Degenerate Seidel’s Formula

• Degenerate Seidel's formula is a generalization of the classical approach that incorporates a lambda parameter to modify the Euler–Seidel matrix method.
• It transforms exponential generating functions via convolution with a lambda-deformed series, yielding novel combinatorial identities for sequences like degenerate Bell numbers.
• Its derived recurrence relations extend classical results, offering a systematic method to generate new identities in combinatorial analysis.

The degenerate Seidel’s formula is a generalization of the classical Seidel’s formula, introduced by incorporating a real parameter $\lambda$ into the Euler–Seidel matrix method. This framework provides a transformation of exponential generating functions via convolution with a lambda-deformed exponential series, yielding new combinatorial identities for sequences such as the degenerate Bell and Fubini numbers and polynomials. The degenerate Seidel’s formula and its underlying recurrence are foundational tools for deriving identities in the theory of special numbers and combinatorial analysis (Kim et al., 14 Dec 2025).

1. Classical Euler–Seidel Matrix Method and Seidel’s Formula

Given an initial sequence $(a_n)_{n\ge0}$, the Euler–Seidel array $(a_{n,k})_{n,k\ge0}$ is constructed by

$$a_{n,0} = a_n, \qquad a_{n,k} = a_{n,k-1} + a_{n+1,k-1} \quad (k\ge 1).$$

The initial sequence is $(a_{n,0})_{n\ge0}$ and the final sequence is $(a_{0,n})_{n\ge0}$. Inductive arguments yield the inversion relations: $$\begin{align*} a_{0,n} &= \sum_{k=0}^n \binom{n}{k} a_{k,0}, \ a_{n,0} &= \sum_{k=0}^n (-1)^{n-k} \binom{n}{k} a_{0,k}. \end{align*}$$ At the generating function level, for the ordinary generating function $A_1(t) = \sum_{n\ge0} a_{n,0} t^n$, the final sequence is given by

$$\overline{A}_1(t) = \frac{1}{1-t}\, A_1\left(\frac{t}{1-t}\right).$$

For the exponential generating function $A_2(t) = \sum_{n\ge 0} a_{n,0} \frac{t^n}{n!}$, Seidel’s formula is

$\overline{A}_2(t) = e^{t} A_2(t).$

2. The Degenerate Euler–Seidel Matrix Construction

The degenerate Euler–Seidel matrix depends on a parameter $\lambda \in \mathbb{R}$. For a sequence $(a_{n,\lambda})_{n\ge 0}$, define recursively: $a_{n,0}(\lambda) = a_{n,\lambda}$

$$a_{n,k}(\lambda) = \bigl(1-(k-n)\lambda\bigr) a_{n,k-1}(\lambda) + a_{n+1,k-1}(\lambda), \quad k\ge 1.$$

The array $\bigl(a_{n,k}(\lambda)\bigr)_{n,k\ge0}$ produces an initial column $\{a_{n,0}(\lambda)\}$ and a final row $\{a_{0,n}(\lambda)\}$. This generalization reduces to the classical case for $\lambda=0$.

3. Lambda-Generalized Binomial Identities

The method utilizes degenerate falling and rising factorials: $$(x)_{m,\lambda} = x(x-\lambda)(x-2\lambda)\cdots (x-(m-1)\lambda), \quad (x)_{0,\lambda}=1$$

$$\langle x \rangle_{m,\lambda} = x(x+\lambda)(x+2\lambda)\cdots (x+(m-1)\lambda), \quad \langle x \rangle_{0,\lambda}=1.$$

Two key identities relate the initial and final rows: $$\begin{aligned} \mathrm{(i)} \quad & a_{0,n}(\lambda) = \sum_{k=0}^n \binom{n}{k} (1-\lambda)_{n-k,\lambda} \, a_{k,0}(\lambda) \ \mathrm{(ii)} \quad & a_{n,0}(\lambda) = \sum_{k=0}^n \binom{n}{k} (-1)^{n-k} \langle 1-\lambda \rangle_{n-k,\lambda} \, a_{0,k}(\lambda) \end{aligned}$$ These are established by induction, using properties such as $$(x+y)_{m,\lambda}=\sum_{j=0}^m\binom{m}{j}(x)_{j,\lambda}(y)_{m-j,\lambda}$$.

