Hsu–Shiue Family & Generalized Stirling Numbers

• The Hsu–Shiue family is a three-parameter system of generalized Stirling numbers defined as connection coefficients between factorial polynomial bases and includes classical and weighted variants.
• Finite-difference techniques and explicit alternating sum formulas yield precise coefficient extraction, underpinning recurrence relations and Riordan array structures.
• This family bridges combinatorial models with quantum operator ordering and analytic extensions, linking generating functions, elliptic functions, and various combinatorial identities.

Searching arXiv for recent and foundational papers on the Hsu–Shiue family and related generalized Stirling-number frameworks. The expression Hsu–Shiue family most commonly denotes the three-parameter family of generalized Stirling numbers introduced by Hsu and Shiue, together with the associated polynomial, functional, and Riordan-array structures built from them. In this usage, the family consists of connection coefficients between generalized factorial bases and includes classical Stirling numbers, Lah-type arrays, and multiple noncentral or weighted variants as specializations. A separate special-function usage applies the Hsu–Shiue name to an elliptic-function family obtained from the inverse of a hypergeometric integral involving $${}_2F_1\!\left(\frac13,\frac23;\frac12;\bullet\right)$$; that construction was developed by Li-Chien Shen and later recast with streamlined notation and new proofs. Across these settings, the common theme is a parametric coefficient system that mediates between natural bases or uniformizations [(He, 2011); (Maier, 2022); (Robinson, 2019)].

1. Foundational definition as generalized Stirling numbers

In the combinatorial literature, the Hsu–Shiue family is introduced through a generalized factorial-basis expansion. One standard form is

$$(z)_{n,-a}=\sum_{k=0}^n S(n,k,a,\beta,r)\,(z-r)_{k,-\beta},$$

with

$(z)_{n,-a}=z(z-a)(z-2a)\cdots(z-(n-1)a).$

A closely related notation used elsewhere is

$$(x)_{n,a}=\sum_{k=0}^n S_{n,k}(a,b;r)\,(x-r)_{k,b}.$$

In both forms, the Hsu–Shiue numbers are connection coefficients between two factorial polynomial bases [(He, 2011); (Maier, 2022)].

The family is normalized by apex and boundary data such as

$S(0,0)=1,\qquad S(n,n)=1,$

and, in the formulation of the unified Stirling framework,

$S(1,0)=r,\qquad S(n,0)=(r)_{n,-a}.$

These coefficients interpolate classical arrays. Two basic specializations are

$S(n,k,1,0,0)=s(n,k),\qquad S(n,k,0,1,0)=S(n,k),$

recovering the classical Stirling numbers of the first and second kinds, respectively. Other specializations listed in the literature include Lah numbers, Riordan’s non-central Stirling numbers, Carlitz’s weighted and degenerate variants, Charalambides–Koutras numbers, Gould–Hopper non-central Lah numbers, Tsylova and Todorov families, Ahuja–Enneking associated Lah numbers, and Broder’s $r$-Stirling numbers (He, 2011).

This definition places the Hsu–Shiue family in the connection-coefficient tradition: the array is determined not by a single recurrence alone, but by how one factorial basis expands in another. That viewpoint is central in later work on GKP triangles, generalized Eulerian numbers, and operator ordering (Maier, 2022, Maier, 2023).

2. Core identities, finite-difference formulas, and recurrence structure

Several papers emphasize that the Hsu–Shiue family admits exact coefficient extraction by generalized divided differences. One form is

$S(n,k,a,\beta,r)=\left.\Delta_\beta^k (z)_{n,-a}\right|_{z=r} = \begin{cases} \beta^k k!\,(z)_{n,-a}[r,r+\beta,\dots,r+k\beta], & \beta\neq 0,\[4pt] D^k (z)_{n,-a}\big|_{z=r}, & \beta=0. \end{cases}$

For $\beta\neq 0$, this yields the explicit alternating sum

$$(z)_{n,-a}=\sum_{k=0}^n S(n,k,a,\beta,r)\,(z-r)_{k,-\beta},$$0

Equivalent finite-difference forms also appear as

$$(z)_{n,-a}=\sum_{k=0}^n S(n,k,a,\beta,r)\,(z-r)_{k,-\beta},$$1

or

$$(z)_{n,-a}=\sum_{k=0}^n S(n,k,a,\beta,r)\,(z-r)_{k,-\beta},$$2

depending on the chosen parametrization [(He, 2011); (Maier, 2022)].

