On Degenerate Simsek and Stirling Numbers

Lahcen Oussi Mathematical Institute Wrocław University Pl. Grunwaldzki 2 50-384 Wrocław Poland

                 lahcen.oussi@math.uni.wroc.pl

Abstract In this paper, we introduce the degenerate Simsek and Stirling numbers. We derive some related properties and identities, including the recurrence relation. In addition, we establish some relations between these numbers and certain other special numbers.

1 Introduction

Stirling numbers and other special numbers occur in combinatorial problems, discrete probability, the theory of partitions, mathematical physics, etc. They have also been used to construct computational physics and mathematical models. In recent years, various well-known degenerate versions of these special numbers have been investigated. There are several motivations for defining the degenerate versions. In particular, by specializing the parameters of the degenerate versions, we obtain the ordinary ones. Moreover, these degenerate versions have many combinatorial applications, and they appear in many interesting results in different branches of mathematics, such as umbral calculus, probability theory, differential equations, operator algebras, etc. (cf. [12, 15, 17, 26]).

Stirling numbers of the second kind S_{2}(n, k) count the number of partitions of a set of size n into k disjoint, non-empty subsets. They appear as coefficients in the expansion of

x^{n} = ∑_{k=0}^{n} S_{2}(n, k)(x)_{k},  (1)

where

(x)_{n} := \begin{cases}x(x − 1)(x − 2) ⋯ (x − n + 1), & if n ≥ 1; \\ 1, & if n = 0, \end{cases}

denotes the falling factorial (cf. [4, 26]). Moreover, Stirling numbers of the first kind s(n, k) count the number of permutations of a set of size n with exactly k cycles. They appear as coefficients in the expansion of the falling factorial (x)_{n} as follows:

(x)_{n} = ∑_{k=0}^{n} s(n, k)x^{k},  (2)

(cf. [4, 26]).

Throughout this paper, for z ∈ ℂ, we assume that log z denotes the principal branch of the many-valued function with the imaginary part Im(log z), constrained by

−π < Im(log z) ≤ π.

The generating functions of s(n, k) and S_{2}(n, k) are given, respectively, by

\frac{(log(1 + t))^{k}}{k!} = ∑_{n≥k} s(n, k)\frac{t^{n}}{n!}  (3)

and

\frac{(e^{t} − 1)^{k}}{k!} = ∑_{n≥k} S_{2}(n, k)\frac{t^{n}}{n!}.  (4)

Recently, several authors [1, 10, 11, 14, 16, 18, 20, 21, 22, 23, 27, 29] have studied different modifications of Stirling numbers. In particular, the λ-Stirling numbers of the second kind S_{2}(n, k; λ) are defined by the following generating function:

\frac{(λe^{t} − 1)^{k}}{k!} = ∑_{n≥0} S_{2}(n, k; λ)\frac{t^{n}}{n!},  (5)

(cf. [19, 31]). The degenerate Stirling numbers of the second kind S_{2}(n, k|α), were introduced by D. S. Kim and T. Kim [14], by means of the generating function

\frac{(e^{\frac{log(1+αt)}{α}} − 1)^{k}}{k!} = ∑_{n≥0} S_{2}(n, k|α)\frac{t^{n}}{n!}.  (6)

They can be expressed in relation to Stirling numbers of the first and second kind as follows [14]:

S_{2}(n, k|α) = ∑_{j=k}^{n} s(n, j)S_{2}(j, k)α^{n−j}, where 0 ≤ k ≤ n.  (7)

It is worth mentioning that Eq. (6) was first defined by T. Kim [13], dealing with the degenerate Stirling polynomials of the second kind.

The degenerate Stirling numbers of the second kind S_{2}(n, k|α) reduce to Stirling numbers of the second kind S_{2}(n, k), when α → 0. Moreover, Stirling numbers of the second kind S_{2}(n, k) are a special case of exponential partial Bell polynomials B_{n,k}, defined as follows [4]:

\frac{1}{k!}( ∑_{j≥1} x_{j} \frac{t^{j}}{j!})^{k} = ∑_{n≥k} B_{n,k}(x_{1}, …, x_{n−k+1})\frac{t^{n}}{n!}.  (8)

In particular, for x_{j} = 1, ∀j ≥ 1, we obtain the following relation between Stirling numbers of the second kind S_{2}(n, k) and exponential partial Bell polynomials B_{n,k}:

B_{n,k}(1, 1, …, 1) = S_{2}(n, k),

(cf. [4]).

Simsek [28] considered a new family of special numbers y_{1}(n, k; λ) as coefficients of the generating function

F_{y_{1}}(t, k; λ) := \frac{(λe^{t} + 1)^{k}}{k!} = ∑_{n≥0} y_{1}(n, k; λ)\frac{t^{n}}{n!},  (9)

which are essentially related to the many well-known special numbers, for instance, Bernoulli numbers, Fibonacci numbers, Stirling numbers of the second kind, Lucas numbers and the central numbers. Using Eq. (9), the numbers y_{1}(n, k; λ) can be expressed explicitly by the following formula:

y_{1}(n, k; λ) = \frac{1}{k!} ∑_{j=0}^{k} \binom{k}{j}j^{n}λ^{j}.  (10)

Substituting λ = 1 in (10) and setting

B(n, k) := k!y_{1}(n, k; 1) = ∑_{j=0}^{k} \binom{k}{j}j^{n},  (11)

the numbers B(n, k) are related to the numbers of the form a_{k}2^{k}, with a_{k} being a sequence of positive integers. We have the following first cases [28]:

B(0, k) = 2^{k},
B(1, k) = k2^{k−1},
B(2, k) = k(k + 1)2^{k−2}.

