A generating function is a device somewhat similar to a bag. Instead of carrying many little objects detachedly, which could be embarrassing, we put them all in a bag, and then we have only one object to carry, the bag.

—George Polya, Mathematics and plausible reasoning (1954)

波利亞在其大作《Mathematics and plausible reasoning》寫到

The name “generating function” is due to Laplace. Yet, without giving it a name, Euler used the device of generating functions long before Laplace [..]. He applied this mathematical tool to several problems in Combinatory Analysis and the Theory of Numbers.

。指出『生成函數』

Generating function

In mathematics, the term generating function is used to describe an infinite sequence of numbers (a_n) by treating them as the coefficients of a series expansion. The sum of this infinite series is the generating function. Unlike an ordinary series, this formal series is allowed to diverge, meaning that the generating function is not always a true function and the “variable” is actually an indeterminate. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem.^[1] One can generalize to formal series in more than one indeterminate, to encode information about arrays of numbers indexed by several natural numbers.

There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series; definitions and examples are given below. Every sequence in principle has a generating function of each type (except that Lambert and Dirichlet series require indices to start at 1 rather than 0), but the ease with which they can be handled may differ considerably. The particular generating function, if any, that is most useful in a given context will depend upon the nature of the sequence and the details of the problem being addressed.

Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations defined for formal series. These expressions in terms of the indeterminate x may involve arithmetic operations, differentiation with respect to x and composition with (i.e., substitution into) other generating functions; since these operations are also defined for functions, the result looks like a function of x. Indeed, the closed form expression can often be interpreted as a function that can be evaluated at (sufficiently small) concrete values of x, and which has the formal series as its series expansion; this explains the designation “generating functions”. However such interpretation is not required to be possible, because formal series are not required to give a convergent series when a nonzero numeric value is substituted for x. Also, not all expressions that are meaningful as functions of x are meaningful as expressions designating formal series; for example, negative and fractional powers of x are examples of functions that do not have a corresponding formal power series.

Generating functions are not functions in the formal sense of a mapping from a domain to a codomain. Generating functions are sometimes called generating series,^[2] in that a series of terms can be said to be the generator of its sequence of term coefficients.

命名的由來。他還說歐拉早就使用此『母函數』法研究數論︰

『巴塞爾問題』是一個著名的『數論問題』，最早由『皮耶特羅‧門戈利』在一六四四年所提出。由於這個問題難倒了以前許多的數學家，因此一七三五年，當『歐拉』一解出這個問題後，他馬上就出名了，當時『歐拉』二十八歲。他把這個問題作了一番推廣，他的想法後來被『黎曼』在一八五九年的論文《論小於給定大數的質數個數》 On the Number of Primes Less Than a Given Magnitude中所採用，論文中定義了『黎曼ζ函數』，並證明了它的一些基本的性質。那麼為什麼今天稱之為『巴塞爾問題』的呢？因為『此處』這個『巴塞爾』，它正是『歐拉』和『伯努利』之家族的『家鄉』。那麼就這麽樣的一個『級數的和』 $\sum \limits_{n=1}^\infty \frac{1}{n^2} = \lim \limits_{n \to +\infty}\left(\frac{1}{1^2} + \frac{1}{2^2} + \cdots + \frac{1}{n^2}\right)$ 能有什麼『重要性』的嗎？即使僅依據『發散級數』 divergent series 的『可加性』 summable 之『歷史』而言，或又得再過了百年的時間之後，也許早已經是『柯西』之『極限觀』天下後『再議論』的了！！因是我們總該看看『歷史』上『歐拉』自己的『論證』的吧！！

