New Degenerate Simsek Numbers

• The paper introduces new type degenerate Simsek numbers defined via a degenerate exponential generating function and the falling factorial, establishing explicit formulas.
• Key methodology employs Stirling numbers and binomial expansions to transform monomials and relate the new numbers to classical combinatorial sequences.
• Implications include potential applications in colored set partitions and deformed probability distributions, with open problems on combinatorial interpretations and asymptotic behavior.

New type degenerate Simsek numbers are a family of special numbers introduced through a degenerate exponential generating function that differs from the degenerate Simsek numbers studied previously. For parameters $\alpha,\lambda \in \mathbb{C}$ and indices $n,k \in \mathbb{N}_0$, the numbers are denoted by $y_{1,\alpha}^*(n,k;\lambda)$ and are defined in terms of the degenerate falling factorial. The construction yields explicit formulas, recurrence relations, and identities linking the new family to Stirling numbers, classical Simsek numbers, higher-order Bernoulli numbers, and degenerate Apostol–Euler numbers (Oussi, 14 Aug 2025).

1. Definition through the degenerate generating function

Let

$$(x)_{0,\alpha}=1,\qquad (x)_{n,\alpha}=x(x-\alpha)(x-2\alpha)\cdots(x-(n-1)\alpha)\quad (n\ge 1).$$

This is the degenerate falling factorial used throughout the theory.

The new type degenerate Simsek numbers $y_{1,\alpha}^*(n,k;\lambda)$ are defined by

$$\frac{(\lambda e^t+1)_{k,\alpha}}{k!} = \sum_{n=0}^\infty y_{1,\alpha}^*(n,k;\lambda)\,\frac{t^n}{n!}.$$

Equivalently, with

$$F_k(t;\alpha,\lambda):=\frac{(\lambda e^t+1)_{k,\alpha}}{k!},$$

one has

$$F_k(t;\alpha,\lambda)=\sum_{n=0}^\infty y_{1,\alpha}^*(n,k;\lambda)\,\frac{t^n}{n!}.$$

The initial values follow immediately:

$$y_{1,\alpha}^*(0,k;\lambda)=\frac{(\lambda+1)_{k,\alpha}}{k!}, \qquad y_{1,\alpha}^*(n,0;\lambda)=\delta_{n,0}.$$

In this form, the family is indexed simultaneously by the coefficient order $n$ and the upper parameter $n,k \in \mathbb{N}_0$0, producing a triangular array analogous to other special-number tables in combinatorial and analytic number theory (Oussi, 14 Aug 2025).

2. Coefficient extraction and explicit formulas

The generating function is expanded by rewriting the degenerate factorial in shifted form:

$n,k \in \mathbb{N}_0$1

where $n,k \in \mathbb{N}_0$2 is the ordinary falling factorial. The ordinary falling factorial is then expanded via Stirling numbers of the first kind:

$n,k \in \mathbb{N}_0$3

Each power is subsequently expanded by the binomial theorem:

$n,k \in \mathbb{N}_0$4

Comparing coefficients yields the principal explicit formula:

$n,k \in \mathbb{N}_0$5

This formula exhibits the new type degenerate Simsek numbers as a double Stirling–binomial transform of the monomials $n,k \in \mathbb{N}_0$6.

A second explicit form follows from the convolution identity

$n,k \in \mathbb{N}_0$7

leading to

$n,k \in \mathbb{N}_0$8

These identities show that the new family is algebraically controlled by the interaction between degeneracy, encoded by $n,k \in \mathbb{N}_0$9, and Stirling-type basis changes (Oussi, 14 Aug 2025).

3. Recurrence structure

The paper establishes two recurrence relations, one in the upper index $y_{1,\alpha}^*(n,k;\lambda)$0 and one in the lower index $y_{1,\alpha}^*(n,k;\lambda)$1.

For $y_{1,\alpha}^*(n,k;\lambda)$2, the recurrence in $y_{1,\alpha}^*(n,k;\lambda)$3 is

$y_{1,\alpha}^*(n,k;\lambda)$4

This is obtained from the functional relation

$y_{1,\alpha}^*(n,k;\lambda)$5

by coefficient comparison.

For $y_{1,\alpha}^*(n,k;\lambda)$6, the recurrence in $y_{1,\alpha}^*(n,k;\lambda)$7 is

$y_{1,\alpha}^*(n,k;\lambda)$8

with boundary condition

$y_{1,\alpha}^*(n,k;\lambda)$9

This relation is derived by differentiating the $$(x)_{0,\alpha}=1,\qquad (x)_{n,\alpha}=x(x-\alpha)(x-2\alpha)\cdots(x-(n-1)\alpha)\quad (n\ge 1).$$0 functional equation with respect to $$(x)_{0,\alpha}=1,\qquad (x)_{n,\alpha}=x(x-\alpha)(x-2\alpha)\cdots(x-(n-1)\alpha)\quad (n\ge 1).$$1 and comparing coefficients.

