Harmonic-binomial Euler-like sums via expansions of $$(\arcsin x)^p$$

Euler and the Partial Sums of the Prime Harmonic Series

Abstract. In a 1737 paper, Euler gave the first proof that the sum of the reciprocals of the prime numbers diverges. That paper can be considered as the founding document of analytic number theory, and its key innovation — socalled Euler products — are now ubiquitous in the field. In this note, we probe Euler’s claim there that “the sum of the reciprocals of the prime numbers” is “as the logari...

Fractional Sums and Euler-like Identities

when the number of terms x is a rather arbitrary real or even complex number. The resulting theory includes the known interpolation of the factorial by the Γ function or Euler’s little-known formula ∑ −1/2 ν=1 1 ν = −2 ln 2 . Many classical identities like the geometric series and the binomial theorem nicely extend to this more general setting. Sums with a fractional number of terms are closely...

Binomial Sums and Identities

Generalising Tuenter’s binomial sums

Tuenter [Fibonacci Quarterly 40 (2002), 175-180] and other authors have considered centred binomial sums of the form Sr(n) = ∑

Multiple binomial sums

Multiple binomial sums form a large class of multi-indexed sequences, closed under partial summation, which contains most of the sequences obtained by multiple summation of products of binomial coefficients and also all the sequences with algebraic generating function. We study the representation of the generating functions of binomial sums by integrals of rational functions. The outcome is two...