Generalized Gregory Coefficients Overview

• Generalized Gregory coefficients are extensions of the classical Gregory sequence that incorporate polynomial, multivariate, and operator-theoretic deformations.
• They arise in asymptotic expansions, series for constants like Euler’s, and enhanced quadrature formulas, linking combinatorial identities with numerical methods.
• Their diverse applications span analytic continuation, zeta-function asymptotics, and univalent function theory, offering versatile computational frameworks.

Generalized Gregory coefficients are extensions of the classical Gregory coefficients $G_n$, the rational numbers defined by

$$\frac{z}{\ln(1+z)}=\sum_{n=0}^{\infty} G_n z^n,\qquad |z|<1.$$

The classical sequence is also known as the Cauchy numbers of the first kind, Bernoulli numbers of the second kind, and reciprocal logarithmic numbers, with initial values

$$G_0=1,\quad G_1=\frac12,\quad G_2=-\frac1{12},\quad G_3=\frac1{24},\quad G_4=-\frac{19}{720},\quad G_5=\frac3{160},\quad G_6=-\frac{863}{60480}.$$

Current work suggests that the phrase “generalized Gregory coefficients” is used in several closely related senses: polynomial deformations of the classical sequence, multivariate coefficients attached to zeta-function asymptotics, and coefficient systems governing generalized summation, quadrature, and operator factorizations (Alabdulmohsin, 2021, Matsusaka et al., 2023, Matsusaka et al., 2 Apr 2026).

1. Classical sequence and basic identities

The classical Gregory coefficients are encoded by the generating function above and admit several equivalent representations. One explicit formula is

$$G_n=\frac{1}{n!}\sum_{j=0}^n \left[{n\atop j}\right]\frac{(-1)^{\,n-j}}{j+1},$$

where $\left[{n\atop j}\right]$ are the unsigned Stirling numbers of the first kind. They also satisfy

$$\sum_{j=0}^{n}\left\{{n\atop j}\right\} j!\,G_j=\frac{1}{n+1},$$

and the recurrence

$$G_n=(-1)^{n+1}\frac{1}{n+1}+\sum_{l=1}^{n-1}\frac{(-1)^{l-1}G_l}{n+1-l},\qquad G_1=\frac12.$$

Integral representations also occur, for example

$G_n=\frac{1}{n!}\int_0^1 x(x-1)\cdots(x-n+1)\,dx.$

These identities connect Gregory coefficients simultaneously to finite differences, Stirling-number combinatorics, and logarithmic generating functions (Moosmüller et al., 2018, Blagouchine et al., 2017).

The classical sequence already appears in several distinct analytic settings. It occurs in Gregory’s quadrature, in series for Euler’s constant, and in zeta-related expansions. One form of Mascheroni’s formula is

$\gamma=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}G_n}{n},$

while the Fontana–Mascheroni series is

$\gamma=\sum_{n=1}^{\infty}\frac{|G_n|}{n}.$

This breadth of occurrence is the immediate background for the later generalizations (Matsusaka et al., 2 Apr 2026, Blagouchine et al., 2017).

2. Polynomial deformations and one-parameter generalizations

One important extension is the family of Gregory polynomials $$\frac{z}{\ln(1+z)}=\sum_{n=0}^{\infty} G_n z^n,\qquad |z|<1.$$0, defined by

$$\frac{z}{\ln(1+z)}=\sum_{n=0}^{\infty} G_n z^n,\qquad |z|<1.$$1

They satisfy the specialization $$\frac{z}{\ln(1+z)}=\sum_{n=0}^{\infty} G_n z^n,\qquad |z|<1.$$2, the recursion

$$\frac{z}{\ln(1+z)}=\sum_{n=0}^{\infty} G_n z^n,\qquad |z|<1.$$3

and the explicit formula

$$\frac{z}{\ln(1+z)}=\sum_{n=0}^{\infty} G_n z^n,\qquad |z|<1.$$4

These polynomials are also known as Fontana–Bessel or second kind Bernoulli polynomials (Matsusaka et al., 2 Apr 2026).

