Asymptotic expansions for the constants of Landau and Lebesgue

The constants of Landau and Lebesgue are defined, for all integers n⩾0n⩾0, in order, byGn=∑k=0n116k(2kk)2andLn=12π∫−ππ|sin((n+12)t)sin(12t)|dt, which play important roles in the theories of complex analysis and Fourier series, respectively. Certain inequalities and asymptotic expansions for the constants GnGn and LnLn have been investigated by many authors. Here we aim at establishing new asymptotic expansions for the constants GnGn and LnLn of Landau and Lebesgue, respectively, by mainly using Bell polynomials and the partition function.