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arXiv · 2603.29973

Various conjectural series identities

Zhi-Wei Sun
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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 $\pi$ or Riemann's zeta function or Dirichlet's $L$-function. For example, we conjecture that $$\sum_{k=0}^\infty\frac{16k+3}{(-202^2)^k}\binom{2k}kT_k(19,-20)T_{2k}(9,-5)=\frac{43\sqrt{101}}{75\pi},$$ where $T_n(b,c)$ denotes the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. The conjectures in this paper might interest some readers and stimulate further research.

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Zhi-Wei Sun. 2026-03-31. Various conjectural series identities. https://arxiv.org/abs/2603.29973

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