Various conjectural series identities

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 $\ddot{I}€$ or Riemann's zeta function or Dirichlet's $L$-function. For example, we conjecture that $$\sum _{k=0}^{\infty }\frac{16k+3}{(-202^2{)}^k}\left(\genfrac{}{}{0ex}{}{2k}{k}\right)T_k(19,-20)T_{2k}(9,-5)=\frac{43\sqrt{101}}{75\ddot{I}€},$$ 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.

27 pages. Mainly add Section 5