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According to Newton’s theorem, it is possible to expand the nth power (with n integer) of a sum of (real) terms, x and y, via a sum of a finite number of products of integer powers of x and y, with exponents adding up to n in every case; more specifically,

(2.236)

Equation (2.236) is also known as binomial formula, or binomial identity – and x and y may represent numbers, or instead functions. The first description of the above formula in its entirety, for nonnegative integer n, was due to Blaise Pascal – a French physicist, mathematician, and philosopher who lived in the seventeenth century; while Greek mathematician Euclid referred to its second order form in the fourth century BCE, and Indian mathematician Pingala mentioned higher orders one century later. However, it was Isaac Newton – who extended it to every real exponent in 1665, that has received credit for such a relationship thereafter.

The simplest case pertains indeed to n = 2, and accordingly reads

(2.237)

– being also known as another notable case of multiplication; for positive values of x and y, this can be graphically illustrated as in Fig. 2.9. The area (x + y)² of the larger square of side x + y may indeed be obtained via addition of the area of two smaller squares of sides x and y, i.e. x² and y², respectively, to the area xy of each of two rectangles of sides x and y.

Figure 2.9 Geometric demonstration of Newton’s binomial formula at two dimensions – resorting to squares of side x and area x², side y and area y², and side x + y and area (x + y)², complemented with two rectangles of sides x and y and area xy.

If y is replaced by −y, then Eq. (2.237) gives rise to

(2.238)

– also known as another notable case of multiplication; however, a simple geometrical interpretation is no longer possible (due to the negative term). A similar proof may be constructed in three dimensions, corresponding to the volume (x + y)³ of a cube of side x + y; it may indeed be decomposed as

(2.239)

i.e. the sum of volume x² y of each of three parallelipipeds of sides x, x, and y, to the volume xy² of each of three parallelipipeds of sides x, y, and y, and finally to the volume y³ of a cube of side y – besides being directly obtainable from Eq. (2.236) after setting n = 3.

The binomial coefficients in Eq. (2.236), of the form , count in how many ways one can pick up a subset with k elements out from a set of n elements in total; in mathematical terms, this is equivalent to writing

(2.240)

and supports the entries of Pascal’s triangle – denoted as Table 2.1. Careful inspection of this table indicates that the outermost values are always unity, whereas every two consecutive numbers in a given row add up to the value placed in between at the next row. In fact, Eq. (2.240) allows one to write

(2.241)

where factoring n! coupled with elimination of inner parenthesis give rise to

(2.242)

the factorials in denominator may, in turn, be rewritten as

(2.243)

based on their definition, thus allowing further factoring out of (k − 1)! and (n − k)! as

(2.244)

Upon lumping the two factors still in parenthesis, Eq. (2.244) becomes

(2.245)

that degenerates to

(2.246)

the outstanding factors may, in turn, be lumped with the existing factorials to yield

(2.247)

– equivalent, in view of Eq. (2.240), to

(2.248)

thus confirming the initial suggestion and being frequently known as Pascal’s rule.

Table 2.1 Pascal’s triangle encompassing coefficients of power of binomial, , for the first values of n – and, in each case, for k = 0, 1, …, n − 1, n.

n
0											1
1										1		1
2									1		2		1
3								1		3		3		1
4							1		4		6		4		1
5						1		5		10		10		5		1
6					1		6		15		20		15		6		1
7				1		7		21		35		35		21		7		1
8			1		8		28		56		70		56		28		8		1
9		1		9		36		84		126		126		84		36		9		1
10	1		10		45		120		210		252		210		120		45		10		1

In order to exactly prove Eq. (2.236), one should start by realizing that

(2.249)

arises when one sets n = 0; definition of power of nil exponent and summation, as well as lead to

(2.250)

that entails a universal condition. Suppose now that Eq. (2.236) is valid for a given n; its left‐hand side would then read

(2.251)

for n + 1, where power splitting and application of the distributive property meanwhile took place; insertion of Eq. (2.236) leads to

(2.252)

since it applies to (x + y)ⁿ by hypothesis. After factoring x and y, Eq. (2.252) becomes

(2.253)

where the last term of the first summation and the first term of the second summation may to advantage be made explicit as

(2.254)

Equation (2.254) may take the simpler form

(2.255)

because y⁰ = x⁰ = 1 and . Replacement of the counting variable k + 1 in the first summation of Eq. (2.255) by k permits transformation to

(2.256)

in view of the similarity of lower and upper limits for the two summations, one may lump them to get

(2.257)

– where x^k y^n+1−k may, in turn, be factored out as

(2.258)

Equation (2.248) may now be invoked to reformulate Eq. (2.258) to

(2.259)

while the first and last terms may be rewritten to get

(2.260)

association of such terms to the outstanding summation is then fully justified, viz.