A different one-parameter deformation arises from the polynomials $$\frac{z}{\ln(1+z)}=\sum_{n=0}^{\infty} G_n z^n,\qquad |z|<1.$$5 defined by

$$\frac{z}{\ln(1+z)}=\sum_{n=0}^{\infty} G_n z^n,\qquad |z|<1.$$6

This family interpolates between Bernoulli and Gregory data: $$\frac{z}{\ln(1+z)}=\sum_{n=0}^{\infty} G_n z^n,\qquad |z|<1.$$7 and includes the alternating-series specialization

$$\frac{z}{\ln(1+z)}=\sum_{n=0}^{\infty} G_n z^n,\qquad |z|<1.$$8

The associated recurrences are

$$\frac{z}{\ln(1+z)}=\sum_{n=0}^{\infty} G_n z^n,\qquad |z|<1.$$9

and

$$G_0=1,\quad G_1=\frac12,\quad G_2=-\frac1{12},\quad G_3=\frac1{24},\quad G_4=-\frac{19}{720},\quad G_5=\frac3{160},\quad G_6=-\frac{863}{60480}.$$0

with $$G_0=1,\quad G_1=\frac12,\quad G_2=-\frac1{12},\quad G_3=\frac1{24},\quad G_4=-\frac{19}{720},\quad G_5=\frac3{160},\quad G_6=-\frac{863}{60480}.$$1. The same paper proves parity and symmetry properties, including that $$G_0=1,\quad G_1=\frac12,\quad G_2=-\frac1{12},\quad G_3=\frac1{24},\quad G_4=-\frac{19}{720},\quad G_5=\frac3{160},\quad G_6=-\frac{863}{60480}.$$2 is even for $$G_0=1,\quad G_1=\frac12,\quad G_2=-\frac1{12},\quad G_3=\frac1{24},\quad G_4=-\frac{19}{720},\quad G_5=\frac3{160},\quad G_6=-\frac{863}{60480}.$$3, and records that nontrivial roots are real and conjecturally lie within $$G_0=1,\quad G_1=\frac12,\quad G_2=-\frac1{12},\quad G_3=\frac1{24},\quad G_4=-\frac{19}{720},\quad G_5=\frac3{160},\quad G_6=-\frac{863}{60480}.$$4 (Alabdulmohsin, 2021).

Taken together, these constructions show two complementary modes of generalization: $$G_0=1,\quad G_1=\frac12,\quad G_2=-\frac1{12},\quad G_3=\frac1{24},\quad G_4=-\frac{19}{720},\quad G_5=\frac3{160},\quad G_6=-\frac{863}{60480}.$$5 extends the sequence in a polynomial variable, while $$G_0=1,\quad G_1=\frac12,\quad G_2=-\frac1{12},\quad G_3=\frac1{24},\quad G_4=-\frac{19}{720},\quad G_5=\frac3{160},\quad G_6=-\frac{863}{60480}.$$6 embeds the Gregory coefficients into a deformation family that also contains Bernoulli numbers (Alabdulmohsin, 2021, Matsusaka et al., 2 Apr 2026).

3. Bivariate generalized Gregory coefficients from multiple zeta asymptotics

A distinct multivariate extension appears in the asymptotic analysis of the Euler–Zagier multiple zeta function at the origin. For the asymptotic coefficients $$G_0=1,\quad G_1=\frac12,\quad G_2=-\frac1{12},\quad G_3=\frac1{24},\quad G_4=-\frac{19}{720},\quad G_5=\frac3{160},\quad G_6=-\frac{863}{60480}.$$7 arising from the expansion of $$G_0=1,\quad G_1=\frac12,\quad G_2=-\frac1{12},\quad G_3=\frac1{24},\quad G_4=-\frac{19}{720},\quad G_5=\frac3{160},\quad G_6=-\frac{863}{60480}.$$8 at $$G_0=1,\quad G_1=\frac12,\quad G_2=-\frac1{12},\quad G_3=\frac1{24},\quad G_4=-\frac{19}{720},\quad G_5=\frac3{160},\quad G_6=-\frac{863}{60480}.$$9, the generalized Gregory coefficients $$G_n=\frac{1}{n!}\sum_{j=0}^n \left[{n\atop j}\right]\frac{(-1)^{\,n-j}}{j+1},$$0 are defined by the two-variable generating function

$$G_n=\frac{1}{n!}\sum_{j=0}^n \left[{n\atop j}\right]\frac{(-1)^{\,n-j}}{j+1},$$1

The principal identification is

$$G_n=\frac{1}{n!}\sum_{j=0}^n \left[{n\atop j}\right]\frac{(-1)^{\,n-j}}{j+1},$$2

This means that the generalized Gregory coefficients directly describe the leading-order asymptotic coefficients at the origin of the multiple zeta function (Matsusaka et al., 2023).

