International Journal of Advanced Multidisciplinary Research and Studies
Volume 2, Issue 6, 2022
Generalised Frobenius Partitions with Colours and Repetitions
Author(s): Mahadevaswamy BS
In this paper we study the partition function cfk,h(n), the number of generalised Frobenius partitions of n with k colours and h repetitions. This provides a common generalisation of the two functions cfl,k(n), and cfk,l(n) of George E. Andrews. In particular, we develop a method leading to representations of all the generating functions of cfk,h(n) as sums of infinite products.
We also obtain the Hardy – Ramanujan – Rademacher series for cfk,h(n) on the lines of L. W. Kolitsch. The existence of such series for cf1,k(n) and cfk,l(n) was asked for by Andrews and later obtained by Kolitsch.
Finally, we extend the results on q-binomial coefficients and q-series representation of Andrews to our function cfk,h(n). Andrews has established the two congruences cf1,2(5n+3) º cf2,1 (5n+3) º O (mod 5). We show that the analogous congruence cf2,2 (5n+3) º O (mod 5) is false for n = 2. We also study generalised Frobenius partitions with some restriction on its parts.
Keywords: Frobenius Partitions, General Principle
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