Integrating across Pascal's triangle (Q607296)

Using the Gamma function the author extends the binomial coefficients to real variables and proposes a continuous version of Pascal's triangle. Integration over various families of lines and curves yields quantities asymptotic to \(c^x\), where \(c\) is determined by the family and \(x\) is a parameter. Going back to the discrete case one obtains results on sums along curves. For example: \[ \limsup_{\sqrt{m^2+n^2}\to\infty}\binom{m+n}{m}^{\frac{1}{\sqrt{m^2+n^2}}}=2^{\sqrt{2}}. \]