The Mahler measure of linear forms as special values of solutions of algebraic differential equations (Q2270590)

Let \(m(L_n)\) be the logarithmic Mahler measure of the polynomial \(L_n=x_1+\dots+x_n\) and let \(J_0(x)\) be the Bessel function of the first kind of order zero. The cases \(n \leq 4\) have been evaluated by \textit{C. J. Smyth} in 1981 [Bull. Aust. Math. Soc. 23, 49--63 (1981; Zbl 0442.10034)]. The author shows that \[ m(L_n)=\log \sqrt{n}-\frac{\gamma}{2}+\sum_{m=2}^{\infty} \frac{c_m(n)}{n^m}, \] where \(\gamma\) is Euler's constant, \[ c_m(n)=-\frac{b_m(n)(m-1)!}{2}, \] and \(b_m(n)\) is the \(m\)th coefficient of the Taylor expansion of the function \(e^{nx} J_0^n(2\sqrt{x})\). Moreover, it is shown that for each \(n \geq 1\) and \(m \geq 2\) the inequality \[ |c_m(n)| \leq C n^m m^{-5/4} \] holds with \[ C=5 \cdot 2^{-5/4} L_{12}(1)=1.043\dots, \] where \(L_{t}(x)\) is the \(t\)th Laguerre polynomial.