Evaluation of terminating and non-terminating sums containing the digamma function

Dmitrii Karp; Sergei Kalmykov; Vinay Shukla

Evaluation of terminating and non-terminating sums containing the digamma function

T0 review · 0 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper proves new identities converting digamma-weighted series into hypergeometric functions, including cases where digamma cancels completely.

desk verdict A careful, incremental special-functions paper: the new digamma and hypergeometric identities look right, and the small convergence gap is easy to patch. read the letter →

arxiv 2608.01099 v1 pith:CMWE4XWM submitted 2026-08-02 math.CA

Sergei Kalmykov , Dmitrii Karp , Vinay Shukla This is my paper · ORCID

classification math.CA MSC 33C2033C0533B15

keywords digammaserieshypergeometricsummationformulaidentitydegenerationprocessBernoullipolynomialsEulertransformationcontiguousrelations

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes transformation and summation formulas that convert infinite and finite series whose terms carry the digamma function ψ into combinations of classical hypergeometric series, which are easier to evaluate and to transform further. The centerpiece is Theorem 2.2, an identity valid for |z|<1 that expresses the sum of two digamma-weighted hypergeometric series as a single Gauss hypergeometric function multiplied by a combination of digamma values at the parameters. The formulas are produced by a degeneration process in which two upper parameters are allowed to coalesce, with the resulting singularities cancelling; several derived limits are remarkable in that every digamma term cancels, leaving purely hypergeometric product identities. If correct, these identities give new closed-form evaluations for sums of harmonic-type series and new tools for reducing hypergeometric expressions.

What carries the argument

The central mechanism is the degeneration of duality relations for generalized hypergeometric functions. Writing the regularized hypergeometric function rϕ_{r−1} in gamma-function form, the authors set α2 = α1 + p + ε and expand in Taylor series; the 1/sin(πε) singularities in the two parts cancel because the coefficients h1(0) and h2(0) are equal, leaving finite limits expressed through ψ. Theorem 2.2 is instead obtained by applying the differential operators ∂_{b1}+∂c and ∂_{b2}+∂c to Euler's transformation, while Theorems 4.2 and 4.4 are proved by elementary contiguous relations between hypergeometric series.

What would settle it

Evaluate both sides of (2.14) numerically for a concrete admissible parameter set, say b1=1/3, b2=1/5, c=2, at z=0.7, truncating the two series at k=100, and check that the computed difference matches the right-hand side to the expected precision; a mismatch would reveal a failure of the limiting process.

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Extended reading notes

Core claim

At its core, the paper derives and proves a family of identities of the form (1−z)^{c−b1−b2} times a digamma-weighted series plus another digamma-weighted series equals [ψ(c−b1)+ψ(c−b2)−ψ(b1)−ψ(b2)] times 2F1(b1,b2;c;z), where both series are ordinary hypergeometric series with an additional ψ factor in each term. Equating coefficients of z^n in this identity yields a finite summation formula (Theorem 2.3) for terminating digamma sums. A second strand of the paper evaluates terminating digamma sums in terms of hypergeometric functions and Bernoulli polynomials (Theorem 3.1), and a third shows that in certain coalescence limits all digamma contributions vanish, producing identities that are purely products of hypergeometric series (Theorems 4.1–4.4), which the authors believe to be new.

Load-bearing premise

The load-bearing step is the interchange of the limit ε→0 with the infinite hypergeometric sums (and the companion analytic continuation to the stated parameter sets); the paper assumes this regularity without giving a uniform-convergence or continuation argument.