The defining recurrence is of Graham–Knuth–Patashnik type. In one notation,

$$(z)_{n,-a}=\sum_{k=0}^n S(n,k,a,\beta,r)\,(z-r)_{k,-\beta},$$3

while in another,

$$(z)_{n,-a}=\sum_{k=0}^n S(n,k,a,\beta,r)\,(z-r)_{k,-\beta},$$4

The same recurrence governs the Hsu–Shiue coefficients when they are used in generalized geometric-polynomial theory, barred preferential arrangements, and boson-ordering expansions [(Shattuck, 2014); (Maier, 2022); (Maier, 18 Aug 2025)].

A further structural property is homogeneity: $$(z)_{n,-a}=\sum_{k=0}^n S(n,k,a,\beta,r)\,(z-r)_{k,-\beta},$$5 The family also admits matrix composition and inversion laws. One explicit identity is

$$(z)_{n,-a}=\sum_{k=0}^n S(n,k,a,\beta,r)\,(z-r)_{k,-\beta},$$6

with inverse

$$(z)_{n,-a}=\sum_{k=0}^n S(n,k,a,\beta,r)\,(z-r)_{k,-\beta},$$7

These formulas place the family within a Riordan-type group structure and explain why binomial transforms and basis changes recur so persistently in applications (Maier, 2022, Maier, 2023).

3. Analytic extensions: generating functions, Riordan arrays, and Stirling functions

The Hsu–Shiue numbers generate a natural row-polynomial sequence, and the resulting array is an exponential Riordan array. In the notation of the Sheffer-operator literature, the row polynomials are

$$(z)_{n,-a}=\sum_{k=0}^n S(n,k,a,\beta,r)\,(z-r)_{k,-\beta},$$8

with Riordan pair

$$(z)_{n,-a}=\sum_{k=0}^n S(n,k,a,\beta,r)\,(z-r)_{k,-\beta},$$9

Their exponential generating function is

$(z)_{n,-a}=z(z-a)(z-2a)\cdots(z-(n-1)a).$0

The standard exponential-Riordan coefficient formula

$(z)_{n,-a}=z(z-a)(z-2a)\cdots(z-(n-1)a).$1

applies directly after specialization to $(z)_{n,-a}=z(z-a)(z-2a)\cdots(z-(n-1)a).$2 and $(z)_{n,-a}=z(z-a)(z-2a)\cdots(z-(n-1)a).$3 (Maier, 18 Aug 2025).

Equivalent vertical generating functions appear in the GKP analysis: $(z)_{n,-a}=z(z-a)(z-2a)\cdots(z-(n-1)a).$4 This identifies $(z)_{n,-a}=z(z-a)(z-2a)\cdots(z-(n-1)a).$5 as an exponential Riordan array and situates the family as the canonical triangle for one of the principal GKP cases (Maier, 2022).

The analytic generalization extends beyond integer row index. A unified Stirling-function theory defines

$(z)_{n,-a}=z(z-a)(z-2a)\cdots(z-(n-1)a).$6

with absolutely convergent series representation

$(z)_{n,-a}=z(z-a)(z-2a)\cdots(z-(n-1)a).$7

For integer $(z)_{n,-a}=z(z-a)(z-2a)\cdots(z-(n-1)a).$8 and $(z)_{n,-a}=z(z-a)(z-2a)\cdots(z-(n-1)a).$9, these recover the generalized Stirling numbers. The same framework relates the family to generalized factorials $$(x)_{n,a}=\sum_{k=0}^n S_{n,k}(a,b;r)\,(x-r)_{k,b}.$$0, the $$(x)_{n,a}=\sum_{k=0}^n S_{n,k}(a,b;r)\,(x-r)_{k,b}.$$1-gamma function $$(x)_{n,a}=\sum_{k=0}^n S_{n,k}(a,b;r)\,(x-r)_{k,b}.$$2, and recurrences such as

$$(x)_{n,a}=\sum_{k=0}^n S_{n,k}(a,b;r)\,(x-r)_{k,b}.$$3

This analytic continuation is one reason the Hsu–Shiue family functions as more than a discrete triangle (He, 2011).

