Somos's quadratic recurrence constant is defined via the sequence(1)with . This has closed-form solution(2)where is a polylogarithm, is a Lerch transcendent. The first few terms are 1, 2, 12, 576, 1658880, 16511297126400, ... (OEIS A052129). The terms of this sequence have asymptotic growth as(3)(OEIS A116603; Finch 2003, p. 446, term corrected), where is known as Somos's quadratic recurrence constant. Here, the generating function in satisfies the functional equation(4)Expressions for include(5)(6)(7)(8)(9)(OEIS A112302; Ramanujan 2000, p. 348;Finch 2003, p. 446; Guillera and Sondow 2005).Expressions for include(10)(11)(12)(13)(14)(OEIS A114124; Finch 2003, p. 446; Guillera and Sondow 2005; J. Borwein, pers. comm., Feb. 6, 2005), where is a polylogarithm. is also given by the unit square integral(15)(16)(Guillera and Sondow 2005).Ramanujan (1911; 2000, p. 323) proposed..
Catalan's constant is a constant that commonly appears in estimates of combinatorial functions and in certain classes of sums and definite integrals. It is usually denoted (this work), (e.g., Borwein et al. 2004, p. 49), or (Wolfram Language).Catalan's constant may be defined by(1)(Glaisher 1877, who however did not explicitly identify the constant in this paper). It is not known if is irrational.Catalan's constant is implemented in the WolframLanguage as Catalan.The constant is named in honor of E. C. Catalan (1814-1894), who first gave an equivalent series and expressions in terms of integrals. Numerically,(2)(OEIS A006752). can be given analytically by the following expressions(3)(4)(5)where is the Dirichlet beta function, is Legendre's chi-function, is the Glaisher-Kinkelin constant, and is the partial derivative of the Hurwitz zeta function with respect to the first argument.Glaisher (1913) gave(6)(Vardi..