Integrating across Pascal's triangle

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Authors
Northshield, Sam
Issue Date
2011
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Article
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Gamma function , Binomial coefficient , Pascal's triangle
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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.
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This article has been published in the Journal of Mathematical Analysis and Applications: doi:10.1016/j.jmaa.2010.09.018
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Northshield, S. (2011). Integrating across Pascal's triangle. Mathematical Analysis and Applications, 374(2). http://doi.org/10.1016/j.jmaa.2010.09.018
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Mathematical Analysis and Applications
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