The discrete and the continuous

I think it is interesting that at the heart of mathematics resides the distinction between the discrete and the continuous [1,2]. On the one hand, the field of discrete mathematics deals with countable data. Continuous mathematics, on the other hand, deals with uncountable data. So, when considering the former we are primarily interested in finite (or countable) objects, examples of which include the integers, finite series, or finite metric spaces. This is very different than the field of continuous mathematics, which is home to the techniques of algebraic theory and the powerful tools of calculus. Here we have the study of continuous functions (i.e., functions that have no gaps or discontinuities), measure theory, topological group theory, and algebraic geometry. This is also the world in which such mathematical objects as manifolds, real or complex numbers, and continuous metric spaces and topological spaces, are utilised.

In nature both languages are exhibited. It is therefore quite easy to motivate some sort of quest in search for a bridge between the discrete and continuous worlds – to discover formulae that relate discrete sums of numerical powers with integration, for example. Such a quest goes back to the Pythagoreans, Archimedes, and other ancients [1, 3]. The Pythagoreans famously mastered the practice of utilising discrete approximations to obtain continuous areas and volumes by the method of exhaustion. They viewed successive positive integers as triangular numbers, which, in modern notation, gives the famous formula for the sum of powers of positive integers. Furthermore, consider the following sums of powers, which we may write as discrete partial sums:

$\displaystyle 1 + 2 + 3 + ... + n = \frac{1}{2}n(n+1) = \frac{1}{2}n + \frac{1}{2}n^2, \ \ \ \ \ (1)$

$\displaystyle 1^2 + 2^2 + 3^2 + ... + n^2 = \frac{1}{6}n(n+1)(2n + 1) = \frac{1}{6} n + \frac{1}{2}n^2 + \frac{1}{3}n^3, \ \ \ \ \ (2)$

$\displaystyle 1^3 + 2^3 + 3^3 + ... + n^3 = \frac{1}{4}n^2(n+1)^2 = \frac{1}{4}n^2 + \frac{1}{2}n^3 + \frac{1}{4}n^4. \ \ \ \ \ (3)$

The first relationship for the sum of the natural numbers is the one that goes back to the Pythagorean school in the 6th century BC. Archimedes of Syracuse (circa 287-212 BC), considered the greatest mathematician of antiquity, discovered the second relationship for the sum of the squares. The third relationship, for the sum of cubes, can be found in the work of Nicomachus of Gerasa (circa 60–120 AD), after which the identity ${\sum \limits_{k=1}^{n} k^3 = (\sum \limits_{k=1}^{n} k)^2}$ is named Nicomachus’ theorem. However, this formula, albeit without proof, is said to have appeared much early in a work by the Indian mathematician Aryabhata (476–550 CE) [4].

These three discrete sums, which we’ve represented as of closed formulae, can be written as the general sum of powers ${\sum \limits_{k=1}^{n} k^i}$ with ${i}$ a fixed natural number. The sum of powers in closed form is interesting to study in-itself; but where it gets really interesting is when we start to think about its infinite series. Indeed, such a curiosity has motivated some famous results. In modern times the formula for sums of integer powers was first given in generalisable form by Thomas Harriot (c. 1560-1621). Johann Faulhaber (1580-1635) also gave formulas for these sums up to the 17th power [1,2]. It was Pierre de Fermat who made the important when he argued that he could use what are called “figurate numbers” to capture generalised formulae for these discrete sums of numerical powers. It was then Blaise Pascal in his extensive treatise Potestatum Numericarum Summa (circa 1654), which produced the first explicit recursive formula for sums of powers in arithmetic progression using binomial coefficients. This work relates to what many will recognise as Pascal’s triangle. One of Pascal’s motivations was to define a recursive formula that could sum higher powers up to infinity.

Observe that in the three cases (1–3), the sum of the sth powers ${n^s}$ of the integers ${k}$ from 1 to ${n}$ is given, for ${s = 1, 2, 3,}$ by a polynomial in ${n}$ , of degree ${s+1 = 2, 3, 4}$ . Therefore let us define for ${s = 1,2,3,...}$

$\displaystyle S(s,n) := 0^s + 1^s + 2^s + ... + n^s = \sum \limits_{k=0}^{\infty} k^s. \ \ \ \ \ (4)$

It was a general expression for this sum ${S(n,s)}$ that Pascal defined in the form

$\displaystyle S(s,n) = \frac{1}{s+1} ((n+1)^{s+1} - \begin{pmatrix} s + 1 \\ s - 1 \\ \end{pmatrix} S(s-1, n) - ... - \begin{pmatrix} s + 1 \\ 1 \\ \end{pmatrix} S(1,n) - \begin{pmatrix} s + 1 \\ 0 \\ \end{pmatrix} S(0,n) - 1). \ \ \ \ \ (5)$

The next important contribution came in the 18th century, with the work of Jacob Bernoulli titled Ars Conjectandi. This is quite a magnificent treatise, which, in many ways, represents the beginnings of probability theory. Bernoulli conjectured the general pattern in the coefficients of the closed-form polynomial solution to the summation problem. The connection between probability theory and sums of powers is via the combinatorial counting numbers (for more on this history, see this note). He also introduced the all-important Bernoulli numbers into mathematics, which will review in a later note. To see a bit of this , consider Bernoulli’s reformulation of Pascal’s solution given below in a form commonly considered today

