Re: Something for sci.math's amateur mathematicians? (Part 2)

Dave L. Renfro wrote:

[snip]

The problem I posed at

http://groups.google.com/group/sci.math/msg/ddd62dd5b9befbb7

and which Gottfried Helms cited some work he had done
on it at

http://go.helms-net.de/math/potenzsummen/potenzsummen_1.htm

seems to be essentially solved in a 1950/51 American
Mathematical Monthly problem.

P. A. Piza posed the following problem:

Monthly Problem #4380 -- posed in American Mathematical
Monthly 57 (1950), p. 119; solution by Roger Lessard in
American Mathematical Monthly 58 (1951), pp. 429-430.

"For arbitrary positive integers n and k let

S_1(n,k) = 1^k + 2^k + ... + n^k,

and put

S_{p+1}(n,k) = S_p(1,k) + S_p(2,k) + ... + S_p(n,k),

with p = 1, 2, . . .. Thus S_p(n,k) is the p'th iterated
sum, the sum of the sum of . . . the sum of the first
n perfect k'th powers.

Show that S_p(n,k) is a polynomial in n of degree p + k
and determine the form of the polynomial for k = 3, 4, 5.
If possible, determine the form for general k. (The case
k = 2 is the subject of problem 22, Mathematics Magazine,
vol. XXII, no. 1, p. 51.)"

The solution to the k = 2 case is in Mathematics Magazine
22 #3 (Jan./Feb. 1949), pp. 162-163.

Roger Lessard gives two forms for the general k case,
neither of which is simple enough for me to want to
reproduce here.

Dave L. Renfro

.

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