)$ is weak discrete-analytic for $a$ up to some valu …