Various conjectural series identities
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866908959398428672 |
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| author | Sun, Zhi-Wei |
| author_facet | Sun, Zhi-Wei |
| contents | 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 $π$ or Riemann's zeta function or Dirichlet's $L$-function. For example, we conjecture that $$\sum_{k=0}^\infty\frac{16k+3}{(-202^2)^k}\binom{2k}kT_k(19,-20)T_{2k}(9,-5)=\frac{43\sqrt{101}}{75π},$$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_29973 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Various conjectural series identities Sun, Zhi-Wei Number Theory Combinatorics 11B65, 05A19, 11A07, 11M06, 33B15 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 $π$ or Riemann's zeta function or Dirichlet's $L$-function. For example, we conjecture that $$\sum_{k=0}^\infty\frac{16k+3}{(-202^2)^k}\binom{2k}kT_k(19,-20)T_{2k}(9,-5)=\frac{43\sqrt{101}}{75π},$$ 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. |
| title | Various conjectural series identities |
| topic | Number Theory Combinatorics 11B65, 05A19, 11A07, 11M06, 33B15 |
| url | https://arxiv.org/abs/2603.29973 |