New series involving binomial coefficients (III)

TL;DR

This paper evaluates series involving binomial coefficients, introduces a Duality Principle for irrational series, and presents new identities and conjectures related to Ramanujan's and Zeilberger's types, connecting series to special functions.

Contribution

It introduces the Duality Principle for irrational series and applies it to discover 26 new irrational series identities, expanding the understanding of series involving binomial coefficients.

Findings

Proved a series sum involving binomial coefficients equals 3π/2.

Introduced the Duality Principle for irrational series of Ramanujan's and Zeilberger's types.

Conjectured new series identities involving Kronecker symbols and L-functions.

Abstract

We evaluate some series with summands involving a single binomial coefficient $\binom{6k}{3k}$. For example, we prove that $$\sum_{k=0}^\infty\frac{(63k^2+78k+22)8^k}{(2k+1)(6k+1)(6k+5)\binom{6k}{3k}}=\frac{3\pi}2.$$ Motivated by Galois theory, we introduce the so-called Duality Principle for irrational series of Ramanujan's type or Zeilberger's type, and apply it to find 26 new irrational series identities. For example, we conjecture that \begin{align*}&\sum_{k=1}^\infty\frac{(32(91\sqrt{33}-523))^{k}}{k^3\binom{2k}k^2\binom{3k}k} \left((91\sqrt{33}+891)k-33\sqrt{33}-225\right) \\&\qquad=320\left(\frac{11}3\sqrt{33}L_{-11}(2)-27L_{-3}(2)\right), \end{align*} where $ L_{d}(2)=\sum_{k=1}^\infty\frac{(\frac{d}k)}{k^2}$ for any integer $d\equiv0,1\pmod4$ with $(\frac{d}k)$ the Kronecker symbol.