Various irrational series involving binomial coefficients

2025/06/02 by Sun, Zhi-Wei · 1 citation
#05A19 #11B65 #11M06 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2506.01870

Abstract

Motivated by Galois theory, we propose 26 new irrational series of Ramanujan's type or Zeilberger's type. For example, we conjecture that amp;∑_k=1^∞\frac(32(91√(33)-523))^kk³\binom2kk²\binom3kk ((91√(33)+891)k-33√(33)-225)
amp; =320(\frac113√(33)L_-11(2)-27L_-3(2)), where L_d(2)=∑_k=1^∞\frac(\fracdk)k² for any integer d≡0,1\pmod4 with (\fracdk) the Kronecker symbol. This provides a quite efficient way to compute the constant L_-11(2).

Citations

• Proofs of conjectures on Ramanujan-type series of level 3
• New series involving binomial coefficients (II)
• Double series for π and their q-analogues
• WZ-proofs of "divergent" Ramanujan-type series

Cited by

• Fast converging irrational series for L(2,(\frac d⋅))

Related

• Fast converging irrational series for L(2,(\frac d⋅))
• Proofs of conjectures on Ramanujan-type series of level 3
• New series involving binomial coefficients (II)
• On the Connection Between Irrationality Measures and Polynomial Continued Fractions
• New type series for powers of π