Editorial extensions

If this is right

The coefficient-wise identity (2.19) gives new finite summations for digamma-weighted hypergeometric terms, specializing to explicit formulas when b1 = b2 = 1/3 as shown in Example 2.2.
The digamma-free identities (4.2), (4.3), (4.8), and (4.9) can be used as reduction rules for products of hypergeometric series in other derivations.
The digamma sums with Bernoulli-polynomial evaluations in Section 3 provide closed forms for finite harmonic-type sums that previously lacked summation formulas.
Every identity yields a numerical evaluation route: truncating the hypergeometric side is typically more stable than summing the original digamma series.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The method of coalescing parameters could be applied to other duality relations, including the basic hypergeometric analogues mentioned in the paper's reference [23], producing digamma-type series in q-calculus.
One could test whether the hypergeometric product identities of Section 4 have combinatorial interpretations as coefficient identities for classical orthogonal polynomials; the similarity to the Meixner–Sorokin identity noted for Corollary 4.1 suggests the perfectness proofs may be replicable.
The formal power-series proofs suggest that identities (4.3) and (4.9) may hold beyond the stated convergence regions by analytic continuation, which would extend their range of applicability if confirmed.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a solid, incremental paper for the special-functions niche. The main new identities are credible and, as far as I can tell, correct. It won't set the field on fire, but it's an honest piece of work.

What is actually new: Theorem 2.2 (the parameter-differentiated Euler transformation) is a clean derivation that yields (2.14), and the purely hypergeometric limits in Section 4 (Theorems 4.1–4.4) are the most interesting part. The authors themselves say these are 'believed to be new,' and they hedge in the abstract with 'presumably new.' That is fair, but it's worth noting the novelty is asserted, not demonstrated by a literature search. What supports them is that Theorems 4.2 and 4.4 are proved independently via contiguous relations, not merely by the same limiting argument, which gives me confidence.

What it does well: the proofs are unusually careful about singular limits. Remarks 2.1, 3.1, and 4.1 deal with poles, negative-integer parameters, and the ψ/Γ convention, and the regularized hypergeometric ϕ notation is used consistently. The treatment of the 0·∞ products in Theorem 3.1 is explicit. This level of care is exactly what this kind of identity-churning paper needs.

Soft spots, in proportion: the main one is that the degeneration proofs expand h1(ε) and h2(ε) in Taylor series and pass the limit ε→0 through infinite sums without spelling out uniform convergence. The stress-test note convinced me this is routine under the stated exclusions: the defining series are absolutely convergent for |z|<1, uniformly in a neighborhood of ε=0, and the index shifts are valid by the regularized hypergeometric convention. Still, a referee should ask for a sentence saying so, especially in the Appendix proof of Theorem 4.1. The other soft spot is minor: the paper leans heavily on the authors' own prior duality relations, but those are published and established, and the new identities do follow from them rather than restating them.

Bottom line: the central results hold up. This is a sound contribution for specialists in hypergeometric functions and for physicists computing Feynman-parameter integrals. I wouldn't cite it myself — not my area — but the right reader will. Give it a serious referee; I'd recommend accept after minor revisions to make the convergence interchanges explicit.

Referee Report

0 major / 6 minor

Summary. The paper derives new transformation and summation identities for series containing the digamma function. The main mechanism is a degeneration limit applied to known duality and contiguous relations for generalized hypergeometric functions, with singular contributions regularized so that finite limits exist. Principal results include Theorem 2.1, a sum-product identity involving regularized hypergeometric and digamma series; Theorem 2.2, a compact identity obtained by parameter differentiation of Euler's transformation; Theorem 2.3, a terminating summation formula; Theorem 3.1, a finite-sum formula involving Bernoulli polynomials; and Theorems 4.1-4.4, purely hypergeometric product identities in which all digamma contributions cancel. The proofs are detailed, and the regularized definitions for singular parameter values are addressed in Remarks 2.1, 3.1, and 4.1.

