Is there a tight upper bound for $$\Gamma\left(\frac{1+a+b}{2}\right)$$ where $a$ and $b$ are nonnegative integers, that is in terms of a product of a function of $a$ and a function of $b$?
For integers and half-integers, the gamma function aligns nicely with factorials and so when and are both even or both odd, an upper bound is $(\frac{+}{2})!\sqrt{\pi}$, and when their parities differ, an upper bound is $\lfloor\frac{+}{2}\rfloor!$. But I want a bound where I don't have to consider a sum of and , but rather something where I can consider and separately.