Algorithm B: λ-Deformed Sequence Transformation

• Algorithm B is a λ-deformed recursive transformation on an initial sequence that produces a final sequence from the first column of a doubly indexed array.
• It employs an explicit formula using degenerate Stirling numbers of the second kind to bridge ordinary and exponential generating functions.
• Special cases of Algorithm B yield classical degenerate sequences such as Bernoulli, Euler, and Bell numbers, underscoring its combinatorial significance.

Algorithm B, in the sense developed for degenerate Bernoulli and Euler numbers, is a recursive transformation on sequences that depends on a real parameter $\lambda \ne 0$. Starting from an initial sequence $(a_n(\lambda))_{n\geq 0}$, it generates a doubly indexed array $(a_{n,m}(\lambda))_{n,m\geq 0}$ and extracts a “final sequence” from the first entry of each row. In the degenerate setting, this construction yields explicit formulas in terms of degenerate Stirling numbers of the second kind, establishes a direct correspondence between ordinary and exponential generating functions, and produces degenerate Bernoulli, Euler, and Bell numbers for specific initial conditions (Kim et al., 1 Jan 2026).

1. Definition and recursive construction

Let $(a_n(\lambda))_{n\geq 0}$ be a given initial sequence, with $\lambda$ a real parameter and $\lambda \ne 0$. The degenerate B-algorithm defines the matrix $(a_{n,m}(\lambda))_{n,m\geq 0}$ by

$a_{0,n}(\lambda) = a_n(\lambda),\quad (n \geq 0),$

and, for $n\geq 1$, $m\geq 0$,

$(a_n(\lambda))_{n\geq 0}$0

The quantity

$(a_n(\lambda))_{n\geq 0}$1

is the first entry in the $(a_n(\lambda))_{n\geq 0}$2-th row and is called the final sequence (Kim et al., 1 Jan 2026).

The parameter $(a_n(\lambda))_{n\geq 0}$3 deforms the classical B-algorithm through the term $(a_n(\lambda))_{n\geq 0}$4. The recurrence reduces to the classical B-algorithm when $(a_n(\lambda))_{n\geq 0}$5. In the presentation of the paper, the degenerate B-algorithm is one of two $(a_n(\lambda))_{n\geq 0}$6-deformed recurrences introduced in parallel with a degenerate A-algorithm, but its distinguishing feature is the row recursion above and the extraction of the first-column sequence as the principal output.

2. Closed form and the role of degenerate Stirling numbers

A central result is an explicit formula for the final sequence: $(a_n(\lambda))_{n\geq 0}$7 Here $(a_n(\lambda))_{n\geq 0}$8 are the degenerate Stirling numbers of the second kind (Kim et al., 1 Jan 2026).

These numbers are defined through the expansion

$(a_n(\lambda))_{n\geq 0}$9

where

$(a_{n,m}(\lambda))_{n,m\geq 0}$0

and

$(a_{n,m}(\lambda))_{n,m\geq 0}$1

The explicit formula shows that the algorithm is not merely recursive: it is a linear transform from the initial sequence $(a_{n,m}(\lambda))_{n,m\geq 0}$2 to the final sequence $(a_{n,m}(\lambda))_{n,m\geq 0}$3, with coefficients given by $(a_{n,m}(\lambda))_{n,m\geq 0}$4. This is the mechanism by which combinatorial information encoded by degenerate Stirling numbers enters the output of the algorithm.

3. Generating-function correspondence

The degenerate B-algorithm also induces a functional relation between the generating functions of the initial and final sequences. If

$(a_{n,m}(\lambda))_{n,m\geq 0}$5

is the ordinary generating function of the initial sequence, and

$(a_{n,m}(\lambda))_{n,m\geq 0}$6

is the exponential generating function of the final sequence, then

$(a_{n,m}(\lambda))_{n,m\geq 0}$7

where

$(a_{n,m}(\lambda))_{n,m\geq 0}$8

is the degenerate exponential function. Conversely,

$(a_{n,m}(\lambda))_{n,m\geq 0}$9

where $(a_n(\lambda))_{n\geq 0}$0 is the compositional inverse of $(a_n(\lambda))_{n\geq 0}$1 (Kim et al., 1 Jan 2026).

This generating-function pair gives the algorithm a functional interpretation: it transforms an ordinary generating function into an exponential generating function by the substitution $(a_n(\lambda))_{n\geq 0}$2, and recovers the initial ordinary generating function through the inverse substitution $(a_n(\lambda))_{n\geq 0}$3. The paper describes this as a $(a_n(\lambda))_{n\geq 0}$4-deformed transformation connecting ordinary and exponential types.

4. Canonical specializations

For several specific initial sequences, the final sequence coincides with named degenerate special numbers. The paper isolates degenerate Bernoulli numbers, degenerate Euler numbers, and degenerate Bell numbers as principal examples (Kim et al., 1 Jan 2026).

Initial sequence $(a_n(\lambda))_{n\geq 0}$5	Final sequence	Formula
$(a_n(\lambda))_{n\geq 0}$6	$(a_n(\lambda))_{n\geq 0}$7	$(a_n(\lambda))_{n\geq 0}$8
$(a_n(\lambda))_{n\geq 0}$9	$\lambda$0	$\lambda$1
$\lambda$2 for $\lambda$3	$\lambda$4	$\lambda$5

For the degenerate Bernoulli numbers,

$\lambda$6

and the final sequence becomes

$\lambda$7

For the degenerate Euler numbers,

$\lambda$8

and hence

$\lambda$9

The corresponding generating function is obtained from

$\lambda \ne 0$0

as

$\lambda \ne 0$1

For the degenerate Bell numbers, if

$\lambda \ne 0$2

then

$\lambda \ne 0$3

These specializations exhibit the algorithm as a uniform production mechanism for several classical degenerate sequences.

5. Additional identities and structural interpretation

The paper also records relations involving the degenerate Stirling numbers of the first kind $\lambda \ne 0$4. For the degenerate Bernoulli numbers,

$\lambda \ne 0$5

and for the degenerate Euler numbers,

$\lambda \ne 0$6

These formulas connect the outputs of the degenerate B-algorithm to the first-kind degenerate Stirling numbers through triangular identities (Kim et al., 1 Jan 2026).

In the summary of the paper, the degenerate B-algorithm is described as recursively transforming an initial sequence using $\lambda \ne 0$7-dependent rules, encoding combinatorial transformations through degenerate Stirling numbers, and generalizing the mapping from an ordinary generating function to an exponential generating function under a $\lambda \ne 0$8-deformed transformation. A plausible implication is that the algorithm should be viewed not only as a recurrence but also as a structured operator on sequence classes, with special-number families arising as distinguished orbits under particular initial data.

6. Terminological scope and disambiguation

The designation “Algorithm B” is not unique across the literature. In a different context, it refers to a hierarchical NeuroBayes-based full reconstruction algorithm for $\lambda \ne 0$9 mesons at Belle, with a 4-stage hierarchy, 1104 exclusive decay channels, and 71 neural networks (Feindt et al., 2011). In the study of binary de Bruijn sequences, “Algorithm B” is identified with the Prefer-One greedy construction and is presented as a special case of the Generalized Prefer-Opposite algorithm (Chang et al., 2019).

This broader usage makes context essential. Within the literature on degenerate Bernoulli and Euler numbers, however, Algorithm B denotes the $(a_{n,m}(\lambda))_{n,m\geq 0}$0-parametrized sequence transformation defined by the recurrence for $(a_{n,m}(\lambda))_{n,m\geq 0}$1 and characterized by explicit Stirling-number expansions and generating-function identities (Kim et al., 1 Jan 2026).