Result: Two classical formulas for the sum of powers of consecutive integers via complex analysis
Title:
Two classical formulas for the sum of powers of consecutive integers via complex analysis
Authors:
Source:
Complex Analysis and its Synergies. 10
Publisher Information:
Springer Science and Business Media LLC, 2024.
Publication Year:
2024
Subject Terms:
contour integration, Nicomachus identity, Bell and Stirling numbers, Bernoulli and Euler numbers and polynomials, Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane, Eulerian numbers, Power series (including lacunary series) in one complex variable, sum of powers, Combinatorial identities, bijective combinatorics, Stirling numbers
Document Type:
Academic journal
Article
File Description:
application/xml
Language:
English
ISSN:
2197-120X
2524-7581
2524-7581
DOI:
10.1007/s40627-024-00131-3
Rights:
Springer Nature TDM
Accession Number:
edsair.doi.dedup.....a9edc75f309a79cfd6177574cf6ff788
Database:
OpenAIRE
Further Information
The authors present a new complex analytic proof of the two classical formulas evaluating the sum of power consecutive integers: \[ 1^m+2^m+3^m+\cdots+n^m=\sum_{j=1}^m j!{\genfrac{\{}{\}}{0pt}{0}{m}{j}}\dbinom{n+1}{j+1} \] and \[ 1^m+2^m+3^m+\cdots+n^m=\sum_{j=1}^m {\genfrac{[}{]}{0pt}{0}{m}{j}} \dbinom{n+1}{j+1}, \] where \({\genfrac{\{}{\}}{0pt}{0}{m}{j}}, \ j=1,2,\ldots,m, \) are Stirling numbers of the second kind and \({\genfrac{[}{]}{0pt}{0}{m}{j}}, \ j=1,2,\ldots,m,\) are Eulerian numbers.