每日彙整: 2017-03-21

樹莓派、樹莓派之學習、樹莓派之教育

時間序列︰生成函數‧漸近展開︰白努利 □○《一》

雖然只想說說『漸近展開』之古法

歐拉-麥克勞林求和公式

歐拉-麥克勞林求和公式在1735年由萊昂哈德·歐拉與科林·麥克勞林分別獨立發現，該公式提供了一個聯繫積分與求和的方法，由此可以導出一些漸進展開式。

公式

[1] 設 {\begin{smallmatrix}f(x)\end{smallmatrix}}為一至少  {\begin{smallmatrix}k+1\end{smallmatrix}}階可微的函數，  {\begin{smallmatrix}a,b\in {\mathbb {Z}}\end{smallmatrix}}，則
{\begin{aligned}\sum _{{a<n\leq b}}f(n)&=\int _{{a}}^{{b}}f(t)\,{\mathrm {d}}t\\&\quad +\sum _{{r=0}}^{{k}}{\frac {(-1)^{{r+1}}B_{{r+1}}}{(r+1)!}}\cdot (f^{{(r)}}(b)-f^{{(r)}}(a))\\&\quad +{\frac {(-1)^{k}}{(k+1)!}}\int _{{a}}^{{b}}{\bar {B}}_{{k+1}}(t)f^{{(k+1)}}(t)\\\end{aligned}}
其中

 

的來歷，實在為難也！並非三百年前已太久，怎知歐拉是否是午夜夢回而得？還是千迴百轉積累至？但要講這個公式，就得從白努利數講起？？！！又誰曉雅各布·白努利如何想來！！？？

Bernoulli number

In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers with deep connections to number theory. The values of the first few Bernoulli numbers are

B0 = 1, B±
1 = ±1/2, B2 = 1/6, B3 = 0, B4 = −1/30, B5 = 0, B6 = 1/42, B7 = 0, B8 = −1/30.

The superscript ± is used by this article to designate the two sign conventions for Bernoulli numbers. They differ only in the sign of the n = 1 term:

  • B−
    n     are the first Bernoulli numbers (OEIS A027641 / OEIS A027642), and is the one prescribed by NIST. In this convention, B−
    1 = −1/2    .
  • B+
    n     are the second Bernoulli numbers (OEIS A164555 / OEIS A027642), which are also called the “original Bernoulli numbers”.[1] In this convention, B+
    1 = +1/2    .

Since Bn = 0 for all odd n > 1, and many formulas only involve even-index Bernoulli numbers, some authors write “Bn” to mean B2n. This article does not follow this notation.

The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in formulas for the sum of powers of the first positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function.

The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jakob Bernoulli, after whom they are named, and independently by Japanese mathematician Seki Kōwa. Seki’s discovery was posthumously published in 1712[2][3] in his work Katsuyo Sampo; Bernoulli’s, also posthumously, in his Ars Conjectandi of 1713. Ada Lovelace‘s note G on the analytical engine from 1842 describes an algorithm for generating Bernoulli numbers with Babbage‘s machine.[4] As a result, the Bernoulli numbers have the distinction of being the subject of the first published complex computer program.

Sum of powers

Main article: Faulhaber’s formula

Bernoulli numbers feature prominently in the closed form expression of the sum of the mth powers of the first n positive integers. For m, n ≥ 0 define

{\displaystyle S_{m}(n)=\sum _{k=1}^{n}k^{m}=1^{m}+2^{m}+\cdots +n^{m}.}

This expression can always be rewritten as a polynomial in n of degree m + 1. The coefficients of these polynomials are related to the Bernoulli numbers by Bernoulli’s formula:

{\displaystyle S_{m}(n)={\frac {1}{m+1}}\sum _{k=0}^{m}{\binom {m+1}{k}}B_{k}^{+}n^{m+1-k},}

where (m + 1

k) denotes the binomial coefficient.

For example, taking m to be 1 gives the triangular numbers 0, 1, 3, 6, … OEIS A000217.

{\displaystyle 1+2+\cdots +n={\frac {1}{2}}\left(B_{0}n^{2}+2B_{1}^{+}n^{1}\right)={\tfrac {1}{2}}\left(n^{2}+n\right).}

Taking m to be 2 gives the square pyramidal numbers 0, 1, 5, 14, … OEIS A000330.

