Mathematics for Enzyme Reaction Kinetics and Reactor Performance. F. Xavier Malcata

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