(2.261)

If Eq. (2.261) is rephrased as

(2.262)

then it becomes clear that Eq. (2.236) will be valid for n + 1 if it is already valid for n; further validity of Eq. (2.236), for the trivial case of n = 0 as per Eqs. (2.249) and (2.250), then suffices to support validity of Eq. (2.236) in general, as per finite induction.

Equation (2.236) obviously applies when a difference rather than a sum is at stake – as already perceived with Eq. (2.238); just replace y by −y, and then apply Newton’s binomial formula to x and −y, according to

(2.263)

– where the minus sign is often taken out to yield

(2.264)

at the expense of (−1)^k = (−1)^−k .

As mentioned previously, Newton generalized the binomial theorem so as to encompass real exponents other than nonnegative integers – and eventually came forward with

(2.265)

where the generalized (binomial) coefficient should then read

(2.266)

en lieu of Eq. (2.240); Pochhammer’s symbol, ((r))_k, stands here for a falling factorial, i.e.

(2.267)

with ((r))₀ set equal to unity by convention – which, if r > k − 1 is an integer, may be reformulated to

(2.268)

following multiplication and division by (r−k)(r − (k + 1))⋯1. For instance, Eqs. (2.265)–(2.267) give rise to

(2.269)

where k → ∞∞ because r = 1/2; Eq. (2.269) degenerates to

(2.270)

or else

(2.271)

following straightforward algebraic manipulation and condensation afterward. If r is instead set equal to −1 and y set equal to 1, then Eq. (2.265) gives rise to

(2.272)

– where algebraic rearrangement supports dramatic simplification to

(2.273)

the right‐hand side is but a geometric series of first term equal to 1 and ratio between consecutive terms equal to −x, so one may retrieve Eq. (2.93) to write

(2.274)

since when ∣x ∣ < 1 – also consistent with (x + 1)⁻¹ representing the reciprocal of x + 1 in the first place, as obtained by long division of 1 by 1 + x following the algorithm depicted in Fig. 2.8. One also finds that

(2.275)

after replacement of x in Eq. (2.265) by its negative, y by 1, and exponent r by a general negative number −z; Eq. (2.275) eventually yields

(2.276)

If a rising factorial, (z)_k, is defined as

(2.277)

with evolution opposite to that entailed by Eq. (2.267), one may condense Eq. (2.276) to

(2.278)

In the case of a trinomial, its square may be calculated via

(2.279)

in agreement with Eq. (2.237) applied to x₁ + x₂ and x₃, rather than x and y; a second application of said formula to (x₁ + x₂)² generates

(2.280)

that may be rearranged to read

(2.281)

upon elimination of parenthesis. The above reasoning may be applied to any (integer) exponent n, and to any number m of terms of polynomial x₁ + x₂ + ⋯ + x_m; the generalized formula looks indeed like

(2.282)

where the summation in the right‐hand side is taken over all sequences of (nonnegative) integer indices k₁ through k_m, such that the sum of all k_i ’s is n. The multinomial coefficients are given by

(2.283)

– and count the number of different ways an n‐element set can be partitioned into disjoint subsets of sizes k₁, k₂, …, k_m . For the example above, one would have been led to

(2.284)

after setting m = 3 and n = 2 in Eq. (2.282), while Eq. (2.283) would yield

(2.285)

the possibilities of integer values for (k₁,k₂,k₃) satisfying the condition placed at the bottom of the summation in Eq. (2.284) encompass (2,0,0), (0,2,0), (0,0,2), (1,1,0), (1,0,1), and (0,1,1) – so Eq. (2.284) becomes

(2.286)

with the aid of Eq. (2.285). After realizing that 0! = 1, 2!/2! = 1, and 2!/1! = 2 – besides – one eventually retrieves Eq. (2.281), using Eq. (2.286) as departure point.