Geodesic
A Generalization of Smith's Determinant
Canadian mathematical bulletin, Tome 29 (1986) no. 1, pp. 109-113

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CMB-1986-020-1
Mots-clés : 10A20 
McCarthy, P. J. A Generalization of Smith's Determinant. Canadian mathematical bulletin, Tome 29 (1986) no. 1, pp. 109-113. doi: 10.4153/CMB-1986-020-1
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[1] 1. Apostol, T. M., Arithmetical properties of generalized Ramanujan sums, Pacific J. Math., 41 (1972), pp. 281–293. Google Scholar

[2] 2. Cohen, E., An extension of Ramanujan's sum, Duke Math. J., 16 (1949), pp. 85–90. Google Scholar

[3] 3. Cohen, E., A class of arithmetical functions, Proc. Nat. Acad. Sci., U.S.A., 41 (1955), pp. 939–944. Google Scholar

[4] 4. Cohen, E., Representations of even functions (mod r), I. Arithmetical identities, Duke Math. J., 25 (1958), pp. 401–422. Google Scholar

[5] 5. Cohen, E., Arithmetical functions associated with the unitary divisors of an integer, Math. Zeit., 74 (1960), pp. 66–80. Google Scholar

[6] 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] 7. McCarthy, P. J., The generation of arithmetical identities, J. reine angew. Math., 203 (1960), pp. 55–63. Google Scholar

[8] 8. McCarthy, P. J., Regular arithmetical convolutions, Portugal. Math., 27 (1968), pp. 1—13. Google Scholar

[9] 9. McCarthy, P. J., Counting restricted solutions of a linear congruence, Nieuw Arch. v. Wisk., 25 (1977), pp. 133–147. Google Scholar

[10] 10. Narkiewicz, W., On a class of arithmetical convolutions, Colloq. Math., 10 (1963), pp. 81—94. Google Scholar

[11] 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] 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

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