English

Various conjectural series identities

Number Theory 2026-04-14 v3 Combinatorics

Abstract

In this paper we collect over 150 new series identities (involving binomial coefficients) conjectured by the author in 2026. The values involved are related to π\pi or Riemann's zeta function or Dirichlet's LL-function. For example, we conjecture that k=016k+3(2022)k(2kk)Tk(19,20)T2k(9,5)=4310175π,\sum_{k=0}^\infty\frac{16k+3}{(-202^2)^k}\binom{2k}kT_k(19,-20)T_{2k}(9,-5)=\frac{43\sqrt{101}}{75\pi}, where Tn(b,c)T_n(b,c) denotes the coefficient of xnx^n in the expansion of (x2+bx+c)n(x^2+bx+c)^n. The conjectures in this paper might interest some readers and stimulate further research.

Keywords

Cite

@article{arxiv.2603.29973,
  title  = {Various conjectural series identities},
  author = {Zhi-Wei Sun},
  journal= {arXiv preprint arXiv:2603.29973},
  year   = {2026}
}

Comments

27 pages. Mainly add Section 5

R2 v1 2026-07-01T11:46:40.852Z