4. Combinatorial realizations and polynomial hierarchies

A major development is the identification of the Hsu–Shiue numbers as weighted counts of extended Lah distributions. In that model, an extended Lah distribution is obtained from a Lah distribution by circling a subset of special elements, with the restriction that $$(x)_{n,a}=\sum_{k=0}^n S_{n,k}(a,b;r)\,(x-r)_{k,b}.$$4 may be circled only if it starts a block. For $$(x)_{n,a}=\sum_{k=0}^n S_{n,k}(a,b;r)\,(x-r)_{k,b}.$$5, the weight is

$$(x)_{n,a}=\sum_{k=0}^n S_{n,k}(a,b;r)\,(x-r)_{k,b}.$$6

and the Hsu–Shiue numbers satisfy

$$(x)_{n,a}=\sum_{k=0}^n S_{n,k}(a,b;r)\,(x-r)_{k,b}.$$7

This interpretation makes the recurrence transparent: the largest label $$(x)_{n,a}=\sum_{k=0}^n S_{n,k}(a,b;r)\,(x-r)_{k,b}.$$8 either forms a new true block, enters as a non-record-low element, begins a true block as a non-minimal record low, or appears as a circled element, producing exactly the recurrence coefficients $$(x)_{n,a}=\sum_{k=0}^n S_{n,k}(a,b;r)\,(x-r)_{k,b}.$$9, $S(0,0)=1,\qquad S(n,n)=1,$0, $S(0,0)=1,\qquad S(n,n)=1,$1, and $S(0,0)=1,\qquad S(n,n)=1,$2 (Shattuck, 2014).

The same combinatorial infrastructure supports a refinement of Xu’s extension of Spivey’s Bell-number formula. Writing

$S(0,0)=1,\qquad S(n,n)=1,$3

one obtains

$S(0,0)=1,\qquad S(n,n)=1,$4

and hence

$S(0,0)=1,\qquad S(n,n)=1,$5

These formulas treat the Hsu–Shiue family as a weighted Stirling system underlying Bell-type convolution identities (Shattuck, 2014).

Another line of work uses the Hsu–Shiue numbers as the coefficient system for generalized geometric polynomials and generalized barred preferential arrangements. The generalized geometric polynomials are

$S(0,0)=1,\qquad S(n,n)=1,$6

with generating function

$S(0,0)=1,\qquad S(n,n)=1,$7

This generating series is interpreted via barred preferential arrangements whose sections have property 1 or property 2, defined by cyclically ordered labelled compartments of sizes $S(0,0)=1,\qquad S(n,n)=1,$8 or $S(0,0)=1,\qquad S(n,n)=1,$9 (Nkonkobe et al., 2019).

The same negative-parameter mechanism produces a second type of higher order generalised geometric polynomials

$S(1,0)=r,\qquad S(n,0)=(r)_{n,-a}.$0

with coefficient formula

$S(1,0)=r,\qquad S(n,0)=(r)_{n,-a}.$1

These polynomials are counted by barred preferential arrangements in which one section has a single cell with $S(1,0)=r,\qquad S(n,0)=(r)_{n,-a}.$2 compartments and the remaining $S(1,0)=r,\qquad S(n,0)=(r)_{n,-a}.$3 sections have cells with $S(1,0)=r,\qquad S(n,0)=(r)_{n,-a}.$4 compartments, all subject to the same splitting rule. Setting $S(1,0)=r,\qquad S(n,0)=(r)_{n,-a}.$5 yields the generalized Euler polynomials discussed in the same paper (Nkonkobea et al., 2020).

5. Transform relations, generalized Eulerian numbers, and boson-operator ordering

The Hsu–Shiue triangles are closely linked to generalized Eulerian triangles by row-wise binomial transforms. In the GKP framework, the generalized Eulerian numbers $S(1,0)=r,\qquad S(n,0)=(r)_{n,-a}.$6 are related to Hsu–Shiue numbers by the upper binomial transform and reflection: $S(1,0)=r,\qquad S(n,0)=(r)_{n,-a}.$7 with inverse

$S(1,0)=r,\qquad S(n,0)=(r)_{n,-a}.$8

The corresponding generalized Worpitzky identity is

$S(1,0)=r,\qquad S(n,0)=(r)_{n,-a}.$9

In this formulation, the Hsu–Shiue family is the one-sided factorial-basis connection array, while the generalized Eulerian family is the two-sided or bifactorial connection array (Maier, 2022).

