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Let $\lceil a\rceil=$ the smallest integer $\geq a$, otherwise known as the ceiling function. When the arguments are real, interpret $\binom{a}b$ using the Euler's gamma function, $\Gamma$. Recently, ...

I was wondering whether there is a Euler–Maclaurin formula of sorts for expressions such as $$ \sum_{x \in [a,b]^d\cap \mathbb{Z}^d} f(x) - \int_{[a,b]^d}f(x) $$ where $d\ge 2$ is an integer, $a,b \...