Geodesic
On Certain Polynomials of Gaussian Type
Canadian journal of mathematics, Tome 31 (1979) no. 2, pp. 274-281

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

Introduction. We shall consider functions of the form where {r i } and {s i } are sets of positive integers. Such functions were studied by E. Grosswald in [2], who took {s i } to be pairwise relatively prime, and asked the following two questions:(a) When is ƒ(t) a polynomial?(b) When does ƒ(t) have positive coefficients?These questions arise naturally from the work of Allday and Halperin, who show in [1] that under suitable circumstance ƒ(t) will be the Poincare polynomial of the orbit space of a certain Lie group action. Grosswald gives a complete answer to (a), but (b) is a much harder question, and a complete answer is provided only for the case m = 2. His treatment involves the representation of the coefficients of ƒ(t) by partition functions, and uses a classical description by Sylvester of the semigroup generated by {s i }. 
Reich, Daniel. On Certain Polynomials of Gaussian Type. Canadian journal of mathematics, Tome 31 (1979) no. 2, pp. 274-281. doi: 10.4153/CJM-1979-029-7
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[1] 1. Allday, C. and Halperin, S., Lie group actions on spaces of finite rank, Quart. J. of Math., Oxford, (to appear). Google Scholar

[2] 2. Grosswald, E., Reducible rational fractions of the type of Gaussian polynomials with only nonnegative coefficientsj Canadian Math. Bull., (to appear). Google Scholar

[3] 3. Nijenhuis, A. and Wilf, H., Representations of integers by linear forms in non-negative integers, J. of Number Theor. 4 (1972), 98–106. Google Scholar

[4] 4. Sylvester, J., A constructive theory of partitions, Amer. J. Math. 5 (1882), 251–330. Google Scholar

[5] 5. Sylvester, J., Mathematical questions with their solutions, Educational Times 41 (1884), p. 21. Google Scholar

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