Significance. If correct, the identities provide a useful and systematic set of tools for evaluating digamma-weighted hypergeometric sums, converting products of digamma series with hypergeometric functions into single hypergeometric functions or finite expressions. A particular strength is that the main formulas are supported by independent proof methods: Theorem 2.2 is proved by differentiating Euler's transformation, and Theorems 4.2 and 4.4 are proved by elementary contiguous relations. The paper also isolates several purely hypergeometric identities (Theorems 4.1, 4.3, and Corollary 4.1) that are likely of independent interest. The treatment of singular parameter cases is careful, and the stated exclusions appear to match the places where the regularization arguments are needed.

minor comments (6)

[Abstract/Title] The title page contains the typo "EV ALUATION" instead of "EVALUATION"; this should be corrected.
[Section 4, before Theorem 4.1] The sentence introducing Theorem 4.1 contains the misspelling "tated" for "stated"; please fix this typo.
[Corollary 2.2 proof] The proof begins by substituting z=1 into (2.6), but Theorem 2.1 is stated for 0<|z|<1. Since the corollary uses Gauss summation to evaluate the resulting hypergeometric functions, a short Abel-limit or analytic-continuation justification for z tending to 1 should be supplied, along with the convergence conditions it requires.
[Eqs. (2.9) and (4.14)] The degeneration proofs interchange the limit epsilon->0 with infinite summation. For fixed |z|<1 and under the stated exclusions, the defining series converge absolutely and uniformly in a neighborhood of epsilon=0, so the interchange is routine; nevertheless, the paper never states this justification. Please add a sentence (or a brief lemma) making the uniform-convergence argument explicit for the proofs of Theorems 2.1, 3.1, 4.1, and 4.3.
[Appendix, proof of Theorem 4.1] In the proof after (4.14), the relation h_2(0)=h_1(0) is asserted to follow "by shifting the index of summation similarly to the proof of Theorem 2.1". Since this equality is load-bearing for the cancellation of the 1/Gamma(epsilon) singularity, please either display the few lines of the index shift or give an exact reference to the corresponding part of the proof of Theorem 2.1.
[Theorem 2.2 statement] The statement says "all expressions below are non-singular", which is somewhat vague. It would be helpful to state explicitly that c is not a non-positive integer and that the digamma arguments avoid poles, or else to refer to the regularization convention of Section 2.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the new identities are genuine consequences of stated prior theorems and independent parameter differentiation.

full rationale

I traced the derivation chain through the paper. Section 2 starts from the published duality relation [22, eq. (3)] and computes the degenerate limit α2 → α1 + p; the key step (2.9) uses h1(0) = h2(0), which is proved in the text by explicit index shifts, and the resulting identity (2.6) is a new limit consequence rather than a restatement of the input. Theorems 2.2 and 2.3 are proved independently by applying ∂b1 + ∂c and ∂b2 + ∂c to Euler's transformation (2.15); no fitted parameter or target identity is fed into the derivation. Section 3 processes [11, Theorem 6.2] with the same degeneration technique, and Corollary 3.4 isolates the purely hypergeometric part by explicit cancellation of prefactors and digamma terms. Section 4 takes b → 1 limits of [11, eq. (6.7)] and [11, Lemma 6.5], which themselves go back to the external source [19]; Theorems 4.2 and 4.4 are then proved by fresh coefficient comparisons of contiguous relations, not by invoking the conclusions. The cited inputs [22] and [11] are published, parameter-free theorems whose assumptions do not include the new formulas, so the self-citations are legitimate premises rather than circular load-bearing. The only mild gap is that the ε → 0 interchange with the infinite hypergeometric sums is not written out in full, but absolute convergence for fixed |z| < 1 makes that interchange routine; this is a rigor gap, not circularity. No equation in the paper reduces to its own input by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims rest on standard hypergeometric theorems plus a regularization convention for singular terms. There are no free parameters and no invented entities. The added conventions are internal to the paper and are accompanied by the continuity and handling arguments.