4. Derivation and Statement of the Degenerate Seidel’s Formula

Define the initial exponential generating function as

$$S_\lambda(t) = \sum_{n=0}^\infty a_{n,0}(\lambda)\, \frac{t^n}{n!}.$$

Consider the series

$$e_\lambda^{1-\lambda}(t) = \sum_{m=0}^\infty (1-\lambda)_{m,\lambda} \frac{t^m}{m!},$$

which plays the role of a "degenerate exponential" [Editor's term]. The convolution product yields: $$e_\lambda^{1-\lambda}(t) S_\lambda(t) = \sum_{n=0}^\infty a_{0,n}(\lambda) \frac{t^n}{n!}.$$ Therefore, the degenerate Seidel’s formula is

$$\boxed{ \overline{S}_\lambda(t) = e_\lambda^{1-\lambda}(t)\, S_\lambda(t) }$$

where $$\overline{S}_\lambda(t) = \sum_{n\ge0} a_{0,n}(\lambda)\, \frac{t^n}{n!}$$. Expanded,

$$\sum_{n=0}^\infty a_{0,n}(\lambda)\, \frac{t^n}{n!} = \left(\sum_{m=0}^\infty (1-\lambda)_{m,\lambda} \frac{t^m}{m!}\right) \left(\sum_{k=0}^\infty a_{k,0}(\lambda)\, \frac{t^k}{k!}\right).$$

5. Limiting Cases and Degenerations

• Classical Limit $\lambda \to 0$: $(1-\lambda)_{m,\lambda} \to 1$ for all $m$, so $e_\lambda^{1-\lambda}(t) \to e^t$, recovering the classical Seidel formula:

$$\overline{S}_\lambda(t) \to e^t S_0(t) = \overline{A}_2(t).$$

• Case $\lambda = 1$: $(1-\lambda)_{m,\lambda}$ becomes $0$ for $m > 0$, i.e., $(0)_{m,1} = \delta_{m,0}$, so $e_1^{0}(t) = 1$. The transformation is trivial and leaves the exponential generating function unchanged, with initial and final sequences coinciding. The recurrence reduces to $a_{n,k}(1) = a_{n,k-1}(1) + a_{n+1,k-1}(1)$ with zero weight.

6. Application: Degenerate Bell Numbers

The degenerate Bell numbers $\{\phi_{n,\lambda}\}_{n\ge0}$ have exponential generating function

$$S_\lambda(t) = \sum_{n\ge0} \phi_{n,\lambda} \frac{t^n}{n!} = \exp\left(e_\lambda(t)-1\right).$$

Applying the degenerate Seidel’s formula gives: $$\sum_{n\ge0} a_{0,n}(\lambda) \frac{t^n}{n!} = e_\lambda^{1-\lambda}(t)\, \exp\left(e_\lambda(t) - 1\right) = \frac{d}{dt} \left[\exp\left(e_\lambda(t) - 1\right)\right] = \sum_{n\ge0} \phi_{n+1,\lambda} \frac{t^n}{n!}.$$ Thus the final row is $a_{0,n}(\lambda) = \phi_{n+1,\lambda}$. Equating coefficients in the binomial identity reproduces the recurrence: $$\phi_{n+1,\lambda} = \sum_{k=0}^n \binom{n}{k} (1-\lambda)_{n-k,\lambda} \,\phi_{k,\lambda}, \quad n\ge0,$$ matching Theorem 2.6 in Kim & Kim (Kim et al., 14 Dec 2025). This outcome demonstrates that the degenerate Seidel’s formula not only generalizes classical transforms but also systematizes the generation of identities for degenerate combinatorial sequences.

7. Significance and Scope of the Degenerate Seidel’s Formula

The degenerate Seidel’s formula extends the classical Euler–Seidel method to a broader class of combinatorial arrays parameterized by $\lambda$. It systematically relates initial and final sequences in the Euler–Seidel framework and provides transformation identities for exponential generating functions involving degenerate factorial polynomials. The formula’s degenerations recover both the classical Seidel's result and trivial transformation cases. Its applications to degenerate Bell numbers illustrate how the approach yields nontrivial recursions and new expressions for degenerate combinatorial numbers and polynomials (Kim et al., 14 Dec 2025). A plausible implication is that analogous transformations may be constructed for other classes of special numbers via suitable choices of initial sequences and deformation parameters.