The recurrences are computationally significant because they provide two complementary ways to generate the triangular array: one by stepping in $$(x)_{0,\alpha}=1,\qquad (x)_{n,\alpha}=x(x-\alpha)(x-2\alpha)\cdots(x-(n-1)\alpha)\quad (n\ge 1).$$2, the other by stepping in $$(x)_{0,\alpha}=1,\qquad (x)_{n,\alpha}=x(x-\alpha)(x-2\alpha)\cdots(x-(n-1)\alpha)\quad (n\ge 1).$$3. The data also states that, in practice, one computes each row either by the recurrence above or by the double-sum explicit formula (Oussi, 14 Aug 2025).

4. Structural identities and links with Stirling-type arrays

A central identity relates the new type degenerate Simsek numbers to Stirling numbers of the first kind:

$$(x)_{0,\alpha}=1,\qquad (x)_{n,\alpha}=x(x-\alpha)(x-2\alpha)\cdots(x-(n-1)\alpha)\quad (n\ge 1).$$4

where

$$(x)_{0,\alpha}=1,\qquad (x)_{n,\alpha}=x(x-\alpha)(x-2\alpha)\cdots(x-(n-1)\alpha)\quad (n\ge 1).$$5

The family is also connected to degenerate Stirling numbers of the second kind $$(x)_{0,\alpha}=1,\qquad (x)_{n,\alpha}=x(x-\alpha)(x-2\alpha)\cdots(x-(n-1)\alpha)\quad (n\ge 1).$$6 and to the original Simsek numbers $$(x)_{0,\alpha}=1,\qquad (x)_{n,\alpha}=x(x-\alpha)(x-2\alpha)\cdots(x-(n-1)\alpha)\quad (n\ge 1).$$7 through

$$(x)_{0,\alpha}=1,\qquad (x)_{n,\alpha}=x(x-\alpha)(x-2\alpha)\cdots(x-(n-1)\alpha)\quad (n\ge 1).$$8

A further identity links the numbers to the new-type degenerate Stirling numbers $$(x)_{0,\alpha}=1,\qquad (x)_{n,\alpha}=x(x-\alpha)(x-2\alpha)\cdots(x-(n-1)\alpha)\quad (n\ge 1).$$9:

$y_{1,\alpha}^*(n,k;\lambda)$0

These formulas place $y_{1,\alpha}^*(n,k;\lambda)$1 within a network of Stirling-type transforms. In particular, the construction is not isolated: it is tied to both first-kind and second-kind degenerate Stirling numbers, and it interpolates back to the original Simsek numbers via explicit coefficient identities rather than only by limiting arguments (Oussi, 14 Aug 2025).

5. Relations with Bernoulli and Apostol–Euler families

The generating function admits a reformulation in terms of Bernoulli polynomials of order $y_{1,\alpha}^*(n,k;\lambda)$2:

$y_{1,\alpha}^*(n,k;\lambda)$3

Accordingly, the numbers $y_{1,\alpha}^*(n,k;\lambda)$4 can be written in terms of the higher-order Bernoulli numbers $y_{1,\alpha}^*(n,k;\lambda)$5.

The paper also introduces degenerate Apostol–Euler numbers of order $y_{1,\alpha}^*(n,k;\lambda)$6 through

$y_{1,\alpha}^*(n,k;\lambda)$7

From a mixed generating function in $y_{1,\alpha}^*(n,k;\lambda)$8 and an auxiliary variable $y_{1,\alpha}^*(n,k;\lambda)$9, it derives bilinear relations between $$\frac{(\lambda e^t+1)_{k,\alpha}}{k!} = \sum_{n=0}^\infty y_{1,\alpha}^*(n,k;\lambda)\,\frac{t^n}{n!}.$$0, $$\frac{(\lambda e^t+1)_{k,\alpha}}{k!} = \sum_{n=0}^\infty y_{1,\alpha}^*(n,k;\lambda)\,\frac{t^n}{n!}.$$1, and the numbers $$\frac{(\lambda e^t+1)_{k,\alpha}}{k!} = \sum_{n=0}^\infty y_{1,\alpha}^*(n,k;\lambda)\,\frac{t^n}{n!}.$$2.

These relations indicate that the new type degenerate Simsek numbers participate in a broader algebra of degenerate special functions. The Bernoulli and Apostol–Euler links are structural rather than merely formal: they arise from direct generating-function manipulations and coefficient comparison (Oussi, 14 Aug 2025).

