Various conjectural series identities

TL;DR

This paper compiles over 150 new conjectural series identities involving binomial coefficients, related to fundamental constants like pi and zeta functions, aiming to inspire further mathematical research.

Contribution

It introduces a large collection of new conjectural series identities involving binomial coefficients and special functions, expanding the landscape of known mathematical series.

Findings

Over 150 new conjectural series identities proposed

Identities involve binomial coefficients and special functions

Some identities relate to pi and zeta functions

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.