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Derived and integral sets of basic sets of polynomials

1953; American Mathematical Society; Volume: 4; Issue: 2 Linguagem: Inglês

10.1090/s0002-9939-1953-0052571-6

1088-6826

M.N. Mikhail,

Advanced Mathematical Theories and Applications

For the general terminology used in this paper the reader is referred to Whittaker's books [2; 3]. I shall investigate the mode of increase and the effectiveness of basic derived and integral sets of any finite order. So far as I know, this is done here for the first time. Let pn(z)= i p iziq n(z) = q qizi be two basic sets; then the product set {Un(z) } = { p(z) }q{(z) }, in this order, is defined by Uiq= Eh Pihqhi. We have zn= i 7rniPi(z) = Ei Xniqi(z). It follows from the definition of a basic set of polynomials that the first derived set: po (z), P1(z), p2' (z), * form a basic set of polynomials with, perhaps, the omission of one of the polynomials. Thus the hth derived set {Dh{pPn(Z)} } will form a basic set {vn(z) }, if through differentiation of the basic set { pn(z) } certain polynomials, at most h of these, are omitted.