The numbers B(n, k) have many combinatorial applications. For instance, Golombek [5] established the following formula:

B(n, k) = \frac{d^{n}}{dt^{n}} (e^{t} + 1)^{k}|_{t=0}.  (12)

Spivey [30] proved a relation between the numbers B(n, k) and Stirling numbers of the second kind S_{2}(n, k) as follows:

B(n, k) = ∑_{j=0}^{k} \binom{k}{j}j!2^{k−j}S_{2}(n, j).  (13)

On the other hand, Simsek [28] derived the following recursive formula:

B(n, k) = \frac{2^{k−n}}{m_{0}} \binom{k}{n} − ∑_{j=1}^{n} \frac{m_{j}}{m_{0}} B(n − j, k),  (14)

where m_{j} ∈ ℚ, for j = 0, 1, …, n. Furthermore, Simsek [28] conjectured that

B(n, k) = 2^{k−n} ∑_{j=0}^{n−1} x_{j}k^{n−j}  (15)

holds for some positive integers x_{0}, x_{1}, …, x_{n−1} , and therefore, stated the following problems:

1. How can we compute the coefficients x_{j}, j = 0, 1, 2, …, n − 1?

2. Is it possible to find functions f_{n}, such that f_{n}(x) = ∑_{k≥0} B(n, k)x^{k}, for |x| > r?

In 2019, Xu [32] solved the above problems using the standard Stirling numbers of the first and second kind, and generalized the result to an arbitrary λ, that is, to the family of numbers B(n, k; λ) := k!y_{1}(n, k; λ).

In this paper, we define the degenerate Simsek numbers y_{1}(n, k; λ|α) and a different version of degenerate Stirling numbers of the second kind S_{2}(n, k; λ|α) (which will be called (α, λ)-Stirling numbers) other than the numbers in (6), as a modification and generalization of Simsek numbers y_{1}(n, k; λ) and Stirling numbers of the second kind S_{2}(n, k). We establish a relation between these numbers and the well-known special numbers: Stirling numbers of the first and second kind, the degenerate Stirling numbers of the second kind S_{2}(n, k|α), first and second kind Apostol-Euler numbers, and λ-Stirling numbers of the second kind. We investigate some basic properties, including a recursive formula for these numbers. Moreover, we extend Xu’s solution for Simsek’s questions [28] to the numbers B(n, k; λ|α) := k!y_{1}(n, k; λ|α).

2 Main results

In this section, we introduce the degenerate Simsek and Stirling numbers by using generating functions. We establish some properties and recurrence relations regarding these numbers. We also introduce the degenerate first and second kind Apostol-type Euler numbers and their relation with the degenerate Simsek numbers.

2.1 Degenerate Simsek and Stirling numbers

For k ∈ ℕ_{0} := {0, 1, 2, …}, λ ∈ ℂ and α ∈ ℝ \ {0}, we define the degenerate Simsek numbers y_{1}(n, k; λ|α) by means of the following generating function:

F_{y_{1}}(t, k; λ|α) := \frac{1}{k!}(λe^{\frac{log(1+αt)}{α}} + 1)^{k} = ∑_{n≥0} y_{1}(n, k; λ|α)\frac{t^{n}}{n!}.  (16)

It follows that, for α → 0, the above identity reduces to (9).

Theorem 1. The degenerate Simsek numbers y_{1}(n, k; λ|α) can be expressed explicitly as follows:

y_{1}(n, k; λ|α) = \frac{1}{k!} ∑_{m=0}^{n} ∑_{j=0}^{k} \binom{k}{j}j^{m}λ^{j}α^{n−m}s(n, m)  (17)

or

y_{1}(n, k; λ|α) = \frac{1}{k!} ∑_{j=0}^{k} \binom{k}{j}λ^{j}α^{n}(\frac{j}{α})_{n}.  (18)

Proof. Using (16), we obtain

∑_{n≥0} y_{1}(n, k; λ|α)\frac{t^{n}}{n!} = \frac{1}{k!}(λe^{\frac{log(1+αt)}{α}} + 1)^{k}  (19)

= \frac{1}{k!} ∑_{j=0}^{k} \binom{k}{j}λ^{j}e^{\frac{j}{α} log(1+αt)}  (20)

= \frac{1}{k!} ∑_{j=0}^{k} \binom{k}{j}λ^{j} ∑_{m≥0} \frac{j^{m}(log(1 + αt))^{m}}{α^{m}m!} .  (21)

By using (3) in (21), we get

∑_{n≥0} y_{1}(n, k; λ|α)\frac{t^{n}}{n!} = \frac{1}{k!} ∑_{n≥0} ∑_{m=0}^{n} ∑_{j=0}^{k} \binom{k}{j}j^{m}λ^{j}α^{n−m}s(n, m)\frac{t^{n}}{n!}.  (22)

Comparing the coefficient of \frac{t^{n}}{n!} yields (17). Similarly, one can prove (18) by writing e^{\frac{log(1+αt)}{α}} = (1 + αt)^{\frac{1}{α}}, and using the binomial expansion.