A parallel treatment in the Weyl–Heisenberg algebra shows that powers of single-annihilator boson strings are governed by Hsu–Shiue coefficients. For

$S(n,k,1,0,0)=s(n,k),\qquad S(n,k,0,1,0)=S(n,k),$0

one normal-ordering identity is

$S(n,k,1,0,0)=s(n,k),\qquad S(n,k,0,1,0)=S(n,k),$1

while the anti-normal companion is

$S(n,k,1,0,0)=s(n,k),\qquad S(n,k,0,1,0)=S(n,k),$2

In the $S(n,k,1,0,0)=s(n,k),\qquad S(n,k,0,1,0)=S(n,k),$3 realization, the key substitution is $S(n,k,1,0,0)=s(n,k),\qquad S(n,k,0,1,0)=S(n,k),$4 together with

$S(n,k,1,0,0)=s(n,k),\qquad S(n,k,0,1,0)=S(n,k),$5

which converts factorial-basis expansions directly into operator identities (Maier, 2023, Maier, 18 Aug 2025).

The exponential generating functions of the Hsu–Shiue polynomials then yield explicit ordered exponentials. One formula is

$S(n,k,1,0,0)=s(n,k),\qquad S(n,k,0,1,0)=S(n,k),$6

with an analogous anti-normal formula. A later extension introduces a two-point Hsu–Shiue family $S(n,k,1,0,0)=s(n,k),\qquad S(n,k,0,1,0)=S(n,k),$7 that interpolates between the $S(n,k,1,0,0)=s(n,k),\qquad S(n,k,0,1,0)=S(n,k),$8 and $S(n,k,1,0,0)=s(n,k),\qquad S(n,k,0,1,0)=S(n,k),$9 endpoints and gives a unified $r$0-ordered expression for

$r$1

This places the Hsu–Shiue family simultaneously in combinatorics, Riordan-array theory, and quantum-operator ordering (Maier, 18 Aug 2025).

6. The distinct Hsu–Shiue/Shen elliptic-function family

A separate usage of the name occurs in elliptic-function theory. For $r$2, one defines an inverse map $r$3 through

$r$4

equivalently

$r$5

The associated functions are

$r$6

They satisfy the immediate relation

$r$7

and, using

$r$8

together with the triplication identity

$r$9

one derives

$S(n,k,a,\beta,r)=\left.\Delta_\beta^k (z)_{n,-a}\right|_{z=r} = \begin{cases} \beta^k k!\,(z)_{n,-a}[r,r+\beta,\dots,r+k\beta], & \beta\neq 0,\[4pt] D^k (z)_{n,-a}\big|_{z=r}, & \beta=0. \end{cases}$0

Differentiation gives

$S(n,k,a,\beta,r)=\left.\Delta_\beta^k (z)_{n,-a}\right|_{z=r} = \begin{cases} \beta^k k!\,(z)_{n,-a}[r,r+\beta,\dots,r+k\beta], & \beta\neq 0,\[4pt] D^k (z)_{n,-a}\big|_{z=r}, & \beta=0. \end{cases}$1

and hence

$S(n,k,a,\beta,r)=\left.\Delta_\beta^k (z)_{n,-a}\right|_{z=r} = \begin{cases} \beta^k k!\,(z)_{n,-a}[r,r+\beta,\dots,r+k\beta], & \beta\neq 0,\[4pt] D^k (z)_{n,-a}\big|_{z=r}, & \beta=0. \end{cases}$2

This differential equation is the structural basis of the family (Robinson, 2019).

