Integrating across Pascal's triangle

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:374

Article

Northshield, Sam

关键词: Gamma function;    Binomial coefficient;    Pascal's triangle;   

DOI  :  10.1016/j.jmaa.2010.09.018

来源: Elsevier

【 摘 要 】

Sums across the rows of Pascal's triangle yield 2(n) while certain diagonal sums yield the Fibonacci numbers which are asymptotic to phi(n) where phi is the golden ratio. Sums across other diagonals yield quantities asymptotic to c(n) where c depends on the directions 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 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. Finally, we revisit the discrete case to get results on sums along curves. (C) 2010 Elsevier Inc. All rights reserved.

【 授权许可】

Free