Remark 2. One can also prove (18) using the combination of (17) and (2).

Example 3. The special cases of the degenerate Simsek numbers are as follows: • y_{1}(0, k; λ|α) = \frac{(λ+1)^{k}}{k!}

• y_{1}(n, 0; λ|α) = δ_{n,0}

• y_{1}(n, 1; λ|α) = δ_{n,0} + λ(\frac{1}{α})_{n}α^{n} Remark 4.

1. If we set λ = −1 in (17) or (18), we obtain the degenerate Stirling numbers of the

second kind

S_{2}(n, k|α) = (−1)^{k}y_{1}(n, k; −1|α).

2. For α → 0, we obtain

y_{1}(n, k; λ|0) := lim_{α→0} y_{1}(n, k; λ|α) = y_{1}(n, k; λ),

which are Simsek numbers given by Eq. (10). If moreover we take λ = −1, then we obtain a relation with Stirling numbers of the second kind as follows:

S_{2}(n, k) = (−1)^{k}y_{1}(n, k; −1|0).

Theorem 5. For any non-negative integers n and k, we have

λ^{k}(\frac{k}{α})_{n}α^{n} = ∑_{l=0}^{k}(−1)^{k−l}l!\binom{k}{l}y_{1}(n, l; λ|α).  (23)

Proof. We have

λ^{k} ∑_{n≥0}(\frac{k}{α})_{n}α^{n} \frac{t^{n}}{n!} = λ^{k}(1 + αt)^{\frac{k}{α}} = ((λe^{\frac{log(1+αt)}{α}} + 1) − 1)^{k}

= ∑_{l=0}^{k} \binom{k}{l}(λe^{\frac{log(1+αt)}{α}} + 1)^{l}(−1)^{k−l}

= ∑_{n≥0} ∑_{l=0}^{k}(−1)^{k−l}\binom{k}{l}l!y_{1}(n, l; λ|α)\frac{t^{n}}{n!},

where in the last equality, we use (16). Hence, the desired result follows from the comparison of the coefficient of \frac{t^{n}}{n!}.

In the following theorem, we establish another explicit formula for the numbers y_{1}(n, k; λ|α) in relation with the degenerate Stirling numbers of the second kind S_{2}(n, k|α).

Theorem 6. Let n be a non-negative integer. Then

y_{1}(n, k; λ|α) = \frac{1}{k!} ∑_{j=0}^{n} \binom{k}{j}j!λ^{j}(λ + 1)^{k−j}S_{2}(n, j|α).  (24)

Proof. By virtue of (6) and (16), we obtain

∑_{n≥0} y_{1}(n, k; λ|α)\frac{t^{n}}{n!} = \frac{1}{k!}(λ(e^{\frac{log(1+αt)}{α}} − 1) + λ + 1)^{k}

= \frac{1}{k!} ∑_{j=0}^{k} \binom{k}{j}λ^{j}(λ + 1)^{k−j}(e^{\frac{log(1+αt)}{α}} − 1)^{j}

=^{by using (6)}\frac{1}{k!} ∑_{n≥0} ∑_{j=0}^{k} \binom{k}{j}λ^{j}j!(λ + 1)^{k−j}S_{2}(n, j|α)\frac{t^{n}}{n!},

and the desired result follows from the comparison of coefficient of \frac{t^{n}}{n!}.

Now, from the viewpoint of Eqs. (5) and (6), we consider the (α, λ)-Stirling numbers of the second kind S_{2}(n, k; λ|α), which are given by the generating function

G_{S_{2}}(t, k; λ|α) := \frac{(λe^{\frac{log(1+αt)}{α}} − 1)^{k}}{k!} = ∑_{n≥0} S_{2}(n, k; λ|α)\frac{t^{n}}{n!}.  (25)

If we substitute λ = 1 in Eq. (25), we obtain the degenerate Stirling numbers of the second kind (6), while for α → 0, we get the λ-Stirling numbers of the second kind (5). Moreover, the (α, λ)-Stirling numbers of the second kind S_{2}(n, k; λ|α) can be reduced to some other well-known special numbers. Let us recall that in this context, the generalized Stirling numbers S(n, k; α, β, r), with αβ ≠ 0, were introduced by Hsu and Shiue [11], and that they satisfy the following identity [11, Thm. 2]:

(1 + αt)^{\frac{r}{α}} (\frac{(1 + αt)^{\frac{β}{α}} − 1}{β}) = k! ∑_{n≥0} S(n, k; α, β, r)\frac{t^{n}}{n!}.  (26)

Letting r = 0 and β = 1 in (26) gives

S(n, k; α, 1, 0) = S_{2}(n, k; 1|α).

The degenerate weighted Stirling numbers of the second kind S(n, k; λ|θ), introduced by Howard [10], by means of the generating function

(1 + θt)^{\frac{λ}{θ}} ((1 + θt)^{\frac{1}{θ}} − 1)^{k} = k! ∑_{n≥k} S(n, k; λ|θ)\frac{t^{n}}{n!},

can be reduced to the (α, λ)-Stirling numbers S_{2}(n, k; λ|α) as follows:

S(n, k; 0|α) = S_{2}(n, k; 1|α).