{\displaystyle 1^{2}+2^{2}+\cdots +n^{2}={\frac {1}{3}}\left(B_{0}n^{3}+3B_{1}^{+}n^{2}+3B_{2}n^{1}\right)={\tfrac {1}{3}}\left(n^{3}+{\tfrac {3}{2}}n^{2}+{\tfrac {1}{2}}n\right).}

Some authors use the alternate convention for Bernoulli numbers and state Bernoulli’s formula in this way:

{\displaystyle S_{m}(n)={\frac {1}{m+1}}\sum _{k=0}^{m}(-1)^{k}{\binom {m+1}{k}}B_{k}^{-}n^{m+1-k}.}

Bernoulli’s formula is sometimes called Faulhaber’s formula after Johann Faulhaber who also found remarkable ways to calculate sums of powers.

Faulhaber’s formula was generalized by V. Guo and J. Zeng to a q-analog (Guo & Zeng 2005).

 

且順著歷史之軌跡︰

Faulhaber’s formula

Johann Faulhaber (5 May 1580 – 10 September 1635) was a German mathematician.

Born in Ulm, Faulhaber was a trained weaver who later took the role of a surveyor of the city of Ulm. He collaborated with Johannes Kepler and Ludolph van Ceulen. Besides his work on the fortifications of cities (notably Basel and Frankfurt), Faulhaber built water wheels in his home town and geometrical instruments for the military. Faulhaber made the first publication of Henry Briggs’s Logarithm in Germany. He died in Ulm.

Faulhaber’s major contribution was in calculating the sums of powers of integers. Jacob Bernoulli makes references to Faulhaber in his Ars Conjectandi.

In mathematics, Faulhaber’s formula, named after Johann Faulhaber, expresses the sum of the p-th powers of the first n positive integers

\sum_{k=1}^n k^p = 1^p + 2^p + 3^p + \cdots + n^p

as a (p + 1)th-degree polynomial function of n, the coefficients involving Bernoulli numbers Bj.

The formula says

  \sum_{k=1}^n k^p = {1 \over p+1} \sum_{j=0}^p (-1)^j{p+1 \choose j} B_j n^{p+1-j},\qquad \mbox{where}~B_1 = -\frac{1}{2}.

For example, the case p = 1 is

{\displaystyle 1+2+3+\cdots +n={1 \over 2}\sum _{j=0}^{1}(-1)^{j}{2 \choose j}B_{j}n^{2-j}}

{\displaystyle ={1 \over 2}\left(B_{0}n^{2}-2B_{1}n\right)={1 \over 2}\left(n^{2}+n\right).}

Faulhaber himself did not know the formula in this form, but only computed the first seventeen polynomials; the general form was established with the discovery of the Bernoulli numbers (see History section below). The derivation of Faulhaber’s formula is available in The Book of Numbers by John Horton Conway and Richard K. Guy.[1]

There is also a similar (but somehow simpler) expression: using the idea of telescoping and the binomial theorem, one gets Pascal‘s identity:[2]

{\displaystyle (n+1)^{k+1}-1=\sum _{m=1}^{n}\left((m+1)^{k+1}-m^{k+1}\right)}

  {\displaystyle =\sum _{p=0}^{k}{\binom {k+1}{p}}(1^{p}+2^{p}+\dots +n^{p})}.

This in particular yields the examples below, e.g., take k = 1 to get the first example.

History

Faulhaber’s formula is also called Bernoulli’s formula. Faulhaber did not know the properties of the coefficients discovered by Bernoulli. Rather, he knew at least the first 17 cases, as well as the existence of the Faulhaber polynomials for odd powers described above.[3]

A rigorous proof of these formulas and his assertion that such formulas would exist for all odd powers took until Carl Jacobi (1834).

 

來趟生成函數應用之旅吧。

【註︰推導練習】

{(n+1)}^{k+1} - 1 = (2^{k+1} - 1) + (3^{k+1} - 2^{k+1}) + ( \cdots ) + \left( {(n+1)}^{k+1} - n^{k+1} \right)

= \sum \limits_{m=1}^{n} \left( {(m+1)}^{k+1} - m^{k+1} \right) 。

依據二項式定理

{(m+1)}^{k+1} - m^{k+1} = \sum \limits_{p=0}^{k+1} \left( \begin{array}{ccc} k+1 \\ p \end{array} \right) m^p - m^{k+1}

= \sum \limits_{p=0}^{k} \left( \begin{array}{ccc} k+1 \\ p \end{array} \right) m^p

\therefore {(n+1)}^{k+1} - 1 = \sum \limits_{m=1}^{n} \sum \limits_{p=0}^{k} \left( \begin{array}{ccc} k+1 \\ p \end{array} \right) m^p

= \sum \limits_{p=0}^{k} \sum \limits_{m=1}^{n} \left( \begin{array}{ccc} k+1 \\ p \end{array} \right) m^p

= \sum \limits_{p=0}^{k} \left( \begin{array}{ccc} k+1 \\ p \end{array} \right) (1^p + 2^p + \cdots + n^p) 。