论文信息 - Truncated Theta Series and Rogers-Ramanujan Functions

Truncated Theta Series and Rogers-Ramanujan Functions

Abstract We consider the squares of the Rogers-Ramanujan functions and for each we obtain a linear recurrence relation for the number of partitions of n into parts congruent to . In this context, we conjecture that for , the theta series 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 an infinite family of linear homogeneous inequalities for the number of partitions of n into parts congruent to . Twenty identities involving Rogers-Ramanujan functions are experimentally discovered considering Jacobi’s triple product identity and Watson’s quintuple product identity.

Mircea Merca | M. Merca

[1] Ae Ja Yee. A truncated Jacobi triple product theorem , 2015, J. Comb. Theory, Ser. A.

[2] Renrong Mao,et al. Proofs of two conjectures on truncated series , 2015, J. Comb. Theory, Ser. A.

[3] B. Berndt. Ramanujan's Notebooks , 1985 .

[4] George E. Andrews,et al. The truncated pentagonal number theorem , 2012, J. Comb. Theory, Ser. A.

[5] George E. Andrews,et al. Shorter Notes: A Simple Proof of Jacobi's Triple Product Identity , 1965 .

[6] E. M. Wright,et al. An Enumerative Proof of An Identity of Jacobi , 1965 .

[7] George E. Andrews,et al. Truncated theta series and a problem of Guo and Zeng , 2018, J. Comb. Theory, Ser. A.

[8] L. Carlitz,et al. A simple proof of the quintuple product identity , 1972 .

[9] L. J. Rogers. Second Memoir on the Expansion of certain Infinite Products , 1893 .

[10] Mathukumalli Vidyasagar,et al. On Watson’s quintuple product identity , 1970 .