In addition, Howard [9] introduced the degenerate associated Stirling numbers of the second kind S_{r}(n, k|λ) as follows:

k! ∑_{n≥(r+1)k} S_{r}(n, k|λ)\frac{t^{n}}{n!} = ((1 + λt)^{\frac{1}{λ}} − 1 − G(r, λ))^{k},  (27)

where G(r, λ) = ∑_{i=1}^{r}(1 − λ)(1 − 2λ) ⋯ (1 − (i − 1)λ)\frac{x^{i}}{i!} . The case r = 0 has been investigated

by Carlitz [3]. Therefore, we have the following relation between the degenerate Stirling numbers S_{r}(n, k|λ) and the (α, λ)-Stirling numbers S_{2}(n, k; λ|α):

S_{0}(n, k|α) = S_{2}(n, k; 1|α).

Theorem 7. The (α, λ)-Stirling numbers of the second kind S_{2}(n, k; λ|α) can be expressed by the following formulas:

S_{2}(n, k; λ|α) = \frac{1}{k!} ∑_{m=0}^{n} ∑_{l=0}^{k} λ^{l}\binom{k}{l}l^{m}α^{n−m}(−1)^{k−l}s(n, m)  (28)

or

S_{2}(n, k; λ|α) = \frac{1}{k!} ∑_{l=0}^{k} λ^{l}\binom{k}{l}(\frac{l}{α})_{n}α^{n}(−1)^{k−l}.  (29)

Proof. According to Eq. (25), we have

∑_{n≥0} S_{2}(n, k; λ|α)\frac{t^{n}}{n!} = \frac{1}{k!} ∑_{l=0}^{k} λ^{l}e^{\frac{l log(1+αt)}{α}} (−1)^{k−l}\binom{k}{l}  (30)

= \frac{1}{k!} ∑_{l=0}^{k} λ^{l} ∑_{m≥0} \frac{l^{m}(log(1 + αt))^{m}}{α^{m}m!} (−1)^{k−l}\binom{k}{l}.  (31)

Using (3) in (31), we get

∑_{n≥0} S_{2}(n, k; λ|α)\frac{t^{n}}{n!} = \frac{1}{k!} ∑_{n≥0} ∑_{m=0}^{n} ∑_{l=0}^{k} λ^{l}\binom{k}{l}l^{m}α^{n−m}(−1)^{k−l}s(n, m)\frac{t^{n}}{n!}.

Hence, Eq. (28) follows by comparing the coefficient of \frac{t^{n}}{n!}.

In the following theorem, we provide a relation between the (α, λ)-Stirling numbers of the second kind S_{2}(n, k; λ|α) and the degenerate Simsek numbers y_{1}(n, k; λ|α).

Theorem 8. Let n and k be non-negative integer. Let λ and α be complex number such that α ≠ 0. Then

S_{2}(n, k; λ^{2}|\frac{α}{2}) = \frac{k!}{2^{n}} ∑_{j=0}^{n} \binom{n}{j}S_{2}(j, k; λ|α)y_{1}(n − j, k; λ|α).  (32)

Proof. From Eqs. (16) and (25), we get the following functional relation between the generating functions of the (α, λ)-Stirling numbers of the second kind S_{2}(n, k; λ|α) and the degenerate Simsek numbers y_{1}(n, k; λ|α):

\frac{(λ^{2}e^{\frac{2}{α} log(1+αt)} − 1)^{k}}{k!} = k!G_{S_{2}}(t, k; λ|α)F_{y_{1}}(t, k; λ|α).  (33)

Thus,

\frac{1}{k!} ∑_{l=0}^{k} \binom{k}{l}λ^{2l}e^{\frac{2l}{α} log(1+αt)}(−1)^{k−l} = k! ∑_{n≥0} S_{2,α}(n, k; λ)\frac{t^{n}}{n!} ∑_{n≥0} y_{1}(n, k; λ|α)\frac{t^{n}}{n!}.

Therefore,

\frac{1}{k!} ∑_{n≥0} ∑_{m=0}^{n} ∑_{l=0}^{k} \binom{k}{l}λ^{2l}(−1)^{k−l}2^{m}l^{m}α^{n−m}s(n, m)\frac{t^{n}}{n!} = k! ∑_{n≥0} ∑_{j=0}^{n} \binom{n}{j}S_{2}(j, k; λ|α)

× y_{1}(n − j, k; λ|α)\frac{t^{n}}{n!}.

Equating the coefficient of \frac{t^{n}}{n!} in both sides of the above identity yields

\frac{1}{k!} ∑_{m=0}^{n} ∑_{l=0}^{k} \binom{k}{l}λ^{2l}(−1)^{k−l}2^{m}l^{m}α^{n−m}s(n, m) = k! ∑_{j=0}^{n} \binom{n}{j}S_{2}(n, k; λ|α)y_{1}(n, k; λ|α).

Then, using (28), the left-hand side of the above identity becomes

2^{n}S_{2}(n, k; λ^{2}|\frac{α}{2}),

and the desired result follows.

