[Paper Review] New proofs of some formulas of Guillera-Ser-Sondow

Vassily Bolbachan|arXiv (Cornell University)|Oct 21, 2009

Advanced Mathematical Identities1 references3 citations

TL;DR

This paper presents new elementary proofs for logarithmic series involving the Euler-Mascheroni constant γ and related constants, using finite difference identities and integral representations. It establishes novel formulas for γ, u, and ln u via binomial coefficient sums, and proves the irrationality of e^γ through a conjectured linear independence condition on integrals of rational functions.

ABSTRACT

We present logarithmic series for u, ln u and the Euler-Mascheroni constant gamma. It was indicated by J. Sondow that Theorem 4 and all proofs are new. All proofs are elementary. We present some conjectures.

Motivation & Objective

• To provide new elementary proofs for known formulas involving the Euler-Mascheroni constant γ, u, and ln u, as previously established by Guillera, Ser, and Sondow.
• To derive novel representations of γ using binomial coefficient sums involving logarithms of factorials and harmonic series.
• To establish a connection between the convergence of certain binomial series and the irrationality of e^γ through a conjectured linear independence condition.
• To extend the scope of known identities by proposing new conjectures involving nested logarithmic expressions and special functions.
• To unify and re-derive existing results in the literature using only elementary calculus and finite difference techniques.

Proposed method

• Utilizes Lemma 7, which expresses the sum ∑ₖ₌₀ᵐ (m choose k)(-1)^k / (k + z) as gₘ(z)/z, where gₘ(z) = m! / [(z+1)(z+2)...(z+m)].
• Applies Lemma 8 to show uniform convergence of gₘ(z)/z to 0 on [z₀, ∞), enabling limit interchange in integrals and series.
• Employs Lemma 9: (m choose k)/k = ∑ₙ₌ₖᵐ (n choose k)/n, to reindex double sums and switch summation order in binomial coefficient expressions.
• Uses Lemma 10: ∑ₖ₌ₙʲ (-1)ᵏ⁺ⁿ (j choose k) = (j-1 choose n-1), to reorganize nested sums involving alternating binomial coefficients.
• Applies integral representations of logarithms, such as ∫₁ᵘ dz/z = ln u and ∫₀¹ᐟʲ gₘ(1/u) du, to connect discrete sums to continuous limits.
• Establishes convergence via comparison with harmonic series and bounds on gₘ(1/u), showing ∫₀¹ᐟʲ gₘ(1/u) du → 0 as m → ∞.

Experimental results

Research questions

• RQ1 Can the formula for the Euler-Mascheroni constant γ = limₘ→∞ ∑ₖ₌₁ᵐ (m choose k)(-1)^k / k ⋅ ln(k!) be re-proven using elementary methods?
• RQ2 What is the connection between the convergence of binomial coefficient sums and the logarithmic representation of u and ln u?
• RQ3 Can the irrationality of e^γ be established from a conjectured linear independence of integrals involving rational functions?
• RQ4 How can nested logarithmic expressions involving multiple layers of logarithms be represented as infinite binomial series?
• RQ5 What is the role of uniform convergence in the limit of binomial sums involving 1/(ku + 1) and related functions?

Key findings

• Theorem 1 establishes that for any u > 0, 1 = limₘ→∞ ∑ₖ₌₁ᵐ (m choose k)(-1)ᵏ⁺¹ / (ku + 1), with uniform convergence on compact intervals.
• Theorem 4 proves that γ = limₘ→∞ ∑ₖ₌₁ᵐ (m choose k)(-1)^k / k ⋅ ln(k!), offering a new elementary derivation of this fundamental constant.
• Corollary 3 shows that u = limₘ→∞ ∏ₖ₌₁ᵐ (k + u)^(m choose k)(-1)ᵏ⁺¹, providing a product representation of real numbers via binomial coefficients.
• Corollary 5 re-proves the formula γ = ∑ⱼ₌₁^∞ ∑ᵢ₌₁ʲ (j-1 choose i-1)(-1)^i / j ⋅ ln i using only elementary identities, bypassing analytic continuation.
• Conjecture 4 provides a general formula for γ involving a sum over distinct positive integers a₁,…,aₘ, with coefficients defined via rational functions.
• The paper concludes that if the numbers yᵢ = ∑ₙ₌₁^∞ ∫₀¹ᐟⁿ dx / (1 + (ix)⁻¹) are linearly independent over ℤ, then e^γ is irrational — a significant number-theoretic implication.