--- title: 'Striling Coefficients: Theory and Applications' url: https://www.emergentmind.com/topics/striling-coefficients type: topic --- # Striling Coefficients: Theory and Applications Searching arXiv for the cited papers and closely related work on Stirling-type coefficients. Searching arXiv: "The Jacobi-Stirling Numbers" 1112.6111; "Close encounters with the Stirling numbers of the second kind" 1806.09468; "Polynomiality of the Striling coefficients of c(Pol^d(C^n)) and Fano schemes" 2509.01725. “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” [2509.01725]. 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 \(c(\Pol^d(\mathbb{C}^n))\) [1806.09468]. ## 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 \[ S(m,n)=0 \ \text{if } m0),\qquad S(0,0)=1. \] 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 [1806.09468]. The same paper emphasizes an analytic interpretation. Stirling’s original coefficients \(A_n\) in the Newton-series expansion of \(z^m\) and Grünert’s coefficients \(B_{m,n}\) from repeated application of \(xD=x\frac{d}{dx}\) to \(e^x\) coincide with the Stirling numbers of the second kind. Writing \[ P_m(x)=\sum_{n=0}^{m}S(m,n)x^n, \] one has \[ (xD)^m e^x=P_m(x)e^x, \qquad \sum_{m=0}^{\infty}P_m(x)\frac{t^m}{m!} = \exp\!\big(x(e^t-1)\big). \] 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 \(n!\) and its Stirling coefficients in the asymptotic sense, so that usage is distinct from the one considered here [1806.09468]. ## 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 \(\alpha,\beta\), but depend only on the combination \(\alpha+\beta+1\), so the natural reparameterization is \[ \gamma:=\frac{\alpha+\beta+1}{2}. \] With this notation, the Jacobi–Stirling numbers of the second kind are denoted \[ \left\{\!{n\atop j}\!\right\}_\gamma \] and are extended to all \(n,j\ge 0\) by \[ \left\{\!{0\atop 0}\!\right\}_\gamma=1,\qquad \left\{\!{n\atop 0}\!\right\}_\gamma=0\ (n>0),\qquad \left\{\!{0\atop j}\!\right\}_\gamma=0\ (j>0). \] They satisfy the Stirling-type triangular recurrence \[ \left\{\!{n\atop j}\!\right\}_\gamma = \left\{\!{n-1\atop j-1}\!\right\}_\gamma + j(j+2\gamma-1)\left\{\!{n-1\atop j}\!\right\}_\gamma, \] which differs from the classical recurrence only by replacing the linear factor \(j\) with the quadratic eigenvalue term \(j(j+2\gamma-1)\). They also satisfy the rational generating function \[ \prod_{r=1}^{j}\frac{1}{1-r(r+2\gamma-1)x} = \sum_{n=0}^{\infty}\left\{\!{n\atop j}\!\right\}_\gamma x^{\,n-j}, \] from which it follows that, for fixed \(n,j\), \(\left\{\!{n\atop j}\!\right\}_\gamma\) is a polynomial in \(\gamma\) with nonnegative integer coefficients. When \(\gamma=1\), these specialize to the Legendre–Stirling numbers [1112.6111]. Their importance as Stirling coefficients comes from operator theory. For the Jacobi differential expression \[ \ell_{\alpha,\beta}[y](x)=-(1-x^2)y''(x)+(\alpha-\beta+(\alpha+\beta+2)x)y'(x)+ky(x), \] with weight \[ w_{\alpha,\beta}(x)=(1-x)^\alpha(1+x)^\beta, \] the \(n\)-th composite power in Lagrangian symmetric form has coefficients \(c_\gamma(n,j)\) given by \[ \ell_{\alpha,\beta}^n[y](x) = w_{\alpha,\beta}(x) \sum_{j=0}^{n} (-1)^j c_\gamma(n,j) (1-x)^{\alpha+j}(1+x)^{\beta+j}y^{(j)}(x). \] For the pure Jacobi operator \(k=0\), these coefficients are exactly the Jacobi–Stirling numbers: \[ \ell_{\alpha,\beta}^n[y](x) = w_{\alpha,\beta}(x) \sum_{j=1}^{n} (-1)^j \left\{\!{n\atop j}\!\right\}_\gamma (1-x)^{\alpha+j}(1+x)^{\beta+j}y^{(j)}(x). \] 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 [1112.6111]. ## 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 \[ (x)^{(0)}_\gamma:=1, \qquad (x)^{(j)}_\gamma:=\prod_{m=0}^{j-1}\bigl(x-m(m+2\gamma-1)\bigr)\quad (j\ge 1). \] Then \[ x^n = \sum_{j=0}^{n} \left\{\!{n\atop j}\!\right\}_\gamma (x)^{(j)}_\gamma, \qquad n\ge 0. \] This is the exact analogue of the classical identity \[ x^n=\sum_{j=0}^{n}\left\{\!{n\atop j}\!\right\}(x)_j, \] with the linear shifts replaced by the quadratic spectral sequence \(m(m+2\gamma-1)\). There is also a vertical recurrence \[ \left\{\!{n\atop j}\!\right\}_\gamma = \sum_{r=1}^{j} \left\{\!{r-1\atop j-1}\!\right\}_\gamma [r(r+2\gamma-1)]^{\,n-r}, \] and a forward-difference positivity property \[ \Delta^k\left\{\!{n\atop j}\!\right\}_\gamma\ge 0 \quad\text{for all }k\in\mathbb N_0,\ n\ge j. \] These features place the Jacobi–Stirling triangle squarely within the class of Stirling-like coefficient arrays [1112.6111]. The first-kind Jacobi–Stirling numbers are defined by inversion. Writing \[ (x)^{(n)}_\gamma = \sum_{j=0}^{n} (-1)^{n+j} \left[{n\atop j}\right]_\gamma x^j, \] one obtains biorthogonality relations between the first- and second-kind arrays: \[ \sum_j (-1)^{n+j}\left\{\!{n\atop j}\!\right\}_\gamma\left[{j\atop m}\right]_\gamma=\delta_{n,m}, \qquad \sum_j (-1)^{j+m}\left[{n\atop j}\right]_\gamma\left\{\!{j\atop m}\!\right\}_\gamma=\delta_{n,m}. \] The first-kind numbers satisfy \[ \left[{n\atop 0}\right]_\gamma=\delta_{n,0}, \qquad \left[{0\atop j}\right]_\gamma=\delta_{j,0}, \] and, for \(n,j\ge 1\), \[ \left[{n\atop j}\right]_\gamma = \left[{n-1\atop j-1}\right]_\gamma + (n-1)(n+2\gamma-2)\left[{n-1\atop j}\right]_\gamma. \] The paper also proves a reciprocity law extending both arrays to integer indices: \[ \left\{\!{-j\atop -n}\!\right\}_\gamma = (-1)^{n+j}\left[{n\atop j}\right]_\gamma. \] In this framework, Jacobi–Stirling coefficients are simultaneously operator coefficients, connection coefficients, and mutually inverse triangular arrays [1112.6111]. ## 4. Combinatorial models and structural properties The combinatorial interpretation of Jacobi–Stirling numbers of the second kind is formulated on the doubled set \[ [n]_2=\{1_1,1_2,2_1,2_2,\dots,n_1,n_2\}. \] A Jacobi–Stirling set partition of \([n]_2\) into \(\gamma\) zero blocks and \(j\) nonzero blocks is a partition into \(\gamma+j\) 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 \[ \left\{\!{n\atop j}\!\right\}_\gamma \] is exactly the number of such partitions. The proof is a direct combinatorial derivation of the recurrence by separating the cases in which \(n_1,n_2\) lie in the same nonzero block or in different blocks [1112.6111]. For first-kind numbers, the paper gives two permutation-pair models. A balanced Jacobi–Stirling permutation pair of length \(n\) is a pair \((\tau_1,\tau_2)\) with \(\tau_1\in S_{n+\gamma}\), \(\tau_2\in S_{n+\gamma-1}\), such that \(\tau_1\) has one more cycle than \(\tau_2\), the cycle maxima of \(\tau_1\) below \(n+\gamma\) are exactly the cycle maxima of \(\tau_2\), and for each non-cycle-maximum \(k\), at least one of \(\tau_1(k),\tau_2(k)\) is \(\le n\). The number of such pairs in which \(\tau_1\) has exactly \(\gamma+j\) cycles is \[ \left[{n\atop j}\right]_\gamma. \] When \(2\gamma\in\mathbb N_0\), there is also an unbalanced model with \(\tau_1\in S_{n+2\gamma-1}\) and \(\tau_2\in S_n\), again counted by \(\left[{n\atop j}\right]_\gamma\). For \(\gamma=1\), these recover Egge’s Legendre–Stirling permutation pairs [1112.6111]. The same work establishes strong structural properties. If \[ A_n(x)=\sum_{j=0}^{n}\left\{\!{n\atop j}\!\right\}_\gamma x^j, \] then for each \(n\ge 1\), \(A_n(x)\) has only real, simple, nonpositive zeros and \(A_n(0)=0\). Consequently, the unsigned Jacobi–Stirling numbers of the first kind and the Jacobi–Stirling numbers of the second kind are unimodal in \(j\) for fixed \(n\), 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 [1112.6111]. ## 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 \[ P(d_0,\dots,d_r,t,x)=\sum_E P_E(d_0,\dots,d_r,t)\,x^E \in \mathbb Q[d_0,\dots,d_r,t][[x]] \] with \(P(d_0,\dots,d_r,t,0)=1\), and let \(K(d_0,\dots,d_r)\) be an integer-valued polynomial. Then \[ S[P,K](d_0,\dots,d_r,x) := \prod_{t=0}^{K(d_0,\dots,d_r)} P(d_0,\dots,d_r,t,x) = \sum_{H\in\mathbb N^n} S[P,K]_H(d_0,\dots,d_r)x^H \] is called a rising product, and the coefficients \(S[P,K]_H\) are called the Stirling coefficients. In the simplest case \(K(d)=d\), this framework contains several classical sequences as direct specializations. Taking \(P(d,t,x)=1+tx\) gives \[ \prod_{t=0}^{d}(1+tx)=\sum_{h=0}^{d+1}S[P]_h(d)x^h, \qquad S[P]_h(d)=\sigma_h(1,2,\dots,d)=\stir{d+1}{d+1-h}, \] so one recovers Stirling numbers of the first kind. Taking \(P(d,t,x)=\frac{1}{1-tx}\) gives \[ \prod_{t=0}^{d}\frac{1}{1-tx} = \sum_{a=0}^{\infty}S[P]_a(d)x^a, \qquad S[P]_a(d)=h_a(1,2,\dots,d)=\begin{Bmatrix} d+a \\ a \end{Bmatrix}, \] so one recovers Stirling numbers of the second kind. Taking \(P(d,t,x)=1+t^s x\) gives generalized Stirling numbers studied by Tweedie and Komatsu as level-\(s\) Stirling numbers [2509.01725]. The central structural theorem is polynomiality. Under the assumptions above, each Stirling coefficient \(S[P,K]_H(d_0,\dots,d_r)\) is a polynomial in the parameters. The proof uses the auxiliary product \[ A[P](d,\underline y_v,x):=\prod_{t=0}^{v}P(d,y_t,x) \] and expresses its coefficients in terms of augmented monomial symmetric polynomials \(\tilde m_\lambda(\underline y_v)\). After the arithmetic specialization \(y_t\mapsto t\), \(v\mapsto K(d)\), one obtains formulae in terms of polynomial specializations \[ \tilde M_\lambda(v)=\tilde m_\lambda(0,1,\dots,v), \] whose leading term is \[ \tilde M_\lambda(v)\sim \prod_i \frac{1}{\lambda_i+1}\,v^{|\lambda|+l(\lambda)} \quad (v\to\infty). \] This yields a finite-sum formula for \(S[P,K]_H(d)\) and proves that the generalized Stirling coefficients are polynomial functions of the discrete parameter [2509.01725]. ## 6. Chern classes, asymptotics, and geometric applications The motivating representation-theoretic example is \(\Pol^d(\mathbb C^n)\), the \(\GL(n)\)-representation of homogeneous degree-\(d\) polynomials in \(n\) variables. If \(x_1,\dots,x_n\) are the Chern roots of the standard representation, then the Chern roots of \(\Pol^d(\mathbb C^n)\) are the weights \[ d_1x_1+\cdots+d_nx_n \qquad (d_1,\dots,d_n)\in\mathbb N^n,\quad d_1+\cdots+d_n=d, \] so \[ c\big(\Pol^d(\mathbb C^n)\big) = \prod_{\substack{(d_1,\dots,d_n)\in\mathbb N^n\\ d_1+\dots+d_n=d}} \bigl(1+d_1x_1+\cdots+d_nx_n\bigr). \] For \(n=2\), this reduces to the single rising product \[ c\big(\Pol^d(\mathbb C^2)\big) = \prod_{t=0}^{d}\bigl(1+t x_1+(d-t)x_2\bigr), \] which is exactly of the generalized Stirling-coefficient type. For general \(n\), the product is iterated rather than single, but the same machinery applies recursively [2509.01725]. The resulting polynomiality theorem states that \[ c\big(\Pol^d(\mathbb C^n)\big)\in \mathbb Q[d][[x_1,\dots,x_n]]^{S_n}. \] Equivalently, for fixed \(n\) and fixed cohomological degree \(k\), every coefficient of \(c_k(\Pol^d(\mathbb C^n))\) in the monomial, Schur, or elementary symmetric basis is a polynomial in \(d\). In the monomial basis, \[ c_k\big(\Pol^d(\mathbb C^n)\big)=\sum_{\mu\vdash k}a_\mu(d)\,m_\mu(x_1,\dots,x_n), \] and \[ a_\mu(d) = \frac{1}{\mu!}\left(\frac{1}{n!}\right)^{|\mu|}d^{n|\mu|} +\text{(lower powers of \(d\))}. \] In the Schur basis, the coefficient \(b_\lambda(d)\) is again polynomial in \(d\), with the explicit leading asymptotic \[ b_\lambda(d)\sim \frac{\prod_{1\le i