The decisive elliptic result is that $S(n,k,a,\beta,r)=\left.\Delta_\beta^k (z)_{n,-a}\right|_{z=r} = \begin{cases} \beta^k k!\,(z)_{n,-a}[r,r+\beta,\dots,r+k\beta], & \beta\neq 0,\[4pt] D^k (z)_{n,-a}\big|_{z=r}, & \beta=0. \end{cases}$3 is a rational function of the Weierstrass $S(n,k,a,\beta,r)=\left.\Delta_\beta^k (z)_{n,-a}\right|_{z=r} = \begin{cases} \beta^k k!\,(z)_{n,-a}[r,r+\beta,\dots,r+k\beta], & \beta\neq 0,\[4pt] D^k (z)_{n,-a}\big|_{z=r}, & \beta=0. \end{cases}$4-function: $S(n,k,a,\beta,r)=\left.\Delta_\beta^k (z)_{n,-a}\right|_{z=r} = \begin{cases} \beta^k k!\,(z)_{n,-a}[r,r+\beta,\dots,r+k\beta], & \beta\neq 0,\[4pt] D^k (z)_{n,-a}\big|_{z=r}, & \beta=0. \end{cases}$5 where $S(n,k,a,\beta,r)=\left.\Delta_\beta^k (z)_{n,-a}\right|_{z=r} = \begin{cases} \beta^k k!\,(z)_{n,-a}[r,r+\beta,\dots,r+k\beta], & \beta\neq 0,\[4pt] D^k (z)_{n,-a}\big|_{z=r}, & \beta=0. \end{cases}$6 has invariants

$S(n,k,a,\beta,r)=\left.\Delta_\beta^k (z)_{n,-a}\right|_{z=r} = \begin{cases} \beta^k k!\,(z)_{n,-a}[r,r+\beta,\dots,r+k\beta], & \beta\neq 0,\[4pt] D^k (z)_{n,-a}\big|_{z=r}, & \beta=0. \end{cases}$7

and satisfies

$S(n,k,a,\beta,r)=\left.\Delta_\beta^k (z)_{n,-a}\right|_{z=r} = \begin{cases} \beta^k k!\,(z)_{n,-a}[r,r+\beta,\dots,r+k\beta], & \beta\neq 0,\[4pt] D^k (z)_{n,-a}\big|_{z=r}, & \beta=0. \end{cases}$8

The discriminant is

$S(n,k,a,\beta,r)=\left.\Delta_\beta^k (z)_{n,-a}\right|_{z=r} = \begin{cases} \beta^k k!\,(z)_{n,-a}[r,r+\beta,\dots,r+k\beta], & \beta\neq 0,\[4pt] D^k (z)_{n,-a}\big|_{z=r}, & \beta=0. \end{cases}$9

so the period lattice is rectangular with fundamental periods $\beta\neq 0$0 and $\beta\neq 0$1. The poles of $\beta\neq 0$2 occur where $\beta\neq 0$3, and the commentary proves that $\beta\neq 0$4 are poles. Since $\beta\neq 0$5 is elliptic of order two, these account for the poles modulo periods (Robinson, 2019).

This theory also clarifies which associated functions are elliptic. From

$\beta\neq 0$6

it follows that $\beta\neq 0$7 and $\beta\neq 0$8 are elliptic; because $\beta\neq 0$9 has simple poles, $$(z)_{n,-a}=\sum_{k=0}^n S(n,k,a,\beta,r)\,(z-r)_{k,-\beta},$$00 and $$(z)_{n,-a}=\sum_{k=0}^n S(n,k,a,\beta,r)\,(z-r)_{k,-\beta},$$01 have triple poles. By contrast, $$(z)_{n,-a}=\sum_{k=0}^n S(n,k,a,\beta,r)\,(z-r)_{k,-\beta},$$02 and $$(z)_{n,-a}=\sum_{k=0}^n S(n,k,a,\beta,r)\,(z-r)_{k,-\beta},$$03 themselves are not elliptic, although the product $$(z)_{n,-a}=\sum_{k=0}^n S(n,k,a,\beta,r)\,(z-r)_{k,-\beta},$$04 is elliptic and satisfies

$$(z)_{n,-a}=\sum_{k=0}^n S(n,k,a,\beta,r)\,(z-r)_{k,-\beta},$$05

In modern language, this family gives an explicit hypergeometric uniformization of a genus-one algebraic curve, with the inverse hypergeometric construction birationally equivalent to the Weierstrass model (Robinson, 2019).