Poly-Numbers: A Unified Mathematical Overview

• Poly-Numbers are a family concept encompassing special sequences like poly-Bernoulli and polytope numbers, defined via polylogarithmic and geometric frameworks.
• They feature explicit Stirling-number formulas, duality relations, and analytic forms such as shifted log-sine integrals and zeta interpolation to elucidate their structure.
• Extensions include multi-indexed, q-deformed, and Schur-type variants, broadening their applications in computational, analytic, and geometric combinatorics.

“Poly-Numbers” is used in contemporary mathematics for several distinct but partially overlapping classes of objects. In one major sense, it refers to poly-Bernoulli, multi-poly-Bernoulli, poly-Cauchy, and allied special-number sequences defined through polylogarithmic or multiple-polylogarithmic generating functions. In another, it refers to polytope numbers, which are integer sequences determined by the facial structure of polytopes. In a third, more computational sense, it appears in Wildberger’s polyseries framework, where “poly-numbers” are represented by truncatable coefficient sequences. The most developed strand is the poly-Bernoulli-centered one, whose recurring features are exponential generating functions, Stirling-transform formulas, duality for negative indices, zeta-type interpolation, and a large combinatorial ecology (Bényi et al., 2015, Kim et al., 2012, Brahimi, 6 Jul 2025).

1. Core polylogarithmic families

The canonical starting point is Kaneko’s poly-Bernoulli numbers $B_n^{(k)}$, defined for $k\in\mathbb Z$ by the exponential generating function

$$\frac{\operatorname{Li}_k(1-e^{-t})}{1-e^{-t}} =\sum_{n=0}^{\infty} B_n^{(k)}\frac{t^n}{n!},$$

where

$$\operatorname{Li}_k(z)=\sum_{m=1}^{\infty}\frac{z^m}{m^k}.$$

In the special case $k=1$, one recovers the classical Bernoulli numbers, $B_n^{(1)}=B_n$ (Kikuchi et al., 16 Mar 2026). This places poly-Bernoulli numbers among the standard polylogarithmic extensions of classical special-number sequences.

The associated polynomial families are obtained by adjoining an exponential factor. One standard form is

$$\sum_{n\ge 0} B_n^{(k)}(z)\frac{t^n}{n!} = \frac{\operatorname{Li}_k(1-e^{-t})}{1-e^{-t}}e^{zt},$$

while another equivalent convention appears as

$$e^{-xt}\frac{\operatorname{Li}_\ell(1-e^{-t})}{1-e^{-t}} = \sum_{m=0}^\infty B_m^{(\ell)}(x)\frac{t^m}{m!}.$$

These polynomial versions are important because they support translation formulas, Appell-type behavior, and parameter deformations (Wakhare et al., 2019, Matsusaka, 2020).

A related bifurcation appears in the type $B$ and type $C$ poly-Bernoulli numbers. In the notation used for Schur-type generalizations, type $k\in\mathbb Z$0 and type $k\in\mathbb Z$1 are generated respectively by

$k\in\mathbb Z$2

This distinction is inherited by several later constructions, including Schur-type and zeta-interpolated variants (Nakamura et al., 2018).

The negative-index sector is singled out throughout the literature. For $k\in\mathbb Z$3, the values $k\in\mathbb Z$4 are emphasized as integer-valued, and negative-index poly-Bernoulli numbers are repeatedly treated as intrinsically combinatorial rather than merely formal coefficients (Matsusaka, 2020, Bényi et al., 2015).

2. Explicit formulas, duality, and analytic realizations

A central structural fact is that poly-Bernoulli numbers admit explicit Stirling-number formulas. For nonnegative index $k\in\mathbb Z$5,

$k\in\mathbb Z$6

while for negative index,

$k\in\mathbb Z$7

These formulas place the theory inside the Stirling-transform apparatus and immediately explain integrality and symmetry phenomena in the negative-index regime (Jolany et al., 2012).

