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Coefficients in Expansions of Certain Rational Functions

  • Ronald Evans (a1), Mourad E. H. Ismail (a2) and Dennis Stanton (a3)

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The constant term of certain rational functions has attracted much attention recently. For example the Dyson conjecture; that the constant term of

is the multinomial coefficient

has spawned many generalizations (see [2], [7]). In this paper we consider some other families of rational functions which have interesting constant terms. For example, Corollary 4 states that the constant term of

(1.1)

is . Here, and throughout this paper, A and B denote fixed positive integers.

In order to prove this result, we consider the rational function in two variables

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References

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1. Andrews, G. E., Identities in combinatorics II: A q-analog of the Lagrange inversion theorem, Proc. Amer. Math. Soc. 53 (1975), 240245.
2. Andrews, G. E., Problems and prospects for basic hyper geometric functions, in Theory and applications of special functions (Academic Press, New York, 1975), 191240.
3. Andrews, G. E., Theory of partitions (Addison-Wesley, Reading, Massachusetts, 1976).
4. Garsia, A., A q-analogue of the Lagrange inversion formula, Houston J. Math. 7 (1981), 205237.
5. Gessel, I., A noncommutative generalization and q-analoque of the Lagrange inversion formula, Transactions Amer. Math. Soc. 257 (1980), 455482.
6. Jacobi, C., De resolution aequationum per series infinitas, J. fur die reine und angewandte Math. 6 (1830), 257286.
7. Macdonald, I., Some conjectures for root systems, SIAM J. Math. Anal., to appear.
8. Mallows, C. L., A formula for expected values, Amer. Math. Monthly 87 (1980), A formula for expected values.
9. Rainville, E. D., Special functions (Macmillan, New York, 1960).
10. Sears, D., On the transformation theory of basic hyper geometric functions, Proc. London Math. Soc. 53 (1951), 158180.
11. Slater, L. J., Generalized hyper geometric functions (Cambridge University Press, Cambridge, 1966).
12. Titchmarsh, E. C., Theory of functions, second edition (Oxford, 1944).
13. Zeilberger, D., A combinatorial proof of Dyson s conjecture, preprint.
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Coefficients in Expansions of Certain Rational Functions

  • Ronald Evans (a1), Mourad E. H. Ismail (a2) and Dennis Stanton (a3)

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