Papers › The Restricted Partition and q-Partial Fractions
The Restricted Partition and q-Partial Fractions
N. Uday Kiran
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The restricted partition function p_N(n) counts the partitions of n into at most N parts. In the nineteenth century Sylvester showed that these partitions can be expressed as a sum of k-periodic quasi-polynomials (1≤k≤N) which he termed as Waves. It is now well-known that one can easily perform a wave decomposition using a special type of partial fraction decomposition (the so-called q-partial fractions) of the generating function of p_N(n). In this paper we show that the coefficients of these q-partial fractions can be expressed as a linear combination of the Ramanujan sums. In particular, we show, for the first time, an appearance of the degenerate Bernoulli numbers, the degenerate Euler numbers and a special generalization of the Ramanujan sums, which we term as a Gaussian-Ramanujan sum, in the formulae for certain waves. These coefficients not only provide a good approximation of p_N(n) but they can also be used for obtaining good bounds. Further, we provide a combinatorial meaning to these sums. Our approach for partial fractions is based on a projection operator on the I-adic completion of the ring of polynomials, where I is an ideal generated by the Cyclotomic polynomial.
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