Remark 9. Similarly, one can prove Theorem 8 using Eqs. (29) and (33), as well as the fact that

\frac{(λ^{2}e^{\frac{2}{α} log(1+αt)} − 1)^{k}}{k!} = \frac{1}{k!}(λ^{2}(1 + αt)^{\frac{2}{α}} − 1)^{k}

= \frac{1}{k!} ∑_{n≥0} ∑_{l=0}^{k} \binom{k}{l}λ^{2l}(−1)^{k−l}(\frac{2l}{α})_{n}α^{n} \frac{t^{n}}{n!}.

It follows that, by letting α → 0, Eq. (32) reduces to the following formula proved by Simsek [28] for λ-Stirling numbers S_{2}(n, k; λ) and Simsek numbers y_{1}(n, k; λ):

S_{2}(n, k; λ^{2}) = \frac{k!}{2^{n}} ∑_{j=0}^{n} \binom{n}{j}S_{2}(j, k; λ)y_{1}(n − j, k; λ).  (34)

Theorem 10. The degenerate Simsek numbers y_{1}(n, k; λ|α) satisfy the following recursive formula:

y_{1}(n + 1, k; λ|α) = (k − αn)y_{1}(n, k; λ|α) − y_{1}(n, k − 1; λ|α).  (35)

Proof. From (16), we have the following partial differential equations:

\frac{∂}{∂t}F_{y_{1}}(t, k; λ|α) = \frac{k}{(1 + αt)}F_{y_{1}}(t, k; λ|α) − \frac{1}{(1 + αt)}F_{y_{1}}(t, k − 1; λ|α)  (36)

and

\frac{∂}{∂t}((1 + αt)F_{y_{1}}(t, k; λ|α)) = ∑_{n≥0} y_{1}(n + 1, k; λ|α)\frac{t^{n}}{n!} + α ∑_{n≥0}(n + 1)y_{1}(n, k; λ|α)\frac{t^{n}}{n!}. (37)

On the other hand, we have

\frac{∂}{∂t}((1 + αt)F_{y_{1}}(t, k; λ|α)) = αF_{y_{1}}(t, k; λ|α) + (1 + αt)\frac{∂}{∂t}F_{y_{1}}(t, k; λ|α)
= α ∑_{n≥0} y_{1}(n, k; λ|α)\frac{t^{n}}{n!} + k ∑_{n≥0} y_{1}(n, k; λ|α)\frac{t^{n}}{n!}  (38)

− ∑_{n≥0} y_{1}(n, k − 1; λ|α)\frac{t^{n}}{n!}.

Comparing the coefficient of \frac{t^{n}}{n!} in both right-hand sides of (37) and (38) completes the proof.

Let us define the numbers B(n, k; λ|α) by the following identity:

B(n, k; λ|α) := k!y_{1}(n, k; λ|α),  (39)

i.e., by means of the generating function

(λe^{\frac{log(1+αt)}{α}} + 1)^{k} = ∑_{n≥0} B(n, k; λ|α)\frac{t^{n}}{n!}.  (40)

Observe that the first terms of B(n, k; 1|α) for fixed n are of the form c_{k}(α)2^{k}, for some function c_{k}(α):

B(0, k; 1|α) = 2^{k},
B(1, k; 1|α) = k2^{k−1},
B(2, k; 1|α) = k(k + 1 − 2α)2^{k−2}.

Remark 11. Note that for α → 0 in Eq. (39), we obtain the numbers B_{0}(n, k; λ) := B(n, k; λ) = k!y_{1}(n, k; λ), considered by Xu [32] and Goubi [7]. If moreover we set λ = 1, then we obtain Eq. (11).

Theorem 12. Let n and k be non-negative integer. Then

B(n, k; λ|α) = λ^{k} ∑_{m=0}^{n} ∑_{j=0}^{m} A_{k,m−j}(\frac{1}{λ})\binom{m}{j}k^{j}s(n, m)α^{n−m},  (41)

where A_{k,m}(a) = ∑_{i=0}^{k} \binom{k}{i}a^{i}(−i)^{m}.

Proof. We have

(λe^{\frac{log(1+αt)}{α}} + 1)^{k} = ∑_{i=0}^{k} \binom{k}{i}λ^{k−i}e^{\frac{k−i}{α} log(1+αt)}

= ∑_{i=0}^{k} \binom{k}{i}λ^{k−i}( ∑_{m≥0} \frac{k^{m}}{α^{m}} \frac{(log(1 + αt))^{m}}{m!} )

× ( ∑_{m≥0} \frac{(−i)^{m}}{α^{m}} \frac{(log(1 + αt))^{m}}{m!} )

= ∑_{n≥0} ∑_{m=0}^{n} ∑_{j=0}^{m} ∑_{i=0}^{k} \binom{k}{i}\binom{m}{j}λ^{k−i}α^{n−m}k^{j}(−i)^{m−j}s(n, m)\frac{t^{n}}{n!}.

Therefore, using (40) and comparing the coefficient of \frac{t^{n}}{n!}, we arrive at (41).

