On Staudt’s Proposition relating to the Bernoullian Numbers

Ludwig Schläfli

A chapter in Gesammelte Mathematische Abhandlungen, 1953, pp 363-364 from Springer

Abstract: Abstract If n be a positive integer number and the variable x less than 1/n, the rational fraction 1/[(1 - x) (1 – 2 x)... (1 - n x)] can be expanded into the series A − n 0 + A − n 1 x + A − n 2 x 2 + A − n 3 x 3 + ⋅ ⋅ ⋅ $${}^{ - n}{A_0} + {}^{ - n}{A_1}x + {}^{ - n}{A_2}{x^2} + {}^{ - n}{A_3}{x^3} + \cdot \cdot \cdot$$ and it will be readily seen that the general coefficient -n A m is a positive integer number2). From ( d d x − n ) ( e x − 1 ) n n ! = ( e x − 1 ) n − 1 ( n − 1 ) ! $$\left( {\frac{d} {{dx}} - n} \right)\frac{{{{\left( {{e^x} - 1} \right)}^n}}} {{n!}} = \frac{{{{\left( {{e^x} - 1} \right)}^{n - 1}}}} {{\left( {n - 1} \right)!}}$$ we may then infer ( e x − 1 ) n n ! = 1 ( d d x − 1 ) ( d d x − 2 ) ... ( d d x − n ) ⋅ 1 = ∑ λ = 0 λ = ∞ A − n λ ( d d x ) − n − λ ⋅ 1 = ∑ λ = 0 λ = ∞ A − n λ x n + 1 ( n + λ ) ! $$\frac{{{{\left( {{e^x} - 1} \right)}^n}}} {{n!}} = \frac{1} {{\left( {\frac{d} {{dx}} - 1} \right)\left( {\frac{d} {{dx}} - 2} \right)...\left( {\frac{d} {{dx}} - n} \right)}} \cdot 1 = \sum\limits_{\lambda = 0}^{\lambda = \infty } {{}^{ - n}{A_\lambda }{{\left( {\frac{d} {{dx}}} \right)}^{ - n - \lambda }} \cdot 1 = \sum\limits_{\lambda = 0}^{\lambda = \infty } {{}^{ - n}{A_\lambda }\frac{{{x^{n + 1}}}} {{\left( {n + \lambda } \right)!}}} }$$ whence ∑ m = 0 m = n ( − 1 ) n − m m ! ( n − m ) ! e m x = ∑ λ = 0 λ = ∞ A − n λ x n + λ ( n + λ ) ! $$\sum\limits_{m = 0}^{m = n} {\frac{{{{\left( { - 1} \right)}^{n - m}}}} {{m!\left( {n - m} \right)!}}{e^{mx}} = \sum\limits_{\lambda = 0}^{\lambda = \infty } {{}^{ - n}{A_\lambda }\frac{{{x^{n + \lambda }}}} {{\left( {n + \lambda } \right)!}}} }$$ therefore n ! A − n m − n = ∑ λ = 0 λ = m ( − 1 ) n − λ ( n λ ) λ m $$n!{}^{ - n}{A_{m - n}} = \sum\limits_{\lambda = 0}^{\lambda = m} {{{\left( { - 1} \right)}^{n - \lambda }}\left( \begin{gathered} n \hfill \\ \lambda \hfill \\ \end{gathered} \right){\lambda ^m}}$$ which expression is zero for m = 0, 1, 2, 3, ..., n - 1 and n! for m = n, and might also have been obtained by the well known method of decomposition of a rational fraction.

Date: 1953
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-0348-4117-7_15

Ordering information: This item can be ordered from
http://www.springer.com/9783034841177

DOI: 10.1007/978-3-0348-4117-7_15

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().