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

(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 ) nby 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 y 0 = x 0 = 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−kmay, 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 )) 0set 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)