The most classical symmetry is Kaneko’s duality

$k\in\mathbb Z$8

or equivalently $k\in\mathbb Z$9 in the notation used by several later papers. This identity persists in many generalizations and is one of the defining structural traits of the subject (Kikuchi et al., 16 Mar 2026, Komaki, 2015).

Analytically, poly-Bernoulli numbers arise as special values of zeta-type objects. A symbolic treatment introduces the poly-Bernoulli umbra

$$\frac{\operatorname{Li}_k(1-e^{-t})}{1-e^{-t}} =\sum_{n=0}^{\infty} B_n^{(k)}\frac{t^n}{n!},$$0

with

$$\frac{\operatorname{Li}_k(1-e^{-t})}{1-e^{-t}} =\sum_{n=0}^{\infty} B_n^{(k)}\frac{t^n}{n!},$$1

This representation yields new iterated integral formulas, Bernoulli-Barnes transforms, recurrences, and a symbolic realization of the Arakawa–Kaneko zeta values as negative moments of the companion umbra $$\frac{\operatorname{Li}_k(1-e^{-t})}{1-e^{-t}} =\sum_{n=0}^{\infty} B_n^{(k)}\frac{t^n}{n!},$$2 (Wakhare et al., 2019).

A newer analytic development identifies a shifted log-sine integral as a more intrinsic zeta-type source for poly-Bernoulli numbers. For $$\frac{\operatorname{Li}_k(1-e^{-t})}{1-e^{-t}} =\sum_{n=0}^{\infty} B_n^{(k)}\frac{t^n}{n!},$$3 and $$\frac{\operatorname{Li}_k(1-e^{-t})}{1-e^{-t}} =\sum_{n=0}^{\infty} B_n^{(k)}\frac{t^n}{n!},$$4,

$$\frac{\operatorname{Li}_k(1-e^{-t})}{1-e^{-t}} =\sum_{n=0}^{\infty} B_n^{(k)}\frac{t^n}{n!},$$5

After analytic continuation, its negative-integer values are governed by Lehmer-type polynomials, and at $$\frac{\operatorname{Li}_k(1-e^{-t})}{1-e^{-t}} =\sum_{n=0}^{\infty} B_n^{(k)}\frac{t^n}{n!},$$6 one obtains the exact identity

$$\frac{\operatorname{Li}_k(1-e^{-t})}{1-e^{-t}} =\sum_{n=0}^{\infty} B_n^{(k)}\frac{t^n}{n!},$$7

Thus anti-diagonal sums of negative-index poly-Bernoulli numbers emerge as special values of a log-sine integral rather than being built into the definition (Matsusaka, 26 Mar 2026).

3. Multi-indexed, Schur-type, and function-field extensions

The one-index theory extends in several directions. A first step is the multi-poly-Bernoulli family

$$\frac{\operatorname{Li}_k(1-e^{-t})}{1-e^{-t}} =\sum_{n=0}^{\infty} B_n^{(k)}\frac{t^n}{n!},$$8

where

$$\frac{\operatorname{Li}_k(1-e^{-t})}{1-e^{-t}} =\sum_{n=0}^{\infty} B_n^{(k)}\frac{t^n}{n!},$$9

Negative-index multi-poly-Bernoulli numbers admit power-sum expansions, special dualities, and explicit relations to ordinary negative-index poly-Bernoulli numbers (Komaki, 2015).

A deeper generalization is the Kaneko–Tsumura multi-indexed theory, defined through the $$\operatorname{Li}_k(z)=\sum_{m=1}^{\infty}\frac{z^m}{m^k}.$$0-type multiple polylogarithm

$$\operatorname{Li}_k(z)=\sum_{m=1}^{\infty}\frac{z^m}{m^k}.$$1

and the generating function

$$\operatorname{Li}_k(z)=\sum_{m=1}^{\infty}\frac{z^m}{m^k}.$$2

For arbitrary depth $$\operatorname{Li}_k(z)=\sum_{m=1}^{\infty}\frac{z^m}{m^k}.$$3, these numbers now have an explicit formula in Stirling numbers of the second kind, and they satisfy the duality

$$\operatorname{Li}_k(z)=\sum_{m=1}^{\infty}\frac{z^m}{m^k}.$$4

The same circle of ideas also supports restricted double-index formulas, $$\operatorname{Li}_k(z)=\sum_{m=1}^{\infty}\frac{z^m}{m^k}.$$5-adic periodicity for negative upper indices, and star-version analogues (Kikuchi et al., 16 Mar 2026, Baba et al., 2022).

