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A Generalization of Smith's Determinant

Published online by Cambridge University Press:  20 November 2018

P. J. McCarthy
P. J. McCarthy*
Affiliation:
Department of Mathematics, University of KansasLawrence, KS 66045, U.S.A.
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Abstract

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We shall evaluate the determinants of n x n matrices of the form [f(i,j)], where f (w, r) is an even function of m (mod r). Among the examples of determinants of this kind are H. J. S. Smith's determinant det [(i,j)], where (m, r) is the greatest common divisor of m and r, and a generalization of Smith's determinant due to T. M. Apostol.

Keywords

10A20
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 29 , Issue 1 , 01 March 1986 , pp. 109 - 113
Copyright
Copyright © Canadian Mathematical Society 1986

References

1. Apostol, T. M., Arithmetical properties of generalized Ramanujan sums, Pacific J. Math., 41 (1972), pp. 281–293.Google Scholar
2. Cohen, E., An extension of Ramanujan's sum, Duke Math. J., 16 (1949), pp. 85–90.Google Scholar
3. Cohen, E., A class of arithmetical functions, Proc. Nat. Acad. Sci., U.S.A., 41 (1955), pp. 939–944.Google Scholar
4. Cohen, E., Representations of even functions (mod r), I. Arithmetical identities, Duke Math. J., 25 (1958), pp. 401–422.Google Scholar
5. Cohen, E., Arithmetical functions associated with the unitary divisors of an integer, Math. Zeit., 74 (1960), pp. 66–80.Google Scholar
6. Jager, H., The unitary analogues of some identities for certain arithmetical functions, Nederl. Akad. Wetensch. Proc. Ser. A, 64 (1961), pp. 508–515.Google Scholar
7. McCarthy, P. J., The generation of arithmetical identities, J. reine angew. Math., 203 (1960), pp. 55–63.Google Scholar
8. McCarthy, P. J., Regular arithmetical convolutions, Portugal. Math., 27 (1968), pp. 1—13.Google Scholar
9. McCarthy, P. J., Counting restricted solutions of a linear congruence, Nieuw Arch. v. Wisk., 25 (1977), pp. 133–147.Google Scholar
10. Narkiewicz, W., On a class of arithmetical convolutions, Colloq. Math., 10 (1963), pp. 81—94.Google Scholar
11. Smith, H. J. S., On the value of a certain arithmetical determinant, Proc. London Math. Soc, 7 (1875/76), pp. 208–212; Collected Mathematical Papers, Vol. II, Chelsea Publishing Co., New York, 1965 reprint, pp. 161-165.Google Scholar
12. Subba, M. V. Rao and Harris, V. C., A new generalization of Ramanujan's sum, J. London Math. Soc., 41 (1966), pp. 595–604.Google Scholar