$\displaystyle S(s,n) = \frac{1}{s + 1}n^{s + 1} + \frac{\sigma^1_1}{1 !}n^s + \frac{\sigma^2_1}{2!}s n^{s-1} + ... + \frac{\sigma^s_{1}}{s!}s(s-1) ... 2n, \ \ \ \ \ (6)$

where ${\sigma^s_m}$ denote the coefficients. This first form of this formula can be more cleanly written in terms of combinatorial factors

$\displaystyle S(s,n) = \frac{1}{s + 1}(\begin{pmatrix} s+1 \\ 0 \\ \end{pmatrix}\sigma^0_1 n^{s+1} + \begin{pmatrix} s + 1 \\ 1 \end{pmatrix}n^s + \begin{pmatrix} s+1 \\ 2 \\ \end{pmatrix}\sigma^2_1 n^{s-1} + ... + \begin{pmatrix} s+1 \\ s \\ \end{pmatrix} \sigma^s_1 n). \ \ \ \ \ (7)$

From Bernoulli’s formula, we see how we may write the polynomials ${S(s,n)}$ in such a way that all coefficients ${\sigma^s_m}$ are expressed in terms of a single sequence of numbers [3]. These ${\sigma^s_m}$ follow from a well-known table of coefficients for the case ${s = 1,2,3,...}$ in which we’re only interested when ${m = 1}$ . When we study the recurrence relations associated with the formula (7) we find for fixed coefficients ${\sigma^s_1}$ the Bernoulli numbers (ignoring the discrepancy in the fact of (-1)); but again, we’ll save these special numbers for a post in their own right, as we will also study their generating function and deeper relation to the sums of powers.

This formula by Bernoulli then takes us to the great Leonhard Euler. To finish the brief historical sketch, we come also to the modern story in the search for a way to make sense (formally and rigorously) of divergent series. It was in this time that Euler displayed his genius and completely broke with the view of his contemporaries, taking a radically new approach to the Basel problem [7]. The main task of the Basel problem was to find the reciprocal squares of the natural numbers ${\sum \limits_{i=1}^{\infty} 1/i^2 = \frac{1}{2} + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \frac{1}{25} + ...}$ . It was first formulated by Gottfried Leibniz and the brothers Jakob and Johann Bernoulli \citep{Peng02b}. Euler’s genius was such that by refitting the problem into a new context, namely in using calculus to relate the discrete sum of an arbitrary function ${\sum \limits f(i)}$ to the antiderivative of the form ${\int_0^{n} \ dx \ f(x)}$ , he could prove the sum of reciprocal squares is exactly ${\pi^2 / 6}$ .

In solving the Basel problem, Euler would conclude a centuries long search for closed expressions for sums of numerical powers ${\sum \limits_{i=1}^{n} i^k \approx \int_0^n x^k \ dx}$ . Since the sum and the integral provide first approximations to each other – from elementary calculus, one will know that the sum can be interpreted as the total area of rectangles forming a step graph along some curve, while the integral, when evaluated between limits, can be interpreted as the area under the same curve – Euler could imagine a bridge connecting continuous and discrete mathematics.

The relation of the first approximation of a discrete sum and an integral that Euler derived was the initial step toward what, today, we know as the Euler-Maclaurin summation formula. The asymptotic nature of the Euler-Maclaurin formula has proven very important. Indeed, so important is this formula that we will spend much time studying it in future posts. (For a historical review of the development of some of Euler’s key ideas, including his motivation to study the problem of interpolation and the connection to expressing the sum of a series using an integral, see also [8]).

I really want to look at other ways the discrete and the continuous are found to engage in a beautiful interplay. Although parallel to my PhD research, it is something I have been thinking about a little bit and taking notes on. Such an investigation includes the study of sums, recurrences, and number theory, including things like generating functions, discrete probability, and asymptotic methods [5]. It also takes us to the heart of divergent series [6] and many important concepts, techniques, and results throughout physics. I even find it interesting that there are some people who speculate that these seemingly disconnected branches of study should be seen as part of a unity of methods and theorems that defines mathematics as a whole – that perhaps there is a more complete foundation of mathematics in which the distinction no longer exists. I’m not entirely sure about this, though I like the picture it represents. Moreover, I do think it is noteworthy that in mathematical physics – indeed, at the heart of quantum gravity – there seems to reside the very same analogous search for a bridge between the discrete. In the stringy picture, sometimes I like to imagine that given the complete non-perturbative theory – or at least a better understanding of brane quantisation – maybe this distinction will be found to fail altogether. It is such a cool thought.

References

[1] David J. Pengelley. Dances between continuous and discrete: Euler’s summation formula. 2019. 1, 2

[2] David J. Pengelley. Dances between continuous and discrete: Euler’s summation formula. 2019. 1, 2

[3] M. Santander. Sumas de potencias y series divergentes un panorama sobre sumaci´on de series, sumas suavizadas, n´umeros y polinomios de bernoulli, la f´ormula de euler-maclaurin y la funci´on de riemann. 2019. 1, 3

[4] C.B. Boyer and U.C. Merzbach. A History of Mathematics. Wiley, 2011. 1

[5] Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik. Concrete mathematics – a foundation for computer science. 1991. 4

[6] G.H Hardy. Divergent Series. Clarendon Press, Oxford, 1973. 4 [7] Leonhard Euler. On the sums of series of reciprocals [arXiv: math/0506415]. [8] Giovanni Ferraro. Some Aspects of Euler’s Theory of Series: Inexplicable Functions and the Euler–Maclaurin Summation Formula. Historia Mathematica, 1998, 25: 290-317.

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