Another extension replaces integer index data by partition data. Schur-type poly-Bernoulli numbers are defined from a Schur-type polylogarithm $$\operatorname{Li}_k(z)=\sum_{m=1}^{\infty}\frac{z^m}{m^k}.$$6 built from semi-standard Young tableaux of shape $$\operatorname{Li}_k(z)=\sum_{m=1}^{\infty}\frac{z^m}{m^k}.$$7. For $$\operatorname{Li}_k(z)=\sum_{m=1}^{\infty}\frac{z^m}{m^k}.$$8, the type $$\operatorname{Li}_k(z)=\sum_{m=1}^{\infty}\frac{z^m}{m^k}.$$9 and type $k=1$0 Schur numbers are generated by

$k=1$1

and

$k=1$2

These admit Arakawa–Kaneko-type and Kaneko–Tsumura-type interpolation theorems, hook-shape recurrences, and Stirling-number descriptions (Nakamura et al., 2018).

In positive characteristic, the function-field analogue is given by the multi-poly-Bernoulli-Carlitz numbers

$k=1$3

where $k=1$4 is the Carlitz exponential, $k=1$5 the Carlitz factorial, and the coefficients $k=1$6 come from Anderson–Thakur polynomials. These numbers satisfy explicit formulas in Stirling-Carlitz numbers and recover the Bernoulli-Carlitz numbers in depth one (Harada, 2018).

4. Parameter deformations, $k=1$7-analogues, and level-two theories

A broad deformation theory surrounds the core poly-Bernoulli family. One parameterized hierarchy passes from $k=1$8 to $k=1$9, then to $B_n^{(1)}=B_n$0, and finally to the fully generalized family $B_n^{(1)}=B_n$1, defined by

$B_n^{(1)}=B_n$2

Its key reduction formula is

$B_n^{(1)}=B_n$3

showing that the fully parameterized family is an affine-logarithmic rescaling of the classical one (Jolany et al., 2012).

A $B_n^{(1)}=B_n$4-deformed branch introduces $B_n^{(1)}=B_n$5-poly-Bernoulli and $B_n^{(1)}=B_n$6-poly-Cauchy polynomials with parameter $B_n^{(1)}=B_n$7. The $B_n^{(1)}=B_n$8-poly-Bernoulli family is defined by

$B_n^{(1)}=B_n$9

and the two $$\sum_{n\ge 0} B_n^{(k)}(z)\frac{t^n}{n!} = \frac{\operatorname{Li}_k(1-e^{-t})}{1-e^{-t}}e^{zt},$$0-poly-Cauchy families are defined by Jackson $$\sum_{n\ge 0} B_n^{(k)}(z)\frac{t^n}{n!} = \frac{\operatorname{Li}_k(1-e^{-t})}{1-e^{-t}}e^{zt},$$1-integrals. These objects admit weighted-Stirling expansions, inversion formulas, and direct transform relations linking poly-Bernoulli and poly-Cauchy theories (Komatsu, 2015).