6. Special cases, limiting behavior, and representative values

Several specializations are given explicitly.

For $$\frac{(\lambda e^t+1)_{k,\alpha}}{k!} = \sum_{n=0}^\infty y_{1,\alpha}^*(n,k;\lambda)\,\frac{t^n}{n!}.$$3,

$$\frac{(\lambda e^t+1)_{k,\alpha}}{k!} = \sum_{n=0}^\infty y_{1,\alpha}^*(n,k;\lambda)\,\frac{t^n}{n!}.$$4

More generally, since

$$\frac{(\lambda e^t+1)_{k,\alpha}}{k!} = \sum_{n=0}^\infty y_{1,\alpha}^*(n,k;\lambda)\,\frac{t^n}{n!}.$$5

one recovers the classical Simsek numbers:

$$\frac{(\lambda e^t+1)_{k,\alpha}}{k!} = \sum_{n=0}^\infty y_{1,\alpha}^*(n,k;\lambda)\,\frac{t^n}{n!}.$$6

For $$\frac{(\lambda e^t+1)_{k,\alpha}}{k!} = \sum_{n=0}^\infty y_{1,\alpha}^*(n,k;\lambda)\,\frac{t^n}{n!}.$$7,

$$\frac{(\lambda e^t+1)_{k,\alpha}}{k!} = \sum_{n=0}^\infty y_{1,\alpha}^*(n,k;\lambda)\,\frac{t^n}{n!}.$$8

which the source describes as a symmetric function in $$\frac{(\lambda e^t+1)_{k,\alpha}}{k!} = \sum_{n=0}^\infty y_{1,\alpha}^*(n,k;\lambda)\,\frac{t^n}{n!}.$$9.

As $$F_k(t;\alpha,\lambda):=\frac{(\lambda e^t+1)_{k,\alpha}}{k!},$$0,

$$F_k(t;\alpha,\lambda):=\frac{(\lambda e^t+1)_{k,\alpha}}{k!},$$1

The source also states a classical recovery: if one further sets $$F_k(t;\alpha,\lambda):=\frac{(\lambda e^t+1)_{k,\alpha}}{k!},$$2 after $$F_k(t;\alpha,\lambda):=\frac{(\lambda e^t+1)_{k,\alpha}}{k!},$$3, one recovers the ordinary Lah numbers or other special triangular arrays. A plausible implication is that the family interpolates among several known combinatorial tables, although the exact route depends on the order of specialization.

Representative values given explicitly include

$$F_k(t;\alpha,\lambda):=\frac{(\lambda e^t+1)_{k,\alpha}}{k!},$$4

The source notes that further entries for $$F_k(t;\alpha,\lambda):=\frac{(\lambda e^t+1)_{k,\alpha}}{k!},$$5 and $$F_k(t;\alpha,\lambda):=\frac{(\lambda e^t+1)_{k,\alpha}}{k!},$$6 may be filled in from the recurrences or from the double-sum formula (Oussi, 14 Aug 2025).

7. Interpretive status, applications, and open problems

The paper distinguishes established identities from conjectural or prospective interpretations. It does not provide a direct combinatorial interpretation of $$F_k(t;\alpha,\lambda):=\frac{(\lambda e^t+1)_{k,\alpha}}{k!},$$7; rather, by analogy with Simsek’s use of negative-order Euler polynomials, it states that one expects these numbers to count, up to weights, a family of colored set-partitions with a degeneracy parameter $$F_k(t;\alpha,\lambda):=\frac{(\lambda e^t+1)_{k,\alpha}}{k!},$$8. A precise bijection remains open.

Potential applications are also framed prospectively. The numbers are said to enter naturally into expansions of degenerate exponential generating functions of the form

$$F_k(t;\alpha,\lambda):=\frac{(\lambda e^t+1)_{k,\alpha}}{k!},$$9

which appear in deformations of Poisson and negative-binomial distributions. The source further indicates that they may be useful in interpolating the Apostol–Euler polynomials of negative order.

The open problems listed are specific: finding a direct combinatorial model, extending the theory to a two-parameter $$F_k(t;\alpha,\lambda)=\sum_{n=0}^\infty y_{1,\alpha}^*(n,k;\lambda)\,\frac{t^n}{n!}.$$0-degenerate version, and investigating the zeros and asymptotics of the triangle $$F_k(t;\alpha,\lambda)=\sum_{n=0}^\infty y_{1,\alpha}^*(n,k;\lambda)\,\frac{t^n}{n!}.$$1. These directions underscore an important point about the present state of the subject: the analytic and algebraic framework is already explicit, whereas the combinatorial semantics and asymptotic theory remain undeveloped (Oussi, 14 Aug 2025).