Integrating across Pascal's triangle

Abstract

Sums across the rows of Pascal's triangle yield powers of 2 while certain diagonal sums yield the Fibonacci numbers which are asymptotic to powers of the golden ratio. Sums across other diagonals yield quantities asymptotic to powers of c where c depends on the direction of the diagonals. We generalize this to the continuous case. Using the gamma function, we generalize the binomial coefficients to real variables and thus form a generalization of Pascal's triangle. Integration of these generalized binomial coefficients over various families of lines and curves yield quantities asymptotic to powers of some c where c can be determined explicitly. Finally, we revisit the discrete case.