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7.1. Interpolation and Extrapolation 211

where, for any nonnegative integer m, and any u,

U

( >

=

U(U-l)(u-2)...(U-7n+l) u(m)

m m. I

=1.

m.

(b) Let ~1, ~2,... , a, be arbitrary complex numbers and suppose

that f(t) is a polynomial over C of degree less than n. Show

that, if p(t) = (t - ui)(t - ~2) ... (t - a,), then

f(t)= k[f (Ui)/#(U~)](t-Ul)*“(tZii)‘ee(t-U,).

i=l

19. (a) Give an example of a polynomial over Q whose coefficients are
    not integers, but which take an integer value for every integer
    value of the variable.
    (b) Suppose that f(t) is a polynomial of degree k over C and that
    f(o), f(l), f(‘%-.>f(k) are integers. Prove that f(n) is an
    integer for each integer n.


20. Find the polynomial h of least degree for which

h(k) = 2k (k = O,l, 2,... ,n).

What is h(n + l)?

Explorations

E.55. Building Up a Polynomial. Consider the following simple-minded
way of building up, step by step, a polynomial whose value at ei is bi
(0 5 i 5 n). Let the first guess at the required polynomial be bo. This has
the correct value at uc, but not at 01. So we add a correction which will keep
the correct value at uo and make the value at ai correct: bo + bol(t - ao),
for a suitably chosen bol. What should the value of bol be? Now we add
a further term to make the value of the polynomial correct at ~2. So as
to not disturb what has been achieved already, this term should be of the
form bo12(t - uo)(t - al).
At the kth stage, we should have a polynomial

bo + bol(t - uo) + bolz(t - uo)(t - UI) + +.

+ bon..k(t - uo)(t - al)---(t - Uk-1).

What should the expressions for the coefficients bolz...n: be? Try to formulate
them in such a way that it is possible to use some analogue of the difference
table.