Truncated Theta Series and Rogers-Ramanujan Functions

Abstract

We consider the squares of the Rogers-Ramanujan functions and for each $S\in \{1,2\}$ we obtain a linear recurrence relation for the number of partitions of n into parts congruent to $\pm S\hspace{0.5em}\mathrm{mod}\hspace{0.5em}5$. In this context, we conjecture that for $1\le S<R,\hspace{0.5em}k\ge 1$, the theta series $$\frac{{(-1)}^k}{{(q^S,q^{R-S};q^R)}_{\infty }}\sum _{j=k}^{\infty }{(-1)}^j{q}^{j(j+1)R/2-jS}(1-q^{(2j+1)S})$$has non-negative coefficients. This improves a conjecture given by G. E. Andrews and M. Merca in 2012, which was proved independently three years later by A. J. Yee using combinatorial methods and R. Mao via partial theta functions. Combinatorial interpretations of this new conjecture give for each $S\in \{1,2,3,4\}$ an infinite family of linear homogeneous inequalities for the number of partitions of n into parts congruent to $\pm S\hspace{0.5em}\mathrm{mod}\hspace{0.5em}5$. Twenty identities involving Rogers-Ramanujan functions are experimentally discovered considering Jacobi’s triple product identity and Watson’s quintuple product identity.

Keywords:

• partitions
• Rogers-Ramanujan functions
• theta series

MSC 2010:

• 05A17
• 05A19
• 05A015
• 05A20