Unknown

Notes on Explorations 407

of Functions: Parts I and II; Problem Book in the Theory of Functions:

Volumes I and II.

E.26. There are many relations which are satisfied by Legendre polynomi-

als. One of the most striking is this formula

2”n! P,(x) = D”(x2 - 1)”

where D is the differentiation operator. The polynomials turn up as coef-

ficients in a generating function expansion

(1 - 2xt + t2)-‘12 = g P,(x)Y.
n=O
A list of properties of Legendre polynomials occurs on pages 50-53 of W.
Magnus & F. Oberhettinger, Formulas and Theorems for the Functions of
Mathematical Physics, (Chelsea, 1949).
To get some idea of the richness of the area of mathematics which includes
the study of these functions, consult Gabor Szegij, Orthogonal Polynomials,
(AMS Colloquium, 1939).
A discussion of Legendre’s use of these polynomials in 1782 in dealing
with a problem of potential theory can be found on pages 525-528 of Morris
Kline, Mathematical Thought from Ancient to Modern Times, (Oxford,
New York, 1972). Legendre’s own paper (referred to in a book review in
Bull. A.M.S. (NS) 19 (1988)) 346-348) is A.M. Legendre, Recherches sur
l’attraction des spheroides homogenes, Me’m. Math. Phys. P&s. ci 1’Acad.
Roy. Sci. (Paris) par divers savants 10 (1785)) 411-434.
For other work, see Mary L. Boas, A formula for the derivatives of Leg-
endre polynomials, Amer. Math. Monthly 70 (1963)) 643-644.

E.28. Although Rolle’s Theorem is a standard topic of a first calculus
course, Rolle himself was interested in using it to locate zeros of polyno
mials. An English translation of an excerpt of his work with commentary
can be found in D.E. Smith, A Source Book In Mathematics, Volume One
(Dover, 1929, 1959).
For a discussion of generalizing Rolle’s Theorem to the complex plane
and an interesting open problem, consult I.J. Schoenberg, A conjectured
analogue of Rolle’s Theorem to polynomials with real or complex coeffi-
cients, Amer. Math. Monthly 93 (1986)) 8-13.

E.29. Modulo the polynomial t2 - t + a we have that

t3 E (1 - a)t - 0

and, in general,
tk = fk(a)t - afk-l(a)