
Applying the multisection series method to the MacLaurin series expansion of arcsin-function, we transform the Apéry–like series involving the central binomial coefficients into systems of linear equations. By resolving the linear systems (for example, by Mathematica), we establish numerous remarkable infinite series formulae for π and logarithm functions, including several recent results due to Almkvist et al. (2003) and Zheng (2008).
We introduce the notion of capacity (ability to contain water) for compositions. Initially the compositions are defined on a finite alphabet
As suggested by Currie, we apply the probabilistic method to problems regarding pattern avoidance. Using techniques from analytic combinatorics, we calculate asymptotic mean pattern occurrence and use them in conjunction with the probabilistic method to establish new results about the Ramsey theory of unavoidable patterns in the abelian full word case and in the nonabelian partial word case.
In this paper, we present several explicit formulas of the sums and hypersums of the powers of the first
We define a generalized class of modified zeta series transformations generating the partial sums of the Hurwitz zeta function and series expansions of the Lerch transcendent function. The new transformation coefficients we define within the article satisfy expansions by generalized harmonic number sequences as the partial sums of the Hurwitz zeta function. These transformation coefficients satisfy many properties which are analogous to known identities and expansions of the Stirling numbers of the first kind and to the known transformation coefficients employed to enumerate variants of the polylogarithm function series. Applications of the new results we prove in the article include new series expansions of the Dirichlet beta function, the Legendre chi function, BBP-type series identities for special constants, alternating and exotic Euler sum variants, alternating zeta functions with powers of quadratic denominators, and particular series defining special cases of the Riemann zeta function constants at the positive integers s ≥ 3.
Let
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