Hankel Determinants of Geometric Moments and Rational Lambert Values

Comparison with earlier criteria

Bundschuh and Väänänen’s Theorem 2 at $\alpha =-1$ [2] gives irrationality for $\log b/\log a<{\theta }_{\mathrm{B}\mathrm{V}}:=1/2-1/{\pi }^2$. Their $q$ is the base $a/b>1$, not its reciprocal. Their logarithmic derivative is $L_q(z)=\sum _{j\ge 1}(q^j+z{)}^{-1}$, so the value at $z=\alpha =-1$ is exactly $F(a/b)$. The rational height is $h(q)=a$, and their parameter $\lambda =\log h(q)/\log q$ is $1/(1-\log b/\log a)$, which gives the displayed cutoff. Since ${\pi }^2<10$, one has ${\theta }_{\mathrm{B}\mathrm{V}}<2/5<\log 4/\log 31$, so $31/4$ lies outside that sufficient region. The two sufficient regions differ on $[{\theta }_{\mathrm{B}\mathrm{V}},{\theta }^{\ast })$. Zudilin also notes an extension to noninteger rational bases $p=r/s$ for the generalized $q$-logarithm when $\log |r|>c\log |s|$, with $c>0$ computable but unspecified [5]. The value $c=\mu$ here comes from the 2004 forms [4], not from evaluating the unspecified constant in the 2016 construction.

The positive-remainder estimates require $x>1$, so they give no statement for negative bases. The series diverges at $x=1$. The companion’s height-criterion section also applies Bundschuh and Väänänen’s cited theorem at $7/2$.