Theorem 13. For any non-negative integers n and k, we have

B(n, k; λ|α) = (λ + 1)^{k−n} ∑_{i=0}^{n} a_{n−i}(α, λ)k^{i},  (42)

where the coefficients a_{n−i}(α, λ) are given by the formula

a_{n−i}(α, λ) = ∑_{j=i}^{n} λ^{j}(λ + 1)^{n−j}s(j, i)S_{2}(n, j|α), for i = 0, …, n.

Proof. It follows from Theorem 6 that

B(n, k; λ|α) = ∑_{j=0}^{n} \binom{k}{j}λ^{j}j!(λ + 1)^{k−j}S_{2}(n, j|α).  (43)

Hence,

B(n, k; λ|α) = ∑_{j=0}^{n}(k)_{j}λ^{j}(λ + 1)^{k−j}S_{2}(n, j|α)

= ∑_{j=0}^{n} ∑_{i=0}^{j} s(j, i)λ^{j}k^{i}(λ + 1)^{k−j}S_{2}(n, j|α)

= (λ + 1)^{k−n} ∑_{i=0}^{n} k^{i} ∑_{j=i}^{n} λ^{j}(λ + 1)^{n−j}S_{2}(n, j|α)s(j, i)

= (λ + 1)^{k−n} ∑_{i=0}^{n} k^{i}a_{n−i}(α, λ),

where in the second equality we use Eq. (2).

Remark 14. Setting λ = 1 in (42) yields

B(n, k; 1|α) = 2^{k−n} ∑_{i=0}^{n} k^{i}a_{n−i}(α, 1),

where the coefficients a_{n−i}(α, 1) can be expressed in terms of Stirling numbers of the first kind, and the degenerate Stirling numbers of the second kind, as follows:

a_{n−i}(α, 1) = ∑_{j=i}^{n} 2^{n−j}s(j, i)S_{2}(n, j|α).

Letting α → 0 in Eq. (42), we obtain

B(n, k; λ|0) = {a_{n}(0, λ) + ka_{n−1}(0, λ) + ⋯ + k^{n}a_{0}(0, λ)}(λ + 1)^{k−n}.

This result was given by Xu [32] as a general answer to the first Simsek open problem [28], regarding the coefficients a_{n}(0, λ).

Let us define

H_{n}(x; λ|α) := ∑_{k≥0} B(n, k; λ|α)x^{k}  (44)

as a generating function of the numbers B(n, k; λ|α). Then we have the resulting theorem.

Theorem 15. For any non-negative integer n, the generating function H_{n}(x; λ|α) is given as follows:

H_{n}(x; λ|α) = \begin{cases}∑_{j=1}^{n} \frac{j!λ^{j}S_{2}(n,j|α)}{(1−x(λ+1))^{j+1}} x^{j}, & for n ≥ 1; \\ \frac{1}{1−x(1+λ)}, & for n = 0. \end{cases}  (45)

For proving the above theorem, we will use the following lemma, given by Goubi [6]. Lemma 16 ([6]). For the formal generating function h(t) := ∑_{n≥0} a_{n}t^{n}, with a_{0} ≠ 0 and a complex number β ≠ 0, we have

h^{β}(t) = a_{0}^{β} + ∑_{n≥1} ∑_{k=1}^{n}(β)_{k}a_{0}^{β−k}B_{n,k}(1!a_{1}, 2!a_{2}, …, (n − k + 1)!a_{n−k+1})\frac{t^{n}}{n!},  (46)

where B_{n,k}(x_{1}, …, x_{n−k+1}) are exponential partial Bell polynomials (8). Proof of Theorem 15. It follows from Eqs. (40) and (44) that

∑_{n≥0} H_{n}(x; λ|α)\frac{t^{n}}{n!} = ∑_{n≥0} ∑_{k≥0} B(n, k; λ|α)x^{k} \frac{t^{n}}{n!}

= ∑_{k≥0}(λ(1 + αt)^{\frac{1}{α}} + 1)^{k}x^{k}

= \frac{1}{1 − x(λ(1 + αt)^{\frac{1}{α}} + 1)}.

Define h(t) := (1 − x(λ + 1)) − λx ∑_{n≥1} (\frac{1}{α})_{n}α^{n} \frac{t^{n}}{n!} and set β = −1. Applying Lemma 16, we obtain

∑_{n≥0} H_{n}(x; λ|α)\frac{t^{n}}{n!} = \frac{1}{1 − x(λ + 1)} + (∑_{n≥1} ∑_{k=1}^{n}(−1)^{k}k!(1 − x(λ + 1))^{−1−k}

× B_{n,k}(−λxα(\frac{1}{α})_{1}, −λxα^{2}(\frac{1}{α})_{2}, …, −λxα^{n−k+1}(\frac{1}{α})_{n−k+1}))\frac{t^{n}}{n!}

= \frac{1}{1 − x(λ + 1)} + ∑_{n≥1}(∑_{k=1}^{n}(−1)^{k}k!(1 − x(λ + 1))^{−1−k}(−λx)^{k}S_{2}(n, k|α))\frac{t^{n}}{n!},

where in the second equality, we use Eq. (8). Hence, Eq. (45) follows. Remark 17. If we let α → 0 in Eq. (45), we get

H_{n}(x; λ|0) = \begin{cases}∑_{j=1}^{n} \frac{j!λ^{j}S_{2}(n,j)}{(1−x(λ+1))^{j+1}} x^{j}, & for n ≥ 1; \\ \frac{1}{1−x(1+λ)}, & for n = 0, \end{cases}

the generating function for B(n, k; λ) := B(n, k; λ|0) given by Goubi [7, 8] and Xu [32] as a general answer to Simsek’s second open question [28], regarding the generating function

H_{n}(x; λ|0) = ∑_{k≥0} B(n, k; λ|0)x^{k}.

