[Paper Review] Parameterized summation relations for the Stieltjes constants

Mark W. Coffey|arXiv (Cornell University)|Feb 25, 2010

Mathematical functions and polynomials10 references3 citations

TL;DR

This paper derives multi-parameter summation identities for the Stieltjes constants γₖ(a) and related functions Aₖ(q) using integral representations and Mellin transforms of the Hurwitz zeta function. The key contribution is a set of exact relations—such as Proposition 1 and 2—linking sums of Stieltjes constants over rational arguments to logarithmic and polygamma functions, with applications to reciprocity laws for Bernoulli polynomials and polygamma functions.

ABSTRACT

The Stieltjes constants $γ_k(a)$ appear in the regular part of the Laurent expansion of the Hurwitz zeta function about its only polar singularity at $s=1$. We present multi-parameter summation relations for these constants that result from identities for the Hurwitz zeta function. We also present multi-parameter summation relations for functions $A_k(x)$ that may be expressed as sums over the Stieltjes constants. Integral representations, especially including Mellin transforms, play an important role. As a byproduct, reciprocity and other summatory relations for polygamma functions and Bernoulli polynomials may be obtained.

Motivation & Objective

• To establish new multi-parameter summation relations for the Stieltjes constants γₖ(a) arising from functional identities of the Hurwitz zeta function.
• To derive integral representations and Mellin transform-based identities that yield exact relations for sums of Stieltjes constants over rational arguments.
• To extend known reciprocity and summatory relations to generalized forms involving parameters p, q, b, and N.
• To connect these results to known special functions such as polygamma functions and Bernoulli polynomials through derived corollaries.
• To provide a systematic framework for evaluating sums of Aₖ(q) functions, which are useful in integral evaluations involving the Hurwitz zeta function.

Proposed method

• Utilizes Lemma 1 to relate sums of Hurwitz zeta functions over rational arguments to transformed sums via parameter exchange (p, q, b).
• Applies analytic continuation to extend identities valid for Re(s) > 1 to the entire complex plane.
• Employs Mellin transform representations of zeta functions associated with rational generating functions to derive functional identities.
• Differentiates zeta function identities with respect to s and evaluates at s = 1 − k to derive relations for Aₖ(q) and Bₖ.
• Uses zeta function identities for rational functions f(T) = p(T)/∏(1 − Tⁿⁱ) to express Z_f(s) via integrals and series, enabling summation identities.
• Applies known results on Bernoulli numbers and polygamma functions to derive corollaries from the main theorems.

Experimental results

Research questions

• RQ1 How can multi-parameter summation identities for Stieltjes constants γₖ(a) be systematically derived from functional equations of the Hurwitz zeta function?
• RQ2 What are the exact relations linking sums of γₖ(a) over rational arguments to logarithmic and polygamma functions?
• RQ3 How do Mellin transform representations of zeta functions associated with rational generating functions yield new summation identities?
• RQ4 Can reciprocity-type relations for Bernoulli polynomials and polygamma functions be derived as corollaries of these identities?
• RQ5 What are the implications of these identities for the functions Aₖ(q) = k ∂/∂z ζ(z,q)|_{z=1−k} in integral evaluations?

Key findings

• Proposition 1 establishes a multi-parameter summation identity: ∑_{r=1}^q γₖ(pr/q − b) = q(−1)^k ln^{k+1}(q/p)/(k+1) + (q/p)∑_{ℓ=0}^{p−1}∑_{j=0}^k (−1)^j (k choose j) ln^j(q/p) γ_{k−j}[1 + (ℓ−b)q/p].
• Proposition 2 gives a relation involving γₘ and logarithmic terms: (p−1)γₘ + ln^{m+1}p/(m+1) − ∑_{k=0}^{m−1} (m choose k) ln^{m−k}p γₖ = (1−p)ln^{m+1}p^{N+1}/(m+1) + p^{−N} ∑_{k=0}^m (m choose k) ln^{m−k}p^{N+1} ∑_{(j,p)=1} γₖ(j/p^{N+1}).
• Corollary 1 provides a reciprocity identity for the digamma function: ln q + (1/q)∑_{r=0}^{q−1} ψ(pr/q − b) = ln p + (1/p)∑_{ℓ=0}^{p−1} ψ[(ℓ−b)q/p].
• Corollary 2 gives a symmetry for polygamma functions: (1/q)∑_{r=0}^{q−1} ψ^{(n−1)}(pr/q − b) = (1/p)(q/p)^{n−1} ∑_{ℓ=0}^{p−1} ψ^{(n−1)}[(ℓ−b)q/p].
• Corollary 3 establishes a summation identity for Bernoulli polynomials: ∑_{r=1}^q B_m(pr/q − b) = (q/p)^{1−m} ∑_{ℓ=0}^{p−1} B_m[1 + (ℓ−b)q/p].
• Proposition 5 yields a relation for Aₖ(q): (1−p^{k−1})Aₖ(1) − (−1)^k (ln p) Bₖ[N(1−p^{k−1})+1] = p^{(N+1)(k−1)} ∑_{(j,p)=1} Aₖ(j/p^{N+1}).