Striling Coefficients: Theory and Applications

• Striling coefficients are generalized arrays arising in the expansion of functions in natural polynomial, differential, or symmetric-function bases, extending classical Stirling numbers.
• They serve as connection coefficients in operator theory, exemplified by the Jacobi–Stirling numbers which incorporate spectral and combinatorial properties with quadratic eigenvalues.
• Modern applications include algebraic geometry and enumerative invariants, where rising products and Chern class calculations illustrate their polynomiality and asymptotic behavior.

Searching arXiv for the cited papers and closely related work on Stirling-type coefficients. Searching "The Jacobi-Stirling Numbers" (Andrews et al., 2011); "Close encounters with the Stirling numbers of the second kind" (Boyadzhiev, 2018); "Polynomiality of the Striling coefficients of c(Pol^d(C^n)) and Fano schemes" (Fehér et al., 1 Sep 2025). “Striling coefficients” is a spelling used in a 2025 paper for a class of coefficient arrays that generalize classical Stirling numbers; the same paper notes that the authors consistently write “Striling coefficients”, but clearly mean “Stirling coefficients” (Fehér et al., 1 Sep 2025). In the literature represented here, the term refers most fundamentally to coefficients that arise when one expands functions or operators in natural polynomial, differential, or symmetric-function bases. In the classical setting these are the Stirling numbers of the first and second kind; in the Jacobi setting they become Jacobi–Stirling numbers attached to powers of the Jacobi differential expression; and in a recent algebraic-geometric setting they are the coefficients of very general rising products, including the total Chern class $c(\Pol^d(\mathbb{C}^n))$ (Boyadzhiev, 2018).

1. Classical meaning: connection coefficients and operator coefficients

Classically, Stirling numbers of the second kind $S(m,n)$ count partitions of an $m$-element set into $n$ nonempty subsets, with

$$S(m,n)=0 \ \text{if } m<n,\qquad S(m,m)=1,\qquad S(m,0)=0 \ (m>0),\qquad S(0,0)=1.$$

They satisfy the standard recurrence

$$\left\{\begin{matrix} m \ n \end{matrix}\right\} = n\left\{\begin{matrix} m-1 \ n \end{matrix}\right\} + \left\{\begin{matrix} m-1 \ n-1 \end{matrix}\right\},$$

and admit the explicit Euler–Stirling formula

$$S(m,n)=\frac{1}{n!}\sum_{k=0}^{n}(-1)^{\,n-k}\binom{n}{k}k^m.$$

A central algebraic role of these numbers is that they are the coefficients in the change of basis from ordinary powers to falling factorials: $$x^n=\sum_{k=0}^{n} S(n,k)\,x^{\underline{k}}, \qquad x^{\underline{k}}=x(x-1)\cdots(x-k+1).$$ The inverse coefficients are Stirling numbers of the first kind, defined by

$x^{\underline{n}}=\sum_{k=0}^{n} s(n,k)\,x^k,$

and the two arrays are inverse matrices: $\sum_{k=0}^{m} S(m,k)s(k,n)=\delta_{m,n}.$ In this precise sense, both families are “Stirling coefficients”: they are connection coefficients between two natural polynomial bases (Boyadzhiev, 2018).

The same paper emphasizes an analytic interpretation. Stirling’s original coefficients $S(m,n)$0 in the Newton-series expansion of $S(m,n)$1 and Grünert’s coefficients $S(m,n)$2 from repeated application of $S(m,n)$3 to $S(m,n)$4 coincide with the Stirling numbers of the second kind. Writing

$S(m,n)$5

one has

$S(m,n)$6

Thus Stirling coefficients are not merely combinatorial counts: they are also differential-operator coefficients and finite-difference coefficients. The same source explicitly notes that it does not focus on Stirling’s asymptotic expansion for $S(m,n)$7 and its Stirling coefficients in the asymptotic sense, so that usage is distinct from the one considered here (Boyadzhiev, 2018).