According to the formula

k!S_{2}(n, k|α) = ∑_{j=1}^{k} \binom{k}{j}(−1)^{k−j}(\frac{j}{α})_{n}α^{n},  (47)

the generating function H_{n}(x; λ|α) can also be expressed as follows:

H_{n}(x; λ|α) = ∑_{k=1}^{n} ∑_{j=0}^{k} \binom{k}{j}(−1)^{k−j}α^{n}(\frac{j}{α})_{n} \frac{λ^{k}x^{k}}{(1 − x(λ + 1))^{j+1}} .  (48)

Let us observe that the generating function H_{n}(x; λ|α) can be written as a rational function:

H_{n}(x; λ|α) = \frac{P_{n}(x; λ|α)}{Q_{n+1}(x; λ)},  (49)

where

P_{n}(x; λ|α) = \begin{cases}∑_{k=1}^{n} λ^{k}k!S_{2}(n, k|α)(1 − x(λ + 1))^{n−k}x^{k}, & for n ≥ 1; \\ 1, & for n = 0, \end{cases}

and

Q_{n}(x; λ) = \frac{1}{(1 − x(λ + 1))^{n}} .

The examples of H_{n}(x; λ|α) for some fixed values of n are given as follows:

H_{0}(x; λ|α) = \frac{1}{1 − x(1 + λ)},

H_{1}(x; λ|α) = \frac{λx}{(1 − x(1 + λ))^{2}} ,

H_{2}(x; λ|α) = \frac{λ(1 + α)x − (λ(α + 1) − λ^{2}(1 − α))x^{2}}{(1 − x(1 + λ))^{3}} .

Theorem 18. Let n be a non-negative integer. The generating function H_{n}(x; λ|α) satisfies the following recursive formula:

H_{0}(x; λ|α) = \frac{1}{1 − x(λ + 1)}  (50)

and

H_{n}(x; λ|α) = \frac{λx}{1 − x(λ + 1)}( ∑_{k=0}^{n−1} \binom{n}{k}(1|α)_{n−k}H_{k}(x; λ|α)) for n ≥ 1,  (51)

where (x|α)_{n} := x(x − α)(x − 2α) ⋯ (x − (n − 1)α).

Proof. From the definition of H_{n}(x; λ|α), we can derive the following:

∑_{n≥0} H_{n}(x; λ|α)\frac{t^{n}}{n!} = \frac{1}{1 − x(λ + 1)} + ∑_{n≥1} ∑_{j=1}^{n} \frac{j!λ^{j}S_{2}(n, j|α)x^{j}}{(1 − x(λ + 1))^{j+1}} \frac{t^{n}}{n!}

= \frac{1}{1 − x(λ + 1)} + ∑_{j=1}^{∞} j!\frac{(λx)^{j}}{(1 − x(λ + 1))^{j+1}} ∑_{n=j}^{∞} S_{2}(n, j|α)\frac{t^{n}}{n!}

= \frac{1}{1 − x(λ + 1)} + ∑_{j=1}^{∞} \frac{(λx)^{j}}{(1 − x(λ + 1))^{j+1}} (e^{\frac{log(1+αt)}{α}} − 1)^{j}

= \frac{1}{1 − x(λ + 1)} + \frac{1}{(1 − x(λ + 1))}(\frac{1}{1 − ^{\frac{λx(e^{\frac{log(1+αt)}{α}} −1)}{1−x(1+λ)}}} − 1)

= \frac{1}{1 − x − λe^{\frac{log(1+αt)}{α}}} ,

where in the third equality, we use Eq. (6). By multiplying both sides of the above identity by (1 − x − xλe^{\frac{log(1+αt)}{α}}) and comparing the coefficient of \frac{t^{n}}{n!}, we obtain the desired result.

Remark 19. Letting α → 0 in Eqs. (50) and (51), we obtain

H_{0}(x; λ|0) = \frac{1}{1 − x(λ + 1)}

and

H_{n}(x; λ|0) = \frac{λx}{1 − x(λ + 1)}( ∑_{k=0}^{n−1} \binom{n}{k}H_{k,0}(x; λ)), for n ≥ 1.

This result has already been obtained by Goubi [7, 8] and Xu [32]. It answered the second Simsek problem [28], regarding the explicit formula for the generating function H_{n}(x; 1|0) = ∑_{k≥0} B(n, k; 1|0)x^{k}. For instance:

H_{0}(x; 1|0) = \frac{1}{1 − 2x},

H_{1}(x; 1|0) = \frac{x}{(1 − 2x)^{2}} ,

H_{2}(x; 1|0) = \frac{x}{(1 − 2x)^{3}} .