A distinct “level two” theory replaces the ordinary polylogarithm by its odd-part analogue

$$\sum_{n\ge 0} B_n^{(k)}(z)\frac{t^n}{n!} = \frac{\operatorname{Li}_k(1-e^{-t})}{1-e^{-t}}e^{zt},$$2

The level-two poly-Bernoulli numbers are defined by

$$\sum_{n\ge 0} B_n^{(k)}(z)\frac{t^n}{n!} = \frac{\operatorname{Li}_k(1-e^{-t})}{1-e^{-t}}e^{zt},$$3

and satisfy the explicit level-two Stirling formula

$$\sum_{n\ge 0} B_n^{(k)}(z)\frac{t^n}{n!} = \frac{\operatorname{Li}_k(1-e^{-t})}{1-e^{-t}}e^{zt},$$4

Here $$\sum_{n\ge 0} B_n^{(k)}(z)\frac{t^n}{n!} = \frac{\operatorname{Li}_k(1-e^{-t})}{1-e^{-t}}e^{zt},$$5 are the Stirling numbers of the second kind with level $$\sum_{n\ge 0} B_n^{(k)}(z)\frac{t^n}{n!} = \frac{\operatorname{Li}_k(1-e^{-t})}{1-e^{-t}}e^{zt},$$6, identified in the paper with the central factorial numbers $$\sum_{n\ge 0} B_n^{(k)}(z)\frac{t^n}{n!} = \frac{\operatorname{Li}_k(1-e^{-t})}{1-e^{-t}}e^{zt},$$7 (Komatsu, 2021).

The same level-two setting also produces polycosecant and polycotangent numbers. They are defined by

$$\sum_{n\ge 0} B_n^{(k)}(z)\frac{t^n}{n!} = \frac{\operatorname{Li}_k(1-e^{-t})}{1-e^{-t}}e^{zt},$$8

with $$\sum_{n\ge 0} B_n^{(k)}(z)\frac{t^n}{n!} = \frac{\operatorname{Li}_k(1-e^{-t})}{1-e^{-t}}e^{zt},$$9. They satisfy dualities analogous to poly-Bernoulli duality,

$$e^{-xt}\frac{\operatorname{Li}_\ell(1-e^{-t})}{1-e^{-t}} = \sum_{m=0}^\infty B_m^{(\ell)}(x)\frac{t^m}{m!}.$$0

but the interpolation theory is asymmetrical: the cited paper explicitly notes that a zeta function interpolating polycotangent numbers at non-positive integers has not yet been constructed (Nishibiro, 2022).

5. Combinatorial models and structural behavior

The negative-index poly-Bernoulli numbers have an unusually rich combinatorial life. They count lonesum $$e^{-xt}\frac{\operatorname{Li}_\ell(1-e^{-t})}{1-e^{-t}} = \sum_{m=0}^\infty B_m^{(\ell)}(x)\frac{t^m}{m!}.$$1-$$e^{-xt}\frac{\operatorname{Li}_\ell(1-e^{-t})}{1-e^{-t}} = \sum_{m=0}^\infty B_m^{(\ell)}(x)\frac{t^m}{m!}.$$2 matrices of size $$e^{-xt}\frac{\operatorname{Li}_\ell(1-e^{-t})}{1-e^{-t}} = \sum_{m=0}^\infty B_m^{(\ell)}(x)\frac{t^m}{m!}.$$3, Callan permutations, max-ascending permutations, Vesztergombi permutations, and acyclic orientations of the complete bipartite graph $$e^{-xt}\frac{\operatorname{Li}_\ell(1-e^{-t})}{1-e^{-t}} = \sum_{m=0}^\infty B_m^{(\ell)}(x)\frac{t^m}{m!}.$$4. The same paper adds a new interpretation in terms of $$e^{-xt}\frac{\operatorname{Li}_\ell(1-e^{-t})}{1-e^{-t}} = \sum_{m=0}^\infty B_m^{(\ell)}(x)\frac{t^m}{m!}.$$5-free $$e^{-xt}\frac{\operatorname{Li}_\ell(1-e^{-t})}{1-e^{-t}} = \sum_{m=0}^\infty B_m^{(\ell)}(x)\frac{t^m}{m!}.$$6-$$e^{-xt}\frac{\operatorname{Li}_\ell(1-e^{-t})}{1-e^{-t}} = \sum_{m=0}^\infty B_m^{(\ell)}(x)\frac{t^m}{m!}.$$7 matrices, where the forbidden $$e^{-xt}\frac{\operatorname{Li}_\ell(1-e^{-t})}{1-e^{-t}} = \sum_{m=0}^\infty B_m^{(\ell)}(x)\frac{t^m}{m!}.$$8-configuration is a $$e^{-xt}\frac{\operatorname{Li}_\ell(1-e^{-t})}{1-e^{-t}} = \sum_{m=0}^\infty B_m^{(\ell)}(x)\frac{t^m}{m!}.$$9 submatrix of the form

