Reduced Denominators and Irrationality Exponents of Cyclotomic and Lucas Layer-Quotient Sums

For integers $d\geq2$ and $a>b\geq1$, put $C_{d,n}(a,b)=(a^{d^{n+1}}-b^{d^{n+1}})/(a^{d^n}-b^{d^n})$ and $\mathcal S_d(a,b)=\sum_{n\geq0}C_{d,n}(a,b)^{-1}$. Let $q_N$ be the denominator in lowest terms of the partial sum $\sum_{n=0}^{N}C_{d,n}(a,b)^{-1}$. We obtain an exact structural description of $q_N$ by evaluating the natural common denominator $Q_N$ and proving that the cancellation factor $Q_N/q_N$ is eventually periodic. Writing $\mu(\xi)$ for the irrationality exponent of $\xi$, we obtain\[\mu\bigl(\mathcal S_d(a,b)\bigr)=\frac{d(d-1)\log a}{d\log a-\log\gcd(a,b)}\qquad(d\geq4).\]For $d=3$, the same formula holds when $\log\gcd(a,b)/\log a\geq3(2-\sqrt3)$, while $\mu(\mathcal S_3(a,1))=2$ for every $a\geq2$. For every fixed integer $m\geq1$ and every periodic sequence $(\omega_n)$ of nonzero integers, the factor cancelled from $Q_N^m$ in the partial sums of $\sum_{n\geq0}\omega_n C_{d,n}(a,b)^{-m}$ is also eventually periodic; for $d\geq4$, the corresponding weighted sum has the irrationality exponent displayed above. For $d=2,3$ and $b=1$, the exponent is $2$ whenever $m$ is a power of $2$ and the weights have a common $2$-adic valuation. Let $(U_r)$ be a first-kind Lucas sequence with coprime integer parameters $P,Q$, where $Q\ne0$, whose real characteristic roots satisfy $\alpha>1$ and $|\beta|<\alpha$. Let $h\geq1$ be an integer. For\[\sum_{n\geq0}\omega_n\left(h^{(d-1)d^n}\frac{U_{d^{n+1}}}{U_{d^n}}\right)^{-m},\]we determine the reduced partial-sum denominators, and the cancelled factors are again eventually periodic. For $d\geq4$, its irrationality exponent is $d(d-1)\log(h\alpha)/\bigl(d\log\alpha+(d-1)\log h\bigr)$. All three families are transcendental for every $d\geq2$.