# On the power-series expansion of a rational function

Acta Arithmetica (1992)

• Volume: 62, Issue: 3, page 229-255
• ISSN: 0065-1036

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## Abstract

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Introduction. The problem of determining the formula for ${P}_{S}\left(n\right)$, the number of partitions of an integer into elements of a finite set S, that is, the number of solutions in non-negative integers, ${h}_{s₁},...,{h}_{{s}_{k}}$, 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 of$xⁿin$[(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 ${s}_{i}=i$). 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 ${P}_{S}\left(n\right)-{P}_{T}\left(n\right)$, where S and T are finite sets, and in particular when this difference is bounded. The differences ${P}_{S}^{\left(0\right)}\left(n\right)-{P}_{T}^{\left(0\right)}\left(n\right)$ and ${P}_{S}^{\left(1\right)}\left(n\right)-{P}_{T}^{\left(1\right)}\left(n\right)$ are also considered, where ${P}_{S}^{\left(0\right)}$ (resp. ${P}_{S}^{\left(1\right)}$) denotes the number of partitions of n into elements of S with an even (resp. odd) number of parts.

## How to cite

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D. V. Lee. "On the power-series expansion of a rational function." Acta Arithmetica 62.3 (1992): 229-255. <http://eudml.org/doc/206491>.

@article{D1992,
abstract = {Introduction. The problem of determining the formula for $P_S(n)$, the number of partitions of an integer into elements of a finite set S, that is, the number of solutions in non-negative integers, $h_\{s₁\},..., h_\{s_k\}$, 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 of$xⁿ in$[(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 $s_i=i$). 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 $P_S(n) - P_T(n)$, where S and T are finite sets, and in particular when this difference is bounded. The differences $P_S^\{(0)\}(n) - P_T^\{(0)\}(n)$ and $P_S^\{(1)\}(n) - P_T^\{(1)\}(n)$ are also considered, where $P_S^\{(0)\}$ (resp. $P_S^\{(1)\}$) denotes the number of partitions of n into elements of S with an even (resp. odd) number of parts.},
author = {D. V. Lee},
journal = {Acta Arithmetica},
keywords = {partitions; growth of difference of two partitions; power series expansion; rational functions},
language = {eng},
number = {3},
pages = {229-255},
title = {On the power-series expansion of a rational function},
url = {http://eudml.org/doc/206491},
volume = {62},
year = {1992},
}

TY - JOUR
AU - D. V. Lee
TI - On the power-series expansion of a rational function
JO - Acta Arithmetica
PY - 1992
VL - 62
IS - 3
SP - 229
EP - 255
AB - Introduction. The problem of determining the formula for $P_S(n)$, the number of partitions of an integer into elements of a finite set S, that is, the number of solutions in non-negative integers, $h_{s₁},..., h_{s_k}$, 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 of$xⁿ in$[(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 $s_i=i$). 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 $P_S(n) - P_T(n)$, where S and T are finite sets, and in particular when this difference is bounded. The differences $P_S^{(0)}(n) - P_T^{(0)}(n)$ and $P_S^{(1)}(n) - P_T^{(1)}(n)$ are also considered, where $P_S^{(0)}$ (resp. $P_S^{(1)}$) denotes the number of partitions of n into elements of S with an even (resp. odd) number of parts.
LA - eng
KW - partitions; growth of difference of two partitions; power series expansion; rational functions
UR - http://eudml.org/doc/206491
ER -

## References

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1. [1] G. E. Andrews, Partitions, originally: Encyclopedia of Mathematics, Vol. 2, Addison-Wesley, reprinted by Cambridge University Press, 1976.
2. [2] H. Davenport, Multiplicative Number Theory, 2nd ed., Springer, 1980. Zbl0453.10002
3. [3] J. W. L. Glaisher, Formulae for partitions into given elements, derived from Sylvester's theorem, Quart. J. Math. 40 (1909), 275-348. Zbl40.0235.04
4. [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.
5. [5] E. M. Wright, Partitions into k parts, Math. Ann. 142 (1961), 311-316. Zbl0100.27301

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