The basic structural properties are explicit. The coefficients satisfy the symmetry

$$G_n=\frac{1}{n!}\sum_{j=0}^n \left[{n\atop j}\right]\frac{(-1)^{\,n-j}}{j+1},$$3

and the specializations

$$G_n=\frac{1}{n!}\sum_{j=0}^n \left[{n\atop j}\right]\frac{(-1)^{\,n-j}}{j+1},$$4

The paper also states that $$G_n=\frac{1}{n!}\sum_{j=0}^n \left[{n\atop j}\right]\frac{(-1)^{\,n-j}}{j+1},$$5 satisfy multiple recursion relations and gives a Stirling polynomial integral representation,

$$G_n=\frac{1}{n!}\sum_{j=0}^n \left[{n\atop j}\right]\frac{(-1)^{\,n-j}}{j+1},$$6

In the same framework, the primitive coefficients $$G_n=\frac{1}{n!}\sum_{j=0}^n \left[{n\atop j}\right]\frac{(-1)^{\,n-j}}{j+1},$$7 reconstruct all coefficients in the asymptotic formula via decomposition (Matsusaka et al., 2023).

The Hurwitz extension replaces Bernoulli numbers by Bernoulli polynomials and introduces generalized Gregory polynomials $$G_n=\frac{1}{n!}\sum_{j=0}^n \left[{n\atop j}\right]\frac{(-1)^{\,n-j}}{j+1},$$8 through

$$G_n=\frac{1}{n!}\sum_{j=0}^n \left[{n\atop j}\right]\frac{(-1)^{\,n-j}}{j+1},$$9

where

$\left[{n\atop j}\right]$0

The corresponding asymptotic coefficients satisfy

$\left[{n\atop j}\right]$1

and the specialization $\left[{n\atop j}\right]$2 recovers the non-Hurwitz case (Matsusaka et al., 2023).

4. Summation formulas, quadrature, and generalized end corrections

Generalized Gregory coefficients also arise from unified summation and quadrature formulas. A general summation formula valid for any polynomial $\left[{n\atop j}\right]$3 and any $\left[{n\atop j}\right]$4 is

$\left[{n\atop j}\right]$5

where $\left[{n\atop j}\right]$6. The dual formula involves $\left[{n\atop j}\right]$7. The specializations $\left[{n\atop j}\right]$8, $\left[{n\atop j}\right]$9, and $$\sum_{j=0}^{n}\left\{{n\atop j}\right\} j!\,G_j=\frac{1}{n+1},$$0 in the dual formula recover, respectively, the Euler–Maclaurin formula, Euler’s acceleration for alternating series, and Gregory’s method (Alabdulmohsin, 2021).

A more explicit quadrature generalization introduces a parameter $$\sum_{j=0}^{n}\left\{{n\atop j}\right\} j!\,G_j=\frac{1}{n+1},$$1, interpreted as the distance from the endpoint of the range of integration to the first node, measured inward in step-lengths. The corresponding endpoint-correction coefficients are determined by

$$\sum_{j=0}^{n}\left\{{n\atop j}\right\} j!\,G_j=\frac{1}{n+1},$$2

followed by

$$\sum_{j=0}^{n}\left\{{n\atop j}\right\} j!\,G_j=\frac{1}{n+1},$$3

Setting $$\sum_{j=0}^{n}\left\{{n\atop j}\right\} j!\,G_j=\frac{1}{n+1},$$4 yields Gregory’s closed Newton–Cotes-like rules, setting $$\sum_{j=0}^{n}\left\{{n\atop j}\right\} j!\,G_j=\frac{1}{n+1},$$5 yields open Newton–Cotes-like rules, and setting $$\sum_{j=0}^{n}\left\{{n\atop j}\right\} j!\,G_j=\frac{1}{n+1},$$6 yields corrected composite midpoint rules. Negative $$\sum_{j=0}^{n}\left\{{n\atop j}\right\} j!\,G_j=\frac{1}{n+1},$$7 samples the integrand outside the range of integration and yields centered finite-difference end-corrections for the trapezoidal rule and the midpoint rule. The same framework also allows different values of $$\sum_{j=0}^{n}\left\{{n\atop j}\right\} j!\,G_j=\frac{1}{n+1},$$8 at the two ends, yielding Adams–Bashforth and Adams–Moulton weights (Putland, 17 Dec 2025).