巴塞爾問題
$\sum_{n=1}^{\infty}\frac{1}{n^2} = \frac{\pi^2}{6}$

邏輯之歐拉圖

假使說『三角函數』 $\sin{x}$ 可以表示為 $\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$ ，那麼『除以』 $x$ 後，將會得到 $\frac{\sin(x)}{x} = 1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \frac{x^6}{7!} + \cdots$ ，然而 $\sin{x}$ 的『根』是 $x = n\cdot\pi$ ，由於『除以』 $x$ 之緣故，因此 $n \neq 0$ ，所以 $n = \pm1, \pm2, \pm3, \dots$ ，那麼 $\frac{\sin(x)}{x}$ 應該會『等於』 $\left(1 - \frac{x}{\pi}\right)\left(1 + \frac{x}{\pi}\right)\left(1 - \frac{x}{2\pi}\right)\left(1 + \frac{x}{2\pi}\right)\left(1 - \frac{x}{3\pi}\right)\left(1 + \frac{x}{3\pi}\right) \cdots$ ，於是也就『等於』 $\left(1 - \frac{x^2}{\pi^2}\right)\left(1 - \frac{x^2}{4\pi^2}\right)\left(1 - \frac{x^2}{9\pi^2}\right) \cdots$ ，若是按造『牛頓恆等式』，考慮 $x^2$ 項的『係數』，就會有 $- \left(\frac{1}{\pi^2} + \frac{1}{4\pi^2} + \frac{1}{9\pi^2} + \cdots \right) = -\frac{1}{\pi^2}\sum_{n=1}^{\infty}\frac{1}{n^2}$ ，然而 $\frac{\sin(x)}{x}$ 之『 $x^2$ 項』的『係數』是『 $- \frac{1}{3!} = -\frac{1}{6}$ 』，所以 $-\frac{1}{6} = -\frac{1}{\pi^2}\sum \limits_{n=1}^{\infty}\frac{1}{n^2}$ ，於是 $\sum \limits_{n=1}^{\infty}\frac{1}{n^2} = \frac{\pi^2}{6}$ 。那麼『歐拉』是『對』的嗎？還是他還是『錯』的呢？？

── 摘自《【Sonic π】電聲學之電路學《四》之《 V!》‧下中》

想必波利亞深知『形式幂級數』

Formal power series

In mathematics, a formal power series is a generalization of a polynomial, where the number of terms is allowed to be infinite; this implies giving up the possibility of replacing the variable in the polynomial with an arbitrary number. Thus a formal power series differs from a polynomial in that it may have infinitely many terms, and differs from a power series, whose variables can take on numerical values. One way to view a formal power series is as an infinite ordered sequence of numbers. In this case, the powers of the variable are used only to indicate the order of the coefficients, so that the coefficient of $x^{5}$ is the fifth term in the sequence. In combinatorics, formal power series provide representations of numerical sequences and of multisets, and for instance allow concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. More generally, formal power series can include series with any finite number of variables, and with coefficients in an arbitrary ring.

歷史上因其緣起引發的爭論︰

《緣起經》玄奘譯

佛言，云何名緣起初義？謂：依此有故彼有，此生故彼生。所謂：無明緣行，行緣識，識緣名色，名色緣六處，六處緣觸，觸緣受，受緣愛，愛緣取，取緣有，有緣生，生緣老死，起愁、歎、苦、憂、惱，是名為純大苦蘊集，如是名為緣起初義。

『邏輯學』上說『有□則有○，無○則無□』，既已『有□』又想『無○』，哪裡能夠不矛盾的啊！過去魏晉時『王弼』講︰一，數之始而物之極也。謂之為妙有者，欲言有，不見其形，則非有，故謂之妙；欲言其無，物由之以生，則非無，故謂之有也。斯乃無中之有，謂之妙有也。假使用『恆等式』 $1 - x^n = (1 - x)(1 + x + \cdots + x^{n-1})$ 來計算 $\frac{1 + x + \cdots + x^{m-1}}{1 + x + \cdots + x^{n-1}}$ ，將等於 $\frac{1 - x^m}{1 - x^n} = (1 - x^m) \left[1 + (x^n) + { (x^n) }^2 + { (x^n) } ^3 + \cdots \right]$ $= 1 - x^m + x^n - x^{n+m} + x^{2n} - \cdots$ ，那麼 $1 - 1 + 1 - 1 + \cdots$ 難道不應該『等於』 $\frac{m}{n}$ 的嗎？一七四三年時，『伯努利』正因此而反對『歐拉』所講的『可加性』說法，『同』一個級數怎麼可能有『不同』的『和』的呢？？作者不知如果在太空裡，乘坐著『加速度』是 $g$ 的太空船，在上面用著『樹莓派』控制的『奈米手』來擲『骰子』，是否一定能得到『相同點數』呢？難道說『牛頓力學』不是只要『初始態』是『相同』的話，那個『骰子』的『軌跡』必然就是『一樣』的嗎？？據聞，法國籍義大利裔大數學家『約瑟夫‧拉格朗日』伯爵 Joseph Lagrange 倒是有個『說法』︰事實上，對於『不同』的 $m,n$ 來講，從『幂級數』來看，那個 $= 1 - x^m + x^n - x^{n+m} + x^{2n} - \cdots$ 是有『零的間隙』的 $1 + 0 + 0 + \cdots - 1 + 0 + 0 + \cdots$ ，這就與 $1 - 1 + 1 - 1 + \cdots$ 『形式』上『不同』，我們怎麼能『先驗』的『期望』結果會是『相同』的呢！！