---
title: New Degenerate Simsek Numbers
url: https://www.emergentmind.com/topics/new-type-degenerate-simsek-numbers
type: topic
---

# New Degenerate Simsek Numbers

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 [2508.11720].

## 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 $k$, producing a triangular array analogous to other special-number tables in combinatorial and analytic number theory [2508.11720].

## 2. Coefficient extraction and explicit formulas

The generating function is expanded by rewriting the degenerate factorial in shifted form:
$$
(\lambda e^t+1)_{k,\alpha}
=
\alpha^k\Big(\frac{\lambda e^t+1}{\alpha}\Big)_k,
$$
where $(x)_k$ is the ordinary falling factorial. The ordinary falling factorial is then expanded via Stirling numbers of the first kind:
$$
\Big(\frac{\lambda e^t+1}{\alpha}\Big)_k
=
\sum_{\ell=0}^k S_1(k,\ell)\Big(\frac{\lambda e^t+1}{\alpha}\Big)^\ell.
$$
Each power is subsequently expanded by the binomial theorem:
$$
(\lambda e^t+1)^\ell
=
\sum_{j=0}^\ell \binom{\ell}{j}\lambda^j e^{jt}
=
\sum_{j=0}^\ell \binom{\ell}{j}\lambda^j \sum_{n=0}^\infty j^n\frac{t^n}{n!}.
$$

Comparing coefficients yields the principal explicit formula:
$$
y_{1,\alpha}^*(n,k;\lambda)
=
\frac{1}{k!}\sum_{\ell=0}^k\sum_{j=0}^\ell
\binom{\ell}{j}\,
\alpha^{k-\ell}\,
S_1(k,\ell)\,
\lambda^j\,
j^n.
$$
This formula exhibits the new type degenerate Simsek numbers as a double Stirling–binomial transform of the monomials $j^n$.

A second explicit form follows from the convolution identity
$$
(x+y)_{k,\alpha}
=
\sum_{j=0}^k \binom{k}{j}(x)_{j,\alpha}(y)_{k-j,\alpha},
$$
leading to
$$
y_{1,\alpha}^*(n,k;\lambda)
=
\frac{1}{k!}
\sum_{\ell=0}^k\sum_{j=0}^\ell
\binom{k}{\ell}\,
\alpha^{\ell-j}\,
\lambda^j\,
j^n\,
(1)_{k-\ell,\alpha}\,
S_1(\ell,j).
$$
These identities show that the new family is algebraically controlled by the interaction between degeneracy, encoded by $\alpha$, and Stirling-type basis changes [2508.11720].

## 3. Recurrence structure

The paper establishes two recurrence relations, one in the upper index $k$ and one in the lower index $n$.

For $k\ge 0$, the recurrence in $k$ is
$$
y_{1,\alpha}^*(n,k+1;\lambda)
=
\frac{1}{k+1}
\left(
\lambda \sum_{\ell=0}^n \binom{n}{\ell} y_{1,\alpha}^*(\ell,k;\lambda)
+
(1-k\alpha)\,y_{1,\alpha}^*(n,k;\lambda)
\right).
$$
This is obtained from the functional relation
$$
F_k(t)=\frac{\lambda e^t+1-k\alpha}{k}\,F_{k-1}(t)
$$
by coefficient comparison.

For $k\ge 1$, the recurrence in $n$ is
$$
\begin{aligned}
y_{1,\alpha}^*(n+1,k;\lambda)
&=
\frac{\lambda}{k}\sum_{j=0}^{n}\binom{n}{j}
\Bigl[
y_{1,\alpha}^*(j,k-1;\lambda)+y_{1,\alpha}^*(j+1,k-1;\lambda)
\Bigr] \\
&\quad+
\frac{1+\alpha-k\alpha}{k}\,
y_{1,\alpha}^*(n+1,k-1;\lambda),
\end{aligned}
$$
with boundary condition
$$
y_{1,\alpha}^*(n,0;\lambda)=\delta_{n0}.
$$
This relation is derived by differentiating the $k\to k-1$ functional equation with respect to $t$ and comparing coefficients.

The recurrences are computationally significant because they provide two complementary ways to generate the triangular array: one by stepping in $k$, the other by stepping in $n$. The data also states that, in practice, one computes each row either by the recurrence above or by the double-sum explicit formula [2508.11720].

## 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:
$$
\sum_{j=0}^k \alpha^{k-j}S_1(k,j)\lambda^j j^n
=
\sum_{\ell=0}^k (-1)_{k-\ell,\alpha}\,\ell!\,\binom{k}{\ell}\,y_{1,\alpha}^*(n,\ell;\lambda),
$$
where
$$
(a)_{m,\alpha}=a(a-\alpha)\cdots(a-(m-1)\alpha).
$$

The family is also connected to degenerate Stirling numbers of the second kind $S_{2,\alpha}(k,\ell)$ and to the original Simsek numbers $y_1(n,j;\lambda)$ through
$$
y_{1,\alpha}^*(n,k;\lambda)
=
\frac{1}{k!}
\sum_{\ell=0}^k\sum_{j=0}^\ell
S_{2,\alpha}(k,\ell)\,S_1(\ell,j)\,j!\,y_1(n,j;\lambda).
$$
A further identity links the numbers to the new-type degenerate Stirling numbers
$S_2^*(n,k\mid \tfrac{\alpha}{\lambda})$:
$$
y_{1,\alpha}^*(n,k;\lambda)
=
\frac{1}{k!}
\sum_{j=0}^k
\binom{k}{j}j!\,\lambda^j\,(\lambda+1)_{k-j,\alpha}\,
S_2^*\Bigl(n,k\Big|\frac{\alpha}{\lambda}\Bigr).
$$

These formulas place $y_{1,\alpha}^*(n,k;\lambda)$ 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 [2508.11720].

## 5. Relations with Bernoulli and Apostol–Euler families

The generating function admits a reformulation in terms of Bernoulli polynomials of order $k$:
$$
F_k(t)
=
\frac{\alpha^k}{k!}
\Bigl(\frac{\lambda e^t+1}{\alpha}\Bigr)_k
=
\frac{\alpha^k}{k!}\,
B_k^{(k+1)}\!\Bigl(\frac{\lambda e^t+1}{\alpha}+1\Bigr).
$$
Accordingly, the numbers $y_{1,\alpha}^*(n,k;\lambda)$ can be written in terms of the higher-order Bernoulli numbers $B_m^{(r)}$.

The paper also introduces degenerate Apostol–Euler numbers of order $r$ through
$$
\Bigl(\frac{2}{\lambda e^{\frac{\log(1+\alpha t)}{\alpha}}+1}\Bigr)^r
=
\sum_{m\ge 0} E_m^{(r)}(\lambda\mid \alpha)\frac{t^m}{m!}.
$$
From a mixed generating function in $n$ and an auxiliary variable $x$, it derives bilinear relations between $y_{1,\alpha}^*(n,k;\lambda)$, $\phi_n^*(x\mid \alpha,\lambda)$, and the numbers $E_m^{(r)}(\lambda\mid \alpha)$.

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 [2508.11720].

## 6. Special cases, limiting behavior, and representative values

Several specializations are given explicitly.

For $\alpha=0$,
$$
y_{1,0}^*(n,k;\lambda)
=
y_1(n,k;\lambda)
=
\frac{1}{k!}\sum_{j=0}^k \binom{k}{j} j^n \lambda^j.
$$
More generally, since
$$
\lim_{\alpha\to 0}(x)_{k,\alpha}=x^k,
$$
one recovers the classical Simsek numbers:
$$
y_{1,\alpha}^*(n,k;\lambda)\xrightarrow[\alpha\to 0]{}\frac{1}{k!}\sum_{j=0}^k \binom{k}{j}j^n\lambda^j=y_1(n,k;\lambda).
$$

For $\lambda=1$,
$$
y_{1,\alpha}^*(n,k;1)
=
\frac{1}{k!}
\sum_{\ell=0}^k\sum_{j=0}^\ell
\binom{\ell}{j}\alpha^{k-\ell}S_1(k,\ell)\,j^n,
$$
which the source describes as a symmetric function in $j$.

As $\lambda\to 0$,
$$
y_{1,\alpha}^*(n,k;0)
=
\frac{1}{k!}\,\alpha^k\,S_1(k,0)\,0^n
=
\delta_{k,0}\,\delta_{n,0}.
$$

The source also states a classical recovery: if one further sets $\alpha\to 0$ after $\lambda\to 1$, 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
$$
y_{1,\alpha}^*(1,1;\lambda)=\lambda+1,
\qquad
y_{1,\alpha}^*(2,2;\lambda)=\frac{1}{2!}[\alpha+2\lambda+1]\lambda.
$$
The source notes that further entries for $0\le n\le 5$ and $0\le k\le n$ may be filled in from the recurrences or from the double-sum formula [2508.11720].

## 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 $y_{1,\alpha}^*(n,k;\lambda)$; 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 $\alpha$. 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
$$
(1+\alpha t)^{(\lambda e^t+1)/\alpha},
$$
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 $(\alpha,\beta)$-degenerate version, and investigating the zeros and asymptotics of the triangle $y_{1,\alpha}^*(n,k;\lambda)$. 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 [2508.11720].

Source: https://www.emergentmind.com/topics/new-type-degenerate-simsek-numbers