$B$0

This model yields a direct combinatorial proof of Kaneko’s negative-index recurrence

$B$1

The same source also stresses the symmetry

$B$2

as visible in all major combinatorial realizations (Bényi et al., 2015).

A more refined construction is the symmetrized poly-Bernoulli number. In that setting, a Stirling-weighted symmetrization restores the $B$3 symmetry uniformly in an auxiliary parameter $B$4, and the alternating diagonal sum

$B$5

is identified with

$B$6

where $B$7 is the Dumont–Foata polynomial. This places symmetrized poly-Bernoulli numbers inside the combinatorics of pistols, Genocchi numbers, and Gandhi polynomials (Matsusaka, 2020).

The broader “poly-number” ecology also includes structural positivity phenomena. For poly-Cauchy numbers, the sign-normalized sequences

$B$8

are log-convex, and the same remains true for multiparameter-poly-Cauchy numbers under the hypotheses

$B$9

The paper additionally proves unimodality for certain eventually constant parameter patterns (Komatsu et al., 2016).

Negative-index multi-poly-Bernoulli numbers also acquire direct combinatorial interpretations. In particular,

$C$0

counts restricted barred preferential arrangements in which one of two fixed restricted sections is empty. The same paper proves that the last digits of restricted barred preferential-arrangement counts, negative-index multi-poly-Bernoulli numbers, and related numbers $C$1 have a four-cycle modulo $C$2 (Nkonkobe et al., 2015).

6. Geometric and computational senses of “poly-number”

Outside the polylogarithmic line, “poly-number” language also appears in geometric combinatorics. Polytope numbers are sequences $C$3 and interior sequences $C$4 attached to a polytope $C$5, defined recursively from facial data and a distinguished vertex. For simplices,

$C$6

A central theorem states that every polytope admits a pointed triangulation, and that its polytope numbers can be decomposed in several ways into sums of simplex numbers, including

$C$7

and

$C$8

This makes polytope numbers a higher-dimensional figurate-number theory controlled by shellings, face lattices, and simplex decompositions (Kim et al., 2012).

A particularly explicit subfamily is given by rectified simplex polytope numbers. If $C$9 denotes the number sequence of the $k\in\mathbb Z$00-rectified $k\in\mathbb Z$01-simplex, then

$k\in\mathbb Z$02

with a parallel formula for the interior sequence. The alternating signs are not formal artifacts: they directly encode the inclusion–exclusion geometry of rectification (Jackson et al., 2015).

A third usage appears in Wildberger’s polyseries framework. There a polyseries is written

$k\in\mathbb Z$03

with truncation

$k\in\mathbb Z$04

The paper emphasizes data-structure operations such as length, index lookup, value extraction, truncation, and sorting, rather than a full general algebra of series operations. Its principal application is a Catalan-number expansion for the quadratic congruence

$k\in\mathbb Z$05

leading to the truncated polyseries solution

$k\in\mathbb Z$06

In this computational usage, a “poly-number” is effectively a coefficient-stream object manipulated through finite truncations (Brahimi, 6 Jul 2025).

Taken together, these strands show that “Poly-Numbers” is best understood as a family resemblance concept rather than a single universally fixed definition. In the strongest and most developed sense, it designates polylogarithm-generated special sequences—especially poly-Bernoulli and poly-Cauchy families—together with their multi-indexed, $k\in\mathbb Z$07-deformed, level-two, Schur-type, and function-field analogues. In parallel, the same term extends to geometric figurate theories and to coefficient-sequence formalisms, indicating that the prefix “poly-” marks not one object but a recurrent pattern of enrichment by higher structure.