Ankush Goswami, Venkata Raghu Tej Pantangi,
"Some formulae for coefficients in restricted $q$-products"
, Serie RISC Report Series, Nummer 20-08, RISC, JKU, Hagenberg, Linz, 5-2020
Original Titel:
Some formulae for coefficients in restricted $q$-products
Sprache des Titels:
Englisch
Original Kurzfassung:
In this paper, we derive some formulae involving coefficients of polynomials which occur quite naturally in the study of restricted partitions. Our method involves a recently discovered sieve technique by Li and Wan (Sci. China. Math. 2010). Based on this method, by considering cyclic groups of different orders we obtain some new results for these coefficients. The general result (see Theorem \ref{main00}) holds for any group of the form $\mathbb{Z}_{N}$ where $N\in\mathbb{N}$ and expresses certain partial sums of coefficients in terms of expressions involving roots of unity. By specializing $N$ to different values, we see that these expressions simplify in some cases and we obtain several nice identities involving these coefficients. We also use a result of Sudler (Quarterly J. Math. 1964) to obtain an asymptotic formula for the maximum absolute value of these coefficients.