Certain properties of pyramidal and figurate numbers

Abstract

In n-point osculatory interpolation of order ri 1 at points xi, i = 1, 2, * * , n, by a rational expression N(x)/D(x), where N(x) and D(x) are polynomials E ajx' and E bjxj, we use the lemma that the system (1) {N(xi)/D(xi)}(m) = f(m)(xi) m = 0,1, * .. , r1, is equivalent to (2) N(m)(xi) = {f(xi)D(xi) (m), m = O, 1, . . . , ri 1, D(xi) # 0. This equivalence does not require N(x) or D(x) to be a polynomial or even a linear combination of given functions. lemma implies that (1), superficially non-linear in aj and bj, being the same as (2), is actually linear. For the n-point interpolation problem, the linear system, of order E.Li ri, which might be large, is replaceable by separate linear Received April 20, 1961. t Much of the material in the present note is contained in an entirely independent (still unpublished) study by Henry C. Thacher, Jr., who was kind enough to send the writer a copy of his preliminary draft. 1. J. C. P. MILLER, & M. F. C. WOOLLETT, Solutions of the Diophantine equation x3 + y3 + z3 = k, London Math. Soc., v. 30, 1955, p. 101-110. 2. L. DICKSON, History of the Theory of Numbers, v. 2, Chap. 1. (Reprint: Stechert, New York, 1934). 3. S. CHOWLA and others, The Diophantine equation x3 + y3 + z3 = x + y + z, submitted to Norske Vid. Selsk. Forh. Trondheim. 4. S. SEGAL, A note on pyramidal numbers, American Math. Monthly, (in press). 5. SWIATOMIR ZABEK, Sur la periodicite modulo m des suites de nombres (k), Ann. Univ. Mariae Curi-Sklodowska, Sect. A, 10 (1956), p. 37-47. This content downloaded from 157.55.39.163 on Wed, 21 Sep 2016 04:43:19 UTC All use subject to http://about.jstor.org/terms