assumptions (5)

standard math Euler's transformation: F(a,b;c;z) = (1-z)^(c-a-b) F(c-a,c-b;c;z).
Used in the proof of Theorem 2.2 (eq. (2.15)) as the starting identity for parameter differentiation.
standard math Gauss summation theorem for 2F1 at unit argument.
Applied in the proof of Corollary 2.2 to sum the two 2phi1(1) series.
standard math Reflection formulas: Gamma(z)Gamma(1-z)=pi/sin(pi z) and psi(1-x)-psi(x)=pi cot(pi x).
Used throughout the degeneration calculations, for example in eq. (2.9) and in the proof of Corollary 2.3.
ad hoc to paper The regularized convention psi(-n)/Gamma(-n)=(-1)^(n+1)n! and the finite difference definition psi(x;j) from (3.3)-(3.4).
The paper introduces these conventions to give meaning to singular terms in the limiting processes; they follow from asymptotics but are not external theorems.
domain assumption Analytic continuation of identities from generic parameters to degenerate cases where singularities cancel.
The paper relies on the standard principle that identities valid for generic parameters extend by continuity; stated before Theorem 2.1: 'the singularities appearing on the left-hand side must cancel out.'

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Pith. "Pith review of Evaluation of terminating and non-terminating sums containing the digamma function." pith.science (2026). https://pith.science/paper/CMWE4XWM

@misc{pith2026260801099,
  author       = {Pith},
  title        = {Pith review of: Evaluation of terminating and non-terminating sums containing the digamma function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CMWE4XWM}},
  note         = {Machine review of arXiv:2608.01099}
}

read the original abstract

We derive transformation and summation formulas for terminating and nonterminating series involving the digamma function. Our principal results are obtained by a limiting process starting with duality relations for the generalized hypergeometric functions and their consequences. Selected formulas are further extended by parameter differentiation of Euler's transformation and by using contiguous relations. Most of our identities express products of hypergeometric and digamma series in terms of hypergeometric functions, and some evaluations of terminating digamma sums involve Bernoulli polynomials. In several cases the digamma contributions cancel, producing identities involving only products of hypergeometric functions.

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

General duality relations for hypergeometric and basic hypergeometric series

Karp, D. and Zhang, Y. , TITLE =. Preprint , VOLUME =. 2026 , NUMBER =. doi:10.48550/arXiv.2606.18904 , NOTE =

work page Pith review arXiv doi:10.48550/arxiv.2606.18904 2026

[2] [2]

Kalmykov, S. I. and Karp, D. and Kuznetsov, A. , TITLE =. Ramanujan J. , FJOURNAL =. 2023 , NUMBER =. doi:10.1007/s11139-022-00598-w , NOTE =

work page doi:10.1007/s11139-022-00598-w 2023

[3] [3]

[2025] 2025 , MRCLASS =

On digamma series convertible into hypergeometric series , BOOKTITLE =. [2025] 2025 , MRCLASS =. doi:10.1090/conm/818/16366 , NOTE =

work page doi:10.1090/conm/818/16366 2025

[4] [4]

Karp, D. B. and Prilepkina, E. G. , TITLE =. Lobachevskii J. Math. , FJOURNAL =. 2022 , NUMBER =

work page 2022

[5] [5]

Shpot, M. A. and Srivastava, H. M. , TITLE =. Appl. Math. Comput. , FJOURNAL =. 2015 , PAGES =. doi:10.1016/j.amc.2015.03.031 , NOTE =

work page doi:10.1016/j.amc.2015.03.031 2015

[6] [6]

Derivatives of the

Greynat, David and Sesma, Javier and Vulvert, Gr\'. Derivatives of the. J. Math. Phys. , FJOURNAL =. 2014 , NUMBER =. doi:10.1063/1.4870619 , NOTE =

doi:10.1063/1.4870619 2014

[7] [7]

Nuclear Phys

Bera, Souvik , TITLE =. Nuclear Phys. B , FJOURNAL =. 2023 , PAGES =. doi:10.1016/j.nuclphysb.2023.116145 , NOTE =

arXiv 2023

[8] [8]

Computer Physics Communications , volume=

A new approach to the epsilon expansion of generalized hypergeometric functions , author=. Computer Physics Communications , volume=. 2014 , publisher=

work page 2014