2. Jacobi–Stirling numbers as Stirling-type coefficients of operator powers

The Jacobi–Stirling numbers were discovered in connection with the spectral theory of powers of the classical second-order Jacobi differential expression. They are originally defined in parameters $S(m,n)$8, but depend only on the combination $S(m,n)$9, so the natural reparameterization is

$m$0

With this notation, the Jacobi–Stirling numbers of the second kind are denoted

$m$1

and are extended to all $m$2 by

$m$3

They satisfy the Stirling-type triangular recurrence

$m$4

which differs from the classical recurrence only by replacing the linear factor $m$5 with the quadratic eigenvalue term $m$6. They also satisfy the rational generating function

$m$7

from which it follows that, for fixed $m$8, $m$9 is a polynomial in $n$0 with nonnegative integer coefficients. When $n$1, these specialize to the Legendre–Stirling numbers (Andrews et al., 2011).

Their importance as Stirling coefficients comes from operator theory. For the Jacobi differential expression

$n$2

with weight

$n$3

the $n$4-th composite power in Lagrangian symmetric form has coefficients $n$5 given by

$n$6

For the pure Jacobi operator $n$7, these coefficients are exactly the Jacobi–Stirling numbers: $n$8 This is the precise operator-theoretic sense in which they are Stirling-type coefficients: they are the coefficients of derivative terms in integral powers of the Jacobi differential operator. The same paper recalls the Laguerre analogue, where classical Stirling numbers of the second kind occur as coefficients of powers of the Laguerre differential expression, making the Jacobi family a direct quadratic-eigenvalue generalization of the classical linear-eigenvalue case (Andrews et al., 2011).

3. Algebraic identities and inversion theory in the Jacobi setting

Jacobi–Stirling numbers of the second kind play the same basis-conversion role as ordinary Stirling numbers. Define the generalized falling factorials

$n$9

Then

$$S(m,n)=0 \ \text{if } m<n,\qquad S(m,m)=1,\qquad S(m,0)=0 \ (m>0),\qquad S(0,0)=1.$$0

This is the exact analogue of the classical identity

$$S(m,n)=0 \ \text{if } m<n,\qquad S(m,m)=1,\qquad S(m,0)=0 \ (m>0),\qquad S(0,0)=1.$$1

with the linear shifts replaced by the quadratic spectral sequence $$S(m,n)=0 \ \text{if } m<n,\qquad S(m,m)=1,\qquad S(m,0)=0 \ (m>0),\qquad S(0,0)=1.$$2. There is also a vertical recurrence

$$S(m,n)=0 \ \text{if } m<n,\qquad S(m,m)=1,\qquad S(m,0)=0 \ (m>0),\qquad S(0,0)=1.$$3

and a forward-difference positivity property

$$S(m,n)=0 \ \text{if } m<n,\qquad S(m,m)=1,\qquad S(m,0)=0 \ (m>0),\qquad S(0,0)=1.$$4

These features place the Jacobi–Stirling triangle squarely within the class of Stirling-like coefficient arrays (Andrews et al., 2011).

The first-kind Jacobi–Stirling numbers are defined by inversion. Writing

$$S(m,n)=0 \ \text{if } m<n,\qquad S(m,m)=1,\qquad S(m,0)=0 \ (m>0),\qquad S(0,0)=1.$$5

one obtains biorthogonality relations between the first- and second-kind arrays: $$S(m,n)=0 \ \text{if } m<n,\qquad S(m,m)=1,\qquad S(m,0)=0 \ (m>0),\qquad S(0,0)=1.$$6 The first-kind numbers satisfy

$$S(m,n)=0 \ \text{if } m<n,\qquad S(m,m)=1,\qquad S(m,0)=0 \ (m>0),\qquad S(0,0)=1.$$7

and, for $$S(m,n)=0 \ \text{if } m<n,\qquad S(m,m)=1,\qquad S(m,0)=0 \ (m>0),\qquad S(0,0)=1.$$8,

$$S(m,n)=0 \ \text{if } m<n,\qquad S(m,m)=1,\qquad S(m,0)=0 \ (m>0),\qquad S(0,0)=1.$$9

The paper also proves a reciprocity law extending both arrays to integer indices: $$\left\{\begin{matrix} m \ n \end{matrix}\right\} = n\left\{\begin{matrix} m-1 \ n \end{matrix}\right\} + \left\{\begin{matrix} m-1 \ n-1 \end{matrix}\right\},$$0 In this framework, Jacobi–Stirling coefficients are simultaneously operator coefficients, connection coefficients, and mutually inverse triangular arrays (Andrews et al., 2011).

