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XXV.—The Generating Function of the Reciprocal of a Determinant
Published online by Cambridge University Press: 06 July 2012
Extract
(1) This is a subject to which very little study has been directed. The first to enunciate any proposition regarding it was Jacobi; but the solitary result which he reached received no attention from mathematicians,—certainly no fruitful attention,—during seventy years following the publication of it.
Jacobi was concerned with a problem regarding the partition of a fraction with composite denominator (u1 − t1) (u2 − t2) … into other fractions whose denominators are factors of the original, where u1, u2, … are linear homogeneous functions of one and the same set of variables. The specific character of the partition was only definable by viewing the given fraction (u1−t1)−1 (u2−t2)−1…as expanded in series form, it being required that each partial fraction should be the aggregate of a certain set of terms in this series. Of course the question of the order of the terms in each factor of the original denominator had to be attended to at the outset, since the expansion for (a1x+b1y+c1z−t)−1 is not the same as for (b1y+c1z+a1x−t)−1. Now one general proposition to which Jacobi was led in the course of this investigation was that the coefficient ofx1−1x2−1x3−1…in the expansion ofy1−1u2−1u3−1…, where
is |a1b2c3…|−1, provided that in energy case the first term of uris that containing xr.
- Type
- Research Article
- Information
- Earth and Environmental Science Transactions of The Royal Society of Edinburgh , Volume 40 , Issue 3 , 1905 , pp. 615 - 629
- Copyright
- Copyright © Royal Society of Edinburgh 1905
References
page 615 note * Jacobi, C. G. J., “Exercitatio algebraica circa discerptionem singularem fractionum, quae plures variabiles involvunt,” Crelle's Journ., v. pp. 344–364 (year 1829)Google Scholar.
page 617 note * In view of the insufficiency in the number of equations it may be noted that in the preceding case there is a redundancy, viz., ten equations to nine unknowns, but no inconsistency.
page 621 note * Had we taken as our vanishing determinant that corresponding to the only other positive term of ∣a 1b 2c 3∣ we should have found the resulting identity objectionable in both forms.
page 621 note † Proceedings Roy. Soc. Edin., xi. pp. 409–418 (Sess. 1881–1882)Google Scholar.
page 623 note * MacMahon, P. A., “A certain class of generating functions in the theory of numbers,” Trans. Roy. Soc., clxxxv. pp. 111–160Google Scholar.
page 625 note * The first to draw attention to determinants of such array was probably Joseph Horner: see his “Notes on Determinants” in the Quart. Journ. of Math., viii. pp. 157–162 (year 1865)Google Scholar.