On the power-series expansion of a rational function
D. V. Lee
Acta Arithmetica, Tome 62 (1992), p. 229-255 / Harvested from The Polish Digital Mathematics Library
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Introduction. The problem of determining the formula for PS(n), the number of partitions of an integer into elements of a finite set S, that is, the number of solutions in non-negative integers, hs₁,...,hsk, of the equation hs₁ s₁ + ... + hsk sk = n, was solved in the nineteenth century (see Sylvester [4] and Glaisher [3] for detailed accounts). The solution is the coefficient ofxⁿin[(1-xs₁)... (1-xsk)]-1, expressions for which they derived. Wright [5] indicated a simpler method by which to find part of the solution (at least in the case si=i). The current paper gives a simple method by which the power-series expansion of a rational function may be derived. Lemma 1 is well known and gives the general form of the solution. Lemma 2 is also well known. See, for example, Andrews [1], Example 2, p. 98. Lemma 3 shows how the recurrence relation of Lemma 2 becomes of bounded degree in certain cases. The recurrence relation is then solved, and the solution is extended from these certain cases to all cases. We then apply the result to investigate the growth of the difference PS(n)-PT(n), where S and T are finite sets, and in particular when this difference is bounded. The differences PS(0)(n)-PT(0)(n) and PS(1)(n)-PT(1)(n) are also considered, where PS(0) (resp. PS(1)) denotes the number of partitions of n into elements of S with an even (resp. odd) number of parts.

Publié le : 1992-01-01
EUDML-ID : urn:eudml:doc:206491 
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D. V. Lee. On the power-series expansion of a rational function. Acta Arithmetica, Tome 62 (1992) pp. 229-255. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav62i3p229bwm/

[000] [1] G. E. Andrews, Partitions, originally: Encyclopedia of Mathematics, Vol. 2, Addison-Wesley, reprinted by Cambridge University Press, 1976.

[001] [2] H. Davenport, Multiplicative Number Theory, 2nd ed., Springer, 1980. | Zbl 0453.10002

[002] [3] J. W. L. Glaisher, Formulae for partitions into given elements, derived from Sylvester's theorem, Quart. J. Math. 40 (1909), 275-348. | Zbl 40.0235.04

[003] [4] J. J. Sylvester, On subinvariants, i.e. semi-invariants to binary quantics of an unlimited order: Excursus on rational functions and partitions, Amer. J. Math. 5 (1882), 119-136.

[004] [5] E. M. Wright, Partitions into k parts, Math. Ann. 142 (1961), 311-316. | Zbl 0100.27301