4. Combinatorial models and structural properties

The combinatorial interpretation of Jacobi–Stirling numbers of the second kind is formulated on the doubled set

$$\left\{\begin{matrix} m \ n \end{matrix}\right\} = n\left\{\begin{matrix} m-1 \ n \end{matrix}\right\} + \left\{\begin{matrix} m-1 \ n-1 \end{matrix}\right\},$$1

A Jacobi–Stirling set partition of $$\left\{\begin{matrix} m \ n \end{matrix}\right\} = n\left\{\begin{matrix} m-1 \ n \end{matrix}\right\} + \left\{\begin{matrix} m-1 \ n-1 \end{matrix}\right\},$$2 into $$\left\{\begin{matrix} m \ n \end{matrix}\right\} = n\left\{\begin{matrix} m-1 \ n \end{matrix}\right\} + \left\{\begin{matrix} m-1 \ n-1 \end{matrix}\right\},$$3 zero blocks and $$\left\{\begin{matrix} m \ n \end{matrix}\right\} = n\left\{\begin{matrix} m-1 \ n \end{matrix}\right\} + \left\{\begin{matrix} m-1 \ n-1 \end{matrix}\right\},$$4 nonzero blocks is a partition into $$\left\{\begin{matrix} m \ n \end{matrix}\right\} = n\left\{\begin{matrix} m-1 \ n \end{matrix}\right\} + \left\{\begin{matrix} m-1 \ n-1 \end{matrix}\right\},$$5 blocks such that the zero blocks are distinguishable, the nonzero blocks are indistinguishable, zero blocks may be empty, nonzero blocks must be nonempty, the union of zero blocks may not contain both copies of any integer, and each nonzero block contains both copies of its smallest element but not both copies of any other element. The counting theorem states that

$$\left\{\begin{matrix} m \ n \end{matrix}\right\} = n\left\{\begin{matrix} m-1 \ n \end{matrix}\right\} + \left\{\begin{matrix} m-1 \ n-1 \end{matrix}\right\},$$6

is exactly the number of such partitions. The proof is a direct combinatorial derivation of the recurrence by separating the cases in which $$\left\{\begin{matrix} m \ n \end{matrix}\right\} = n\left\{\begin{matrix} m-1 \ n \end{matrix}\right\} + \left\{\begin{matrix} m-1 \ n-1 \end{matrix}\right\},$$7 lie in the same nonzero block or in different blocks (Andrews et al., 2011).

For first-kind numbers, the paper gives two permutation-pair models. A balanced Jacobi–Stirling permutation pair of length $$\left\{\begin{matrix} m \ n \end{matrix}\right\} = n\left\{\begin{matrix} m-1 \ n \end{matrix}\right\} + \left\{\begin{matrix} m-1 \ n-1 \end{matrix}\right\},$$8 is a pair $$\left\{\begin{matrix} m \ n \end{matrix}\right\} = n\left\{\begin{matrix} m-1 \ n \end{matrix}\right\} + \left\{\begin{matrix} m-1 \ n-1 \end{matrix}\right\},$$9 with $$S(m,n)=\frac{1}{n!}\sum_{k=0}^{n}(-1)^{\,n-k}\binom{n}{k}k^m.$$0, $$S(m,n)=\frac{1}{n!}\sum_{k=0}^{n}(-1)^{\,n-k}\binom{n}{k}k^m.$$1, such that $$S(m,n)=\frac{1}{n!}\sum_{k=0}^{n}(-1)^{\,n-k}\binom{n}{k}k^m.$$2 has one more cycle than $$S(m,n)=\frac{1}{n!}\sum_{k=0}^{n}(-1)^{\,n-k}\binom{n}{k}k^m.$$3, the cycle maxima of $$S(m,n)=\frac{1}{n!}\sum_{k=0}^{n}(-1)^{\,n-k}\binom{n}{k}k^m.$$4 below $$S(m,n)=\frac{1}{n!}\sum_{k=0}^{n}(-1)^{\,n-k}\binom{n}{k}k^m.$$5 are exactly the cycle maxima of $$S(m,n)=\frac{1}{n!}\sum_{k=0}^{n}(-1)^{\,n-k}\binom{n}{k}k^m.$$6, and for each non-cycle-maximum $$S(m,n)=\frac{1}{n!}\sum_{k=0}^{n}(-1)^{\,n-k}\binom{n}{k}k^m.$$7, at least one of $$S(m,n)=\frac{1}{n!}\sum_{k=0}^{n}(-1)^{\,n-k}\binom{n}{k}k^m.$$8 is $$S(m,n)=\frac{1}{n!}\sum_{k=0}^{n}(-1)^{\,n-k}\binom{n}{k}k^m.$$9. The number of such pairs in which $$x^n=\sum_{k=0}^{n} S(n,k)\,x^{\underline{k}}, \qquad x^{\underline{k}}=x(x-1)\cdots(x-k+1).$$0 has exactly $$x^n=\sum_{k=0}^{n} S(n,k)\,x^{\underline{k}}, \qquad x^{\underline{k}}=x(x-1)\cdots(x-k+1).$$1 cycles is

$$x^n=\sum_{k=0}^{n} S(n,k)\,x^{\underline{k}}, \qquad x^{\underline{k}}=x(x-1)\cdots(x-k+1).$$2

When $$x^n=\sum_{k=0}^{n} S(n,k)\,x^{\underline{k}}, \qquad x^{\underline{k}}=x(x-1)\cdots(x-k+1).$$3, there is also an unbalanced model with $$x^n=\sum_{k=0}^{n} S(n,k)\,x^{\underline{k}}, \qquad x^{\underline{k}}=x(x-1)\cdots(x-k+1).$$4 and $$x^n=\sum_{k=0}^{n} S(n,k)\,x^{\underline{k}}, \qquad x^{\underline{k}}=x(x-1)\cdots(x-k+1).$$5, again counted by $$x^n=\sum_{k=0}^{n} S(n,k)\,x^{\underline{k}}, \qquad x^{\underline{k}}=x(x-1)\cdots(x-k+1).$$6. For $$x^n=\sum_{k=0}^{n} S(n,k)\,x^{\underline{k}}, \qquad x^{\underline{k}}=x(x-1)\cdots(x-k+1).$$7, these recover Egge’s Legendre–Stirling permutation pairs (Andrews et al., 2011).

The same work establishes strong structural properties. If

$$x^n=\sum_{k=0}^{n} S(n,k)\,x^{\underline{k}}, \qquad x^{\underline{k}}=x(x-1)\cdots(x-k+1).$$8

then for each $$x^n=\sum_{k=0}^{n} S(n,k)\,x^{\underline{k}}, \qquad x^{\underline{k}}=x(x-1)\cdots(x-k+1).$$9, $x^{\underline{n}}=\sum_{k=0}^{n} s(n,k)\,x^k,$0 has only real, simple, nonpositive zeros and $x^{\underline{n}}=\sum_{k=0}^{n} s(n,k)\,x^k,$1. Consequently, the unsigned Jacobi–Stirling numbers of the first kind and the Jacobi–Stirling numbers of the second kind are unimodal in $x^{\underline{n}}=\sum_{k=0}^{n} s(n,k)\,x^k,$2 for fixed $x^{\underline{n}}=\sum_{k=0}^{n} s(n,k)\,x^k,$3, with either a single peak or a plateau of length two. The paper also records Mongelli’s result that the Jacobi–Stirling triangle is totally positive, meaning that all minors of the infinite matrix are nonnegative (Andrews et al., 2011).

5. Rising products and the modern generalized notion of “Striling coefficients”

A much broader notion is introduced in the study of Chern classes of polynomial representations. Let

$x^{\underline{n}}=\sum_{k=0}^{n} s(n,k)\,x^k,$4

with $x^{\underline{n}}=\sum_{k=0}^{n} s(n,k)\,x^k,$5, and let $x^{\underline{n}}=\sum_{k=0}^{n} s(n,k)\,x^k,$6 be an integer-valued polynomial. Then

$x^{\underline{n}}=\sum_{k=0}^{n} s(n,k)\,x^k,$7

is called a rising product, and the coefficients $x^{\underline{n}}=\sum_{k=0}^{n} s(n,k)\,x^k,$8 are called the Stirling coefficients. In the simplest case $x^{\underline{n}}=\sum_{k=0}^{n} s(n,k)\,x^k,$9, this framework contains several classical sequences as direct specializations. Taking $\sum_{k=0}^{m} S(m,k)s(k,n)=\delta_{m,n}.$0 gives

$\sum_{k=0}^{m} S(m,k)s(k,n)=\delta_{m,n}.$1

so one recovers Stirling numbers of the first kind. Taking $\sum_{k=0}^{m} S(m,k)s(k,n)=\delta_{m,n}.$2 gives

$\sum_{k=0}^{m} S(m,k)s(k,n)=\delta_{m,n}.$3

so one recovers Stirling numbers of the second kind. Taking $\sum_{k=0}^{m} S(m,k)s(k,n)=\delta_{m,n}.$4 gives generalized Stirling numbers studied by Tweedie and Komatsu as level-$\sum_{k=0}^{m} S(m,k)s(k,n)=\delta_{m,n}.$5 Stirling numbers (Fehér et al., 1 Sep 2025).

The central structural theorem is polynomiality. Under the assumptions above, each Stirling coefficient $\sum_{k=0}^{m} S(m,k)s(k,n)=\delta_{m,n}.$6 is a polynomial in the parameters. The proof uses the auxiliary product

$\sum_{k=0}^{m} S(m,k)s(k,n)=\delta_{m,n}.$7

and expresses its coefficients in terms of augmented monomial symmetric polynomials $\sum_{k=0}^{m} S(m,k)s(k,n)=\delta_{m,n}.$8. After the arithmetic specialization $\sum_{k=0}^{m} S(m,k)s(k,n)=\delta_{m,n}.$9, $S(m,n)$00, one obtains formulae in terms of polynomial specializations

$S(m,n)$01

whose leading term is

$S(m,n)$02

This yields a finite-sum formula for $S(m,n)$03 and proves that the generalized Stirling coefficients are polynomial functions of the discrete parameter (Fehér et al., 1 Sep 2025).

6. Chern classes, asymptotics, and geometric applications

The motivating representation-theoretic example is $S(m,n)$04, the $S(m,n)$05-representation of homogeneous degree-$S(m,n)$06 polynomials in $S(m,n)$07 variables. If $S(m,n)$08 are the Chern roots of the standard representation, then the Chern roots of $S(m,n)$09 are the weights

$S(m,n)$10

so

$S(m,n)$11

For $S(m,n)$12, this reduces to the single rising product

$S(m,n)$13

which is exactly of the generalized Stirling-coefficient type. For general $S(m,n)$14, the product is iterated rather than single, but the same machinery applies recursively (Fehér et al., 1 Sep 2025).

The resulting polynomiality theorem states that

$S(m,n)$15

Equivalently, for fixed $S(m,n)$16 and fixed cohomological degree $S(m,n)$17, every coefficient of $S(m,n)$18 in the monomial, Schur, or elementary symmetric basis is a polynomial in $S(m,n)$19. In the monomial basis,

$S(m,n)$20

and

$S(m,n)$21

In the Schur basis, the coefficient $S(m,n)$22 is again polynomial in $S(m,n)$23, with the explicit leading asymptotic

$S(m,n)$24

The leading term of the full class in degree $S(m,n)$25 is

$S(m,n)$26

For the elementary basis there is a non-uniform degree pattern; if $S(m,n)$27, then

$S(m,n)$28

For $S(m,n)$29, the corresponding asymptotic is proved sharply in Proposition 3.16 (Fehér et al., 1 Sep 2025).

These polynomiality and asymptotic results feed directly into enumerative geometry. For the variety $S(m,n)$30 of hypersurfaces in $S(m,n)$31 containing some $S(m,n)$32-plane, the degree is expressed as an integral of a Chern class over a Grassmannian, and the paper proves that $S(m,n)$33 is a polynomial in $S(m,n)$34 with leading term

$S(m,n)$35

This proves Manivel’s conjectured exponent of $S(m,n)$36. For Fano schemes of lines, the same framework yields formulas for degrees and Euler characteristics, and in the case $S(m,n)$37 the Euler class coefficients in the Schur basis are expressed explicitly in terms of classical Stirling numbers of the first kind. A plausible implication is that the generalized notion of Stirling coefficients functions as a bridge between symmetric-function combinatorics and concrete calculations of characteristic classes and enumerative invariants (Fehér et al., 1 Sep 2025).