Earlier work on Lubbock’s summation formulae identifies another closely related coefficient system. In that setting, the Lubbock coefficients are values of generalized Bernoulli polynomials, for example

$$\sum_{j=0}^{n}\left\{{n\atop j}\right\} j!\,G_j=\frac{1}{n+1},$$9

$$G_n=(-1)^{n+1}\frac{1}{n+1}+\sum_{l=1}^{n-1}\frac{(-1)^{l-1}G_l}{n+1-l},\qquad G_1=\frac12.$$0

and

$$G_n=(-1)^{n+1}\frac{1}{n+1}+\sum_{l=1}^{n-1}\frac{(-1)^{l-1}G_l}{n+1-l},\qquad G_1=\frac12.$$1

That paper states that the generalized Gregory coefficients appear as special cases of these expressions for $$G_n=(-1)^{n+1}\frac{1}{n+1}+\sum_{l=1}^{n-1}\frac{(-1)^{l-1}G_l}{n+1-l},\qquad G_1=\frac12.$$2, linking Lubbock coefficients, generalized Bernoulli polynomials, and generalized Gregory coefficients in a single operator framework (Dowker, 2013).

5. Operator-theoretic generalization in Hermite subdivision

In Hermite subdivision theory, the relevant generalization is not a new scalar sequence but a new family of operators built from the classical Gregory coefficients. The Gregory operator of order $$G_n=(-1)^{n+1}\frac{1}{n+1}+\sum_{l=1}^{n-1}\frac{(-1)^{l-1}G_l}{n+1-l},\qquad G_1=\frac12.$$3 is defined by

$$G_n=(-1)^{n+1}\frac{1}{n+1}+\sum_{l=1}^{n-1}\frac{(-1)^{l-1}G_l}{n+1-l},\qquad G_1=\frac12.$$4

and for $$G_n=(-1)^{n+1}\frac{1}{n+1}+\sum_{l=1}^{n-1}\frac{(-1)^{l-1}G_l}{n+1-l},\qquad G_1=\frac12.$$5 it coincides with the complete Taylor operator of dimension $$G_n=(-1)^{n+1}\frac{1}{n+1}+\sum_{l=1}^{n-1}\frac{(-1)^{l-1}G_l}{n+1-l},\qquad G_1=\frac12.$$6. The coefficients entering the off-diagonal part are precisely $$G_n=(-1)^{n+1}\frac{1}{n+1}+\sum_{l=1}^{n-1}\frac{(-1)^{l-1}G_l}{n+1-l},\qquad G_1=\frac12.$$7 (Moosmüller et al., 2018).

The factorization result is

$$G_n=(-1)^{n+1}\frac{1}{n+1}+\sum_{l=1}^{n-1}\frac{(-1)^{l-1}G_l}{n+1-l},\qquad G_1=\frac12.$$8

for Hermite subdivision operators satisfying the spectral condition of order $$G_n=(-1)^{n+1}\frac{1}{n+1}+\sum_{l=1}^{n-1}\frac{(-1)^{l-1}G_l}{n+1-l},\qquad G_1=\frac12.$$9. The paper states that spectral order $G_n=\frac{1}{n!}\int_0^1 x(x-1)\cdots(x-n+1)\,dx.$0 allows for $G_n=\frac{1}{n!}\int_0^1 x(x-1)\cdots(x-n+1)\,dx.$1 factorizations of the subdivision operator with respect to the Gregory operators, and that the $G_n=\frac{1}{n!}\int_0^1 x(x-1)\cdots(x-n+1)\,dx.$2-th factorization provides a “convergence from contractivity” method for proving $G_n=\frac{1}{n!}\int_0^1 x(x-1)\cdots(x-n+1)\,dx.$3-convergence. It further emphasizes that, whereas previously $G_n=\frac{1}{n!}\int_0^1 x(x-1)\cdots(x-n+1)\,dx.$4 factorization steps were needed to prove $G_n=\frac{1}{n!}\int_0^1 x(x-1)\cdots(x-n+1)\,dx.$5-convergence, the new method requires only one step, independently of $G_n=\frac{1}{n!}\int_0^1 x(x-1)\cdots(x-n+1)\,dx.$6 (Moosmüller et al., 2018).

A common misconception is that this literature introduces a distinct generalized scalar sequence. In this context, the paper explicitly states that the generalization concerns the systematic use of the classical Gregory coefficients in progressively higher-order operator factorizations, not a different “generalized Gregory coefficient” sequence (Moosmüller et al., 2018).

6. Zeta-function constants, finite analogues, and arithmetic variants

Gregory coefficients and Gregory polynomials also organize series for zeta-related constants. New series with rational terms are given for the Stieltjes constants, Euler’s constant, and $G_n=\frac{1}{n!}\int_0^1 x(x-1)\cdots(x-n+1)\,dx.$7. For example,

$G_n=\frac{1}{n!}\int_0^1 x(x-1)\cdots(x-n+1)\,dx.$8

and

$G_n=\frac{1}{n!}\int_0^1 x(x-1)\cdots(x-n+1)\,dx.$9

The same paper introduces generalized Euler constants

$\gamma=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}G_n}{n},$0

together with integral and nested-integral representations, and notes that almost all the constants considered admit simple representations via the Ramanujan summation (Blagouchine et al., 2017).

Gregory polynomials provide a further arithmetic extension in finite analogues of Dobiński’s formula and of Euler’s constant. The paper defines the finite analogue

$\gamma=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}G_n}{n},$1

as well as a Kluyver-type analogue

$\gamma=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}G_n}{n},$2

The paper states that these analogues differ from the Wilson-type finite Euler constant only by linear combinations of $\gamma=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}G_n}{n},$3 and special values of $\gamma=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}G_n}{n},$4 (Matsusaka et al., 2 Apr 2026).

This arithmetic strand reinforces a broader pattern: generalized Gregory coefficients and polynomials are effective not only in asymptotic or numerical contexts but also in finite and $\gamma=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}G_n}{n},$5-adic-style settings. A plausible implication is that their logarithmic generating structure makes them unusually well suited to both analytic continuation problems and finite-arithmetic analogues (Blagouchine et al., 2017, Matsusaka et al., 2 Apr 2026).

7. Geometric function theory and coefficient problems

The generating function

$\gamma=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}G_n}{n},$6

also defines new classes of starlike functions in the Ma–Minda framework: $\gamma=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}G_n}{n},$7 Equivalently, for every $\gamma=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}G_n}{n},$8, there exists an analytic function $\gamma=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}G_n}{n},$9 with $\gamma=\sum_{n=1}^{\infty}\frac{|G_n|}{n}.$0 and $\gamma=\sum_{n=1}^{\infty}\frac{|G_n|}{n}.$1 such that

$\gamma=\sum_{n=1}^{\infty}\frac{|G_n|}{n}.$2

This construction does not define a new generalized Gregory sequence, but it does use the Gregory generating function as the target of subordination (Kazımoğlu et al., 2023, Ahamed et al., 2024).

Sharp coefficient bounds were obtained for this class. One paper proves

$\gamma=\sum_{n=1}^{\infty}\frac{|G_n|}{n}.$3

together with

$\gamma=\sum_{n=1}^{\infty}\frac{|G_n|}{n}.$4

It also records the open problem that for $\gamma=\sum_{n=1}^{\infty}\frac{|G_n|}{n}.$5, the conjectured bound $\gamma=\sum_{n=1}^{\infty}\frac{|G_n|}{n}.$6 remains open beyond $\gamma=\sum_{n=1}^{\infty}\frac{|G_n|}{n}.$7 (Kazımoğlu et al., 2023).

A second paper establishes sharp inequalities for nonlinear logarithmic and Zalcman-type functionals in the same class. It proves

$\gamma=\sum_{n=1}^{\infty}\frac{|G_n|}{n}.$8

and the piecewise sharp Fekete–Szegő inequality

$\gamma=\sum_{n=1}^{\infty}\frac{|G_n|}{n}.$9

It also gives the sharp bounds

$$\frac{z}{\ln(1+z)}=\sum_{n=0}^{\infty} G_n z^n,\qquad |z|<1.$$00

These results show that the Gregory generating function can serve as a precise analytic datum in coefficient problems for univalent-function subclasses (Ahamed et al., 2024).