Affiliation:
1. Department of Mathematical Methods and Models, Fundamental Sciences Applied in Engineering Research Center, University Politehnica of Bucharest, 060042 Bucharest, Romania
Abstract
For each s∈{1,3,5}, we consider Rs(n) to be the number of the partitions of n into parts not congruent to 0, ±s(mod12). In recent years, some relations for computing the value of R3(n) were studied. In this paper, we investigate the parity of Rs(n) when s∈{1,5} and derive the following congruence identity: ∑n=1∞(−q;q)n−12(1+qn)qn2(q;q)2n≡∑n=1∞qn2+q3n2(mod2). For each s∈{1,5}, the number of the partitions of n into parts not congruent to 0, ±s(mod12) is connected with two truncated theta series. Some open problems involving R1(n) and R5(n) are introduced in this context.
Subject
Geometry and Topology,Logic,Mathematical Physics,Algebra and Number Theory,Analysis
Reference13 articles.
1. Euler, L. (1988). Introduction to Analysis of the Infinite, Book I. Blanton, J.D., Translator, Springer.
2. Andrews, G.E. (1998). The Theory of Partitions, Cambridge Mathematical Library, Cambridge University Press.
3. Hardy, G.H., and Wright, E.M. (1979). An Introduction to the Theory of Numbers, Clarendon Press.
4. A short proof of an identity of Euler;Shanks;Proc. Amer. Math. Soc.,1951
5. Gasper, G., and Rahman, M. (1990). Encyclopedia of Mathematics And Its Applications 35, Cambridge University Press.