The authors prove via the WZ method that a quintuple central binomial sum equals 3/π and that the sum of a fourth derivative of a gamma ratio equals 1959/2 ζ(6) minus 432 ζ(3)^2.
Various conjectural series identities
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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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math.CO 1years
2026 1verdicts
UNVERDICTED 1representative citing papers
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Evaluations of some series via the WZ method
The authors prove via the WZ method that a quintuple central binomial sum equals 3/π and that the sum of a fourth derivative of a gamma ratio equals 1959/2 ζ(6) minus 432 ζ(3)^2.