A Short Proof Of An Identity Of Sylvester

We present two short proofs of an identity found by Sylvester and rediscovered by Louck. The first proof is an elementary version of Knuth's proof 
and is analogous to Macdonald's proof of a related identity of Milne. The second is Sylvester's own proof of his identity.

1. Sylvester's identity.Our purpose in this paper is to present two proofs of a fundamental identity found in Sylvester's work, which is in Sylvester's own words [17, p. 90], "a simple theorem for expressing, by means of partial fractions, the sum of the homogeneous powers and products of any number of quantities." The identity in question is where q is a nonnegative integer and the complete homogeneous symmetric function h m (x) in the variables x ≡ (x 1 ,...,x n ) is defined by means of the generating function Further, if m < 0, then h m (x) is defined to be 0. Sylvester [16, p. 42] uses the fact that the sum in (1.1) is 0 when q = 0,...,n − 2, and is a polynomial when q ≥ n − 1.In his later work on partitions [17], he uses (1.1) again.But the identity is most clearly formulated only in the lectures he gave in 1859 [18, p. 156].A little more than a hundred years later, (1.1) was rediscovered by Louck [8].Chen and Louck [2] have pointed out that for q = 0, 1,...,n − 1, the identity was known to Waring [20] in 1779.The q = n − 1 case of (1.1) was rediscovered by Good [4] in his elegant proof of Dyson's [3] conjecture.
In Section 2, we will present two short proofs of Sylvester's theorem.Both involve partial fraction expansions.The first proof succeeds in finding the left hand side of (1.1) by starting from the sum in the right hand side and is analogous to Macdonald's proof of a related identity.The second is Sylvester's own proof of his identity and, as suggested by his description above, transforms the left hand side of (1.1) into the sum on the right.Finally, in Section 3, we comment briefly upon the importance of these identities.

Partial fractions.
We first consider Macdonald's clever proof of an identity found by Milne [12]: Macdonald (see [13]) proved (2.1) by setting t = 0 in the partial fraction expansion Our first proof of Sylvester's identity is analogous to Macdonald's proof of (2.1).To prove (1.1), we consider the partial fraction expansion where p q (z) = 0 if q = 0, 1,...,n−1.Further, if q ≥ n, then it is a polynomial of degree q − n.Let F q (x 1 ,...,x n ) represent the sum on the right hand side of (1.1).Next, set z = 0 in (2.3) to obtain Our proof will be complete once we compute p q (0).But p q (0) is nothing but the constant term in the quotient obtained when z q is divided by (2.5) by comparing with (1.2), it follows that This completes the derivation of Sylvester's identity.Macdonald's proof of (2.1) is very simple, but the choice of the particular rational function on the left hand side of (2.2) is unmotivated.A similar remark holds for (2.3).However, a simple observation remedies this situation.
Once again, consider the sum side of (1.1), where n is replaced by n + 1, and x n+1 is renamed z.In this manner, we obtain It is clear that (2.7) is the same as (2.3), our starting point in the proof of Sylvester's identity.The particular choice of the rational function considered is now transparent.The same observation applies to Macdonald's proof of Milne's identity.This observation is also relevant to Askey's proof of Milne's identity, which is reproduced by Milne [13].Askey first proved that the sum side of (2.1) is independent of x 1 ,...,x n .Suppressing even the dependence on y 1 ,...,y n , we let f n denote the left hand side of (2.1).To complete his proof, Askey found a simple recursion for f n : from which (2.1) follows quite easily.Instead, we find another recursion for f n by replacing n by n + 1 in (2.1) and taking the limit as x n+1 → 0. In this manner, we obtain (2.9) We also have the initial condition f 1 = 1 − y 1 .Milne's identity follows by noting that (2.10) by telescoping.Recursion (2.9) is perhaps even simpler than Askey's recursion.The proof of (1.1) presented above is also related to Sylvester's proof of his identity.In Sylvester's notes [18], where (1.1) appears explicitly, he does not include his proof.But based on his remarks reproduced above and some of his work in his previous paper [17], it seems likely that he obtained (1.1) by considering the partial fraction expansion

.11)
By equating the coefficients of z −q−1 on both sides of the equation, we immediately obtain (1.1).Compare this with our computation of p q (0) above.It is interesting to note that setting z = 0 in (2.11) and replacing x i by x −1 i , we obtain Good's identity, the q = n − 1 case of (1.1).

Concluding remarks.
Our first proof of Sylvester's identity is an elementary version of the proof given by Knuth [7, §1.2.3, problem 33], who found it necessary to use Cauchy's residue theorem.Variations of Sylvester's proof are given by Chen and Louck [2] and Strehl and Wilf [15], though these authors prefer to use the Lagrange interpolation formula rather than partial fractions.Knuth mentions that special cases of (1.1) are useful in the theory of divided differences.Indeed, (1.1) has been rediscovered by Verde-Star [19] in this context.It appears in the context of mathematical physics in the work of Louck and Biedenharn [9,10].Far reaching generalizations of (1.1) have been found by Gustafson and Milne [6] and by Chen and Louck [2].

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation