Spivey-Type Recurrence Relations

• Spivey-type recurrence relations are unified double-sum formulations that generalize classical partition identities via operator methods and combinatorial constructions.
• They extend foundational recurrences such as the Bell number identity to deformed settings including degenerate, q-analogue, and r-generalized polynomial families.
• Their framework underpins applications in quantum optics, statistical mechanics, and algebraic combinatorics by enabling the systematic derivation of weighted, combinatorial recurrences.

A Spivey-type recurrence relation is a unified, double-sum recurrence for combinatorial sequences such as Bell numbers, Bell polynomials, and their generalizations, arising from the interplay between algebraic operator theory and combinatorics. These recurrences generalize classical identities by encoding the way higher-index objects can be recursively constructed from lower-index counterparts through combinatorial or operator-theoretic means. Significant recent research has extended Spivey-type recurrences to numerous polynomial families, including degenerate, $q$-analogue, $r$-generalized, probabilistic, Dowling, Fubini, Lah-Bell, and second-order recurrence sequences, providing a unified perspective on the structural decomposition of combinatorial polynomials.

1. Classical Spivey Recurrence and Operator Methods

The classical Spivey recurrence, proved in 2008, expresses the Bell numbers in terms of a double sum involving Stirling numbers of the second kind: $$B_{n+m} = \sum_{j=0}^m \sum_{k=0}^n \binom{m}{j} \,S(n,k)\, j^{m-k}\, B_k$$ This identity generalizes both the three-term and Touchard/Dobinski recurrences by encompassing the full range of partition refinements indexed by $(m, n)$.

A fundamental insight underpinning many generalizations is the use of linear operators, notably $X$ (multiplication by $x$) and $D$ (differentiation), which satisfy the Heisenberg–Weyl commutator $[D,X]=1$. This operator algebra encodes the combinatorial construction of set partitions, with the action of $(XD)^n$ on suitable generating functions producing Bell and related polynomials. The operator-based approach provides a general machinery for deriving Spivey-type recurrences in classical, degenerate, and $q$-deformed settings (Kim et al., 18 Feb 2025, Kim et al., 13 Mar 2025, Mangontarum et al., 2017, Mangontarum, 2017).

2. Degenerate and Generalized Bell Polynomials

When combinatorial sequences are deformed by additional parameters, Spivey-type recurrences acquire deformations in both coefficients and combinatorial weights:

• Degenerate Bell polynomials: Defined with parameter $\lambda$, using the degenerate falling factorial $$(x)_{n,\lambda} = x(x-\lambda)\cdots(x-(n-1)\lambda)$$, and degenerate Stirling numbers $S_\lambda(n,k)$ (Kim et al., 18 Feb 2025). The recurrence is

$$B_{m+n,\lambda}(x) = \sum_{j=0}^m \sum_{k=0}^n \binom{m}{j} (j-m\lambda)_{n-k,\lambda} x^j B_{k,\lambda}(x)$$

where $(j-m\lambda)_{n-k,\lambda}$ is the degenerate falling factorial.

• Fully degenerate Bell polynomials and multivariate versions are defined via modified generating functions, with recurrences that further incorporate extra parameters and shifts in variables, following the same operator-theoretic strategies (Kim et al., 16 Nov 2025, Kim et al., 6 Sep 2025).
• Degenerate $r$-Bell polynomials incorporate additional "shift" parameters (analogous to the $r$-Stirling numbers), producing recurrences of the form

$$B_{m+n,\lambda}^{(r)}(x) = \sum_{j=0}^m \sum_{k=0}^n \binom{m}{j} \binom{n+r}{k+r} (k - m\lambda)_{n-j,\lambda} x^k B_{j,\lambda}^{(r)}(x)$$

(Kim et al., 18 Feb 2025, Kim et al., 16 Nov 2025).

The degenerate regimes interpolate to the classical cases as $\lambda\to 0$, recovering ordinary Stirling numbers and Bell polynomials.

3. Spivey-Type Recurrences for Other Sequences and Structures

Spivey-type recurrences have been extended far beyond the Bell numbers:

• Dowling, Fubini, and Whitney polynomials: Recurrences appear with double sums over Whitney or degenerate Whitney numbers, and key step replaced by normal-ordering arguments for boson or $q$-boson operators (Kim et al., 4 Mar 2025, Mangontarum, 2017, Mangontarum et al., 2017).
• Lah-Bell and $r$-Lah-Bell polynomials: The recurrence for the Lah-Bell polynomials involves Lah numbers in place of Stirling numbers and falling factorials replacing powers (Kim et al., 13 Mar 2025):

$$B^{n+m}_L(x) = \sum_{j=0}^m \sum_{k=0}^n L(m,j)\binom{n}{k}(m+j)_{n-k} x^j B^k_L(x)$$

with suitable extensions to $r$- and $\lambda$-analogues.

• q-analogues and (q,r)-Dowling polynomials: Mangontarum and Katriel derived operator-based $q$-Spivey recurrences using $q$-bosonic commutation relations (Mangontarum, 2017). In these,

$$B_{n+m,q}(x) = \sum_{k=0}^n \sum_{j=0}^m \binom{n}{k}_q [j]_q^{n-k} q^{jk} B_{k,q}(x) [x]_{q,j}$$

where $[j]_q$ and $\binom{n}{k}_q$ are $q$-analogues of integers and binomials.

• Second-order recurrence sequences: Spivey-type binomial-sum identities (single or double sum) exist for all second-order recurrences (Fibonacci, Lucas, Pell, Jacobsthal, etc.) (Adegoke, 2019).

4. Probabilistic and Analytic Generalizations

Probabilistic Spivey-type recurrences generalize standard combinatorial identities to situations where partition blocks carry random marks, modeled by independent identically distributed random variables $Y$ with well-defined moment generating functions (Kim et al., 24 Aug 2025, Kim et al., 3 Apr 2024). The Spivey recurrence then involves expectations of degenerate Pochhammer symbols and multinomial compositional sums: $$\phi_{l+n,\lambda}^Y(y) = \sum_{k=0}^n \sum_{m=0}^l \binom{l}{m} \frac{y^k}{k!} \sum_{\ell_1+\cdots+\ell_k=n} \binom{n}{\ell_1,\dots,\ell_k} E\left[(S_k-n\lambda)_{l-m,\lambda} \prod_{i=1}^k (Y_i)_{\ell_i,\lambda}\right] \phi_{m,\lambda}^Y(y)$$ where $S_k$ is the sum of $k$ i.i.d.\ $Y$'s. For $Y\equiv1$ one recovers the degenerate Spivey recurrence; more generally, these recurrences encode deformed or weighted set partitions, urn models, or Poisson mixtures.

Such analytic generalizations underpin further developments in quantum optics (normal ordering), statistical mechanics (partition functions with marked blocks), and enumerative combinatorics (generating function manipulations).

5. Structural Patterns and Algebraic Unification

Across combinatorial families, Spivey-type recurrences share a canonical double-sum form, with the structure: $$T_{n+m} = \sum_{j=0}^{m} \sum_{k=0}^{n} \text{Coeff}(m,j) \cdot \text{Weight}(j,n,k) \cdot \text{Monomial}(j/k) \cdot T_k$$ The specifics are dictated by the algebra—classical, degenerate, $q$-deformed, $r$-shifted, or probabilistic—but the unifying principle involves:

• Interpretations in terms of set partitions or other combinatorial decompositions,
• Operator representations (normal ordering, shifted operators),
• The replacement of powers by falling/degenerate/factorial weights,
• Special numbers (Stirling, Whitney, Lah, etc.) encoding the combinatorial part.

Table 1 summarizes key instances.

Family	Recurrence Structure	Coefficients/Weights
Bell polynomials	$\sum_{j,k} \binom{m}{j} S(n,k) j^{m-k} B_k$	Stirling, powers
Degenerate Bell polynomials	$$\sum_{j,k} \binom{m}{j} (j-m\lambda)_{n-k,\lambda} x^j B_{k,\lambda}(x)$$	Degenerate factorials
$r$-Bell/degen. $r$-Bell polynomials	$$\sum_{j,k} \binom{m}{j} \binom{n+r}{k+r}(k-m\lambda)_{n-j,\lambda} x^k B^{(r)}$$	Shifted factorials
Fubini/Dowling/Lah-Bell families	$$\sum_{j,k} \text{Whitney/Lah}(m,j)\,\binom{n}{k}\,\text{weight} \cdot T_k$$	Family-specific
Second-order recursions	$\sum_{k=0}^n \binom{n}{k} U_{m-k} U_k$	Recurrence kernel
Probabilistic extensions	$$\sum_{k=0}^n \sum_m \binom{l}{m}\text{mom.e.n.t.} \cdot T_m$$	Moments, expectations

The operator, algebraic, and combinatorial frameworks are interchangeable and serve to reveal deep unifying themes in discrete mathematics and mathematical physics.

6. Connections, Applications, and Future Directions

Spivey-type recurrences provide a uniform description for decomposing partition-like combinatorial structures. Their generalizations are critical in:

• $q$-analogues and deformed combinatorics (quantum groups, statistical physics),
• Probabilistic partition structures (random partitions, occupancy problems),
• Algebraic combinatorics (Hopf algebras, normal ordering in Fock spaces),
• Applications in combinatorial enumeration with weightings, e.g., set partitions with weighted or colored blocks, compositions, or marked structures.

Current research themes include further generalization to multivariate and noncommutative settings, analytic properties in asymptotic regimes, and interpretation within algebraic operad and Hopf algebra frameworks. The methodology is also being exported to higher-order and mixed recurrence families, suggesting a broad algebraic underpinning for Spivey-type identities.

Key references include Kim & Kim's operator derivations for degenerate and fully degenerate cases (Kim et al., 18 Feb 2025, Kim et al., 16 Nov 2025), probabilistic generalizations (Kim et al., 24 Aug 2025, Kim et al., 3 Apr 2024), $q$- and $(q,r)$-analogue frameworks (Mangontarum, 2017, Kim et al., 4 Mar 2025), Lah and r-Lah polynomials (Kim et al., 13 Mar 2025), and the algorithmic unification of binomial-sum recurrences for sequences of arbitrary order (Adegoke, 2019).