時間序列︰生成函數‧漸近展開︰白努利多項式之根《五》

如果承前篇,知道偶次白努利多項式在開區間裡 (0, \frac{1}{2}) 唯有一根,在開區間 (\frac{1}{2}, 1)中也只有一根,意味什麼呢?只是它的極大、極小恰與奇次白努利多項式三個根 0, \ , \frac{1}{2}, \ 1 互因乎 ?難道不令人好奇耶??若從它的積分表達式

\int_{0}^1 B_n (x) dx = 0, \ n \ge 1

,以及 x = \frac{1}{2} 是其『對稱軸』來看, B_0 (x) =1 就是『開宗明義』的第一命題哩!

第一原理

第一原理英語:First principle),哲學與邏輯名詞,是一個最基本的命題假設,不能被省略或刪除,也不能被違反。第一原理相當於是在數學中的公理。最早由亞里斯多德提出。

First principle

A first principle is a basic, foundational, self-evident proposition or assumption that cannot be deduced from any other proposition or assumption.

In philosophy, first principles are taught by Aristotelians and a nuanced version of first principles are referred to as postulates by Kantians.[1]

In mathematics, first principles are referred to as axioms and postulates.

In physics and other sciences, theoretical work is said to be from first principles, or ab initio, if it starts directly at the level of established science and does not make assumptions such as empirical model and fitting parameters.

In formal logic

In a formal logical system, that is, a set of propositions that are consistent with one another, it is probable that some of the statements can be deduced from one another. For example, in the syllogism, “All men are mortal; Socrates is a man; Socrates is mortal” the last claim can be deduced from the first two.

A first principle is one that cannot be deduced from any other. The classic example is that of Euclid‘s (see Euclid’s Elements) geometry; its hundreds of propositions can be deduced from a set of definitions, postulates, and common notions: all three types constitute first principles.

Aristotle’s contribution

Terence Irwin writes:

When Aristotle explains in general terms what he tries to do in his philosophical works, he says he is looking for “first principles” (or “origins”; archai):

In every systematic inquiry (methodos) where there are first principles, or causes, or elements, knowledge and science result from acquiring knowledge of these; for we think we know something just in case we acquire knowledge of the primary causes, the primary first principles, all the way to the elements. It is clear, then, that in the science of nature as elsewhere, we should try first to determine questions about the first principles. The naturally proper direction of our road is from things better known and clearer to us, to things that are clearer and better known by nature; for the things known to us are not the same as the things known unconditionally (haplôs). Hence it is necessary for us to progress, following this procedure, from the things that are less clear by nature, but clearer to us, towards things that are clearer and better known by nature. (Phys. 184a10–21)

The connection between knowledge and first principles is not axiomatic as expressed in Aristotle’s account of a first principle (in one sense) as “the first basis from which a thing is known” (Met. 1013a14–15). The search for first principles is not peculiar to philosophy; philosophy shares this aim with biological, meteorological, and historical inquiries, among others. But Aristotle’s references to first principles in this opening passage of the Physics and at the start of other philosophical inquiries imply that it is a primary task of philosophy.[2]

 

再輔之以『推步公式』

B_n (1+x) = B_n (x) + n \cdot x^{n-1}, \ n \ge 1

等同的定義了整體白努利多項式也!!

因此白努利多項式序列內在邏輯聯繫密且深,不同觀點光照下不但數理呈現多采多姿︰

\int_{0}^{1} B_1 (x) dx = \int_{0}^{1} x - \frac{1}{2} dx = ( \frac{1}{2} x^2 - \frac{1}{2} x ) |_{0}^{1} = 0

\frac{d \ }{dx} B_1 (x) =  \frac{d \ }{dx} (x - \frac{1}{2}) = 1  = B_0 (x)  。

 

探索其『根』 B_n (x) = 0 者,或許能借此豐富另闢蹊徑吧☆