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Some determinants that should be better known

Published online by Cambridge University Press:  09 April 2009

A. J. van der Poorten
A. J. van der Poorten
Affiliation:
School of Mathematics, The University of New South Wales, New South Wales, 2033, Australia
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Since it is no longer fashionable to publish per se results which depend only on algebraic manipulation, many useful and complicated results of quite general interest languish unseen hidden as lemmata in specialist papers. In particular, the theory of transcendental numbers is rich in ingenious techniques for evaluating determinants. These techniques are apparently not well known even to workers in the field of transcendental numbers, let alone to researchers on other areas where the results might find application. This note accordingly discusses a variety of interesting results on determinants and is to be viewed as an appendix to the encyclopaedic volumes of Muir (1911, 1933) which the researcher might approach in order to obtain information in this area. For reasons of motivation a brief mention is made of the context in which the determinants arise.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 21 , Issue 3 , May 1976 , pp. 278 - 288
Copyright
Copyright © Australian Mathematical Society 1976

References

Baker, A. (1966), ‘A note on the Pade table’, K. Nederl. Akad. Wetensch. Proc. Ser. A. 69, 596–601.CrossRefGoogle Scholar
Coates, J. (1966), ‘On the algebraic approximation of functions I, II, III, IV’, K. Nederl. Akad. Wetensch. Proc. Ser. A. 69, 421–461; 70 (1967), 205–212.CrossRefGoogle Scholar
Fel'dman, N. I. (1951), ‘Approximation of certain transcendental numbers I: the approximation of logarithms of algebraic numbers’, Izv. Akad. Nauk. SSSR Ser. Mat., 15, 53–74;Google Scholar
Amer. Math. Soc. Transt. Ser. 2, 59, 224–245.Google Scholar
Fel'dman, N. I. (1960), ‘Approximation of the logarithms of algebraic numbers by algebraic numbers’, Izv. Akad. Nauk. SSSR Ser. Nat. 24, 475–492;Google Scholar
Amer. Math. Soc. Transl. Ser. 2, 58, 125–142.Google Scholar
Fel'dman, N. I. (1951), ‘Approximation of certain transcendental numbers II: The approximation of certain numbers associated with the Weierstrass function’, Izv. Akad. Nauk. SSSR Ser. Mat. 15, 153–176;Google Scholar
Amer. Math. Soc. Transl. Ser. 2, 59, 246–270.Google Scholar
Hermite, C. (1873), ‘Sur la fonction exponentielle’, Oeuvers III, 151–181.Google Scholar
Hermite, C. (1893), ‘Sur la generalization des fractions continues algebriques’, Oeuvres IV, 357–377.Google Scholar
Hua, L. K., (1963), ‘Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains’, Transl. Math. Mon. Vol 6. Amer. Math. Soc.Google Scholar
Jager, H. (1964), ‘A multidimensional generalisation of the Pade table’, K. Nederl. Akad. Wetensch. Proc. Ser. A, 67, 192–249.Google Scholar
Mahler, K., (1931, 2), ‘Zur Approximation der Exponentialfunktion und des Logarithmus’, J. Reine Angew. Math., 165–6118–150.Google Scholar
Mahler, K. (1931), ‘Ein Beweis des Thue-Siegelschen Satzes uber die Approximation algebraischer Zahlen fur binomische Gleichungen’, Math. Annalen., 105, 267–276.CrossRefGoogle Scholar
Mahler, K. (1953), ‘On the Approximation of logarithms of algebraic numbers’, Phil. Trans. Roy. Soc. Lond. Ser. A, 245, 371–398.Google Scholar
Mahler, K. (1968), ‘Perfect Systems’, Compositeo Math. 19, 95–166.Google Scholar
Muir, T. (1933), A Treatise on the theory of determinants. Longmans.Google Scholar
Muir, T. (1911), Theory of determinants in the historical order of development. MacMillan.Google Scholar
Van der Poorten, A. J. (1970), ‘Generalisations of Turan's main theorems on lowere bounds for sums of powers’, Bull. Austral. Math. Soc. 2, 15–37.CrossRefGoogle Scholar
Van der Poorten, A. J. (1970), ‘A generalisation of Turan's main theorems to binomials and logarithms’, Bull. Austral. Math. Soc. 2, 183–195.CrossRefGoogle Scholar
Van der Poortem, A. J. (1971), ‘Perfect approximation of function’, Bull. Austral. Math. Soc. 5, 117–126.CrossRefGoogle Scholar