2.2 Degenerate Apostol-type Euler numbers

In this section, we introduce the degenerate first and second kind Apostol-type Euler numbers, and establish a relation between these numbers and the degenerate Simsek numbers y_{1}(n, k; λ|α).

For k ∈ ℕ_{0}, we define the degenerate second kind Apostol-type Euler polynomials E_{n}^{∗(k)}(x; λ|α) of order k as the coefficients of the following Taylor power series:

F_{P} (t, x; k, λ|α) : = (\frac{2}{λe^{\frac{log(1+αt)}{α}} + λ^{−1}e^{\frac{log(1−αt)}{α}}} )^{k}e^{\frac{x}{α} log(1+αt)}

= ∑_{n≥0} E_{n}^{∗(k)}(x; λ|α)\frac{t^{n}}{n!}.

(52)

Substituting x = 0 in Eq. (52), we obtain the degenerate second kind Apostol-Euler numbers of order k, denoted by E_{n}^{∗(k)}(λ|α) and given by the means of the generating function:

F_{E^{∗}}(t; k, λ|α) := (\frac{2}{λe^{\frac{log(1+αt)}{α}} + λ^{−1}e^{\frac{log(1−αt)}{α}}} )^{k} = ∑_{n≥0} E_{n}^{∗(k)}(λ|α)\frac{t^{n}}{n!}.  (53)

Moreover, we define the degenerate first kind Apostol-Euler numbers of order k, denoted by E_{n}^{(k)}(λ|α), as follows:

G_{E}(t; k, λ|α) := (\frac{2}{λe^{\frac{log(1+αt)}{α}} + 1})^{k} = ∑_{n≥0} E_{n}^{(k)}(λ|α)\frac{t^{n}}{n!}.  (54)

Remark 20. It follows that for α → 0, the numbers E_{n}^{(k)}(λ|α) and E_{n}^{∗(k)}(λ|α) reduce to first and second kind Apostol-Euler numbers, respectively (cf. [24, 28]).

The relation between the numbers E_{n}^{(k)}(λ|α) and E_{n}^{∗(k)}(λ|α) is given by the following theorem:

Theorem 21. Let k be a non-negative integer. Then

E_{n}^{∗(−k)}(λ|α) = ∑_{j=0}^{k} \binom{k}{j}2^{j−k}(k − j)! ∑_{m=k−j}^{∞} s(m, k − j)α^{m+j−k}(n)_{m} ∑_{l=0}^{n−m}(−1)^{n−l}  (55)

× \binom{n − m}{l}E_{l}^{(−j)}(λ|α)B_{n−m−l}^{(−k+j)}(λ^{−1}|α),

where

H_{B}(t; k, λ|α) := (\frac{log(1 + αt)}{α(λe^{\frac{log(1+αt)}{α}} − 1)})^{k} = ∑_{n≥0} B_{n}^{(k)}(λ|α)\frac{t^{n}}{n!},

and B_{n}^{(k)}(λ|α) can be considered as the degenerate Apostol-Bernoulli numbers of order k, which reduce, for α → 0, to the well-known Apostol-Bernoulli numbers B_{n}^{(k)}(λ) of order k, defined by the generating function:

H_{B}(t; k, λ) := (\frac{t}{λe^{t} − 1})^{k} = ∑_{n≥0} B_{n}^{(k)}(λ)\frac{t^{n}}{n!},

(cf. [19, 25, 31]).

Proof. According to Eqs. (53) and (54), we get

F_{E^{∗}}(t; −k, λ|α) = ∑_{j=0}^{k} \binom{k}{j}2^{j−k}α^{j−k}( log(1 − αt))^{k−j}G_{E}(t; −j, λ|α)
× H_{B}(−t; −k + j, λ^{−1}|α).

Therefore,

∑_{n≥0} E_{n}^{∗(−k)}(λ|α)\frac{t^{n}}{n!} = ∑_{j=0}^{k} 2^{j−k}α^{j−k}( log(1 − αt))^{k−j}( ∑_{n≥0} E_{n}^{(−j)}(λ|α)\frac{t^{n}}{n!})

× ( ∑_{n≥0} B_{n}^{(−k+j)}(λ^{−1}|α)\frac{(−t)^{n}}{n!} )

= ∑_{j=0}^{k} 2^{j−k}α^{j−k}((k − j)! ∑_{m=k−j}^{∞} s(m, k − j)\frac{(−αt)^{m}}{m!} )

× ( ∑_{n≥0} E_{n}^{(−j)}(λ|α)\frac{t^{n}}{n!})( ∑_{n≥0} B_{n}^{(−k+j)}(λ^{−1}|α)\frac{(−t)^{n}}{n!} )

= ∑_{n≥0} ∑_{j=0}^{k} 2^{j−k}(k − j)! ∑_{m≥0} s(m, k − j)α^{m+j−k}(n)_{m}

× ∑_{l=0}^{n−m}(−1)^{n−l}\binom{n − m}{l}E_{l}^{(−j)}(λ|α)B_{n−m−l}^{(−k+j)}(λ^{−1}|α)\frac{t^{n}}{n!}.

Thus, by equating the coefficient of \frac{t^{n}}{n!}, we immediately arrive at Eq. (55).

Using Eqs. (16) and (54), we obtain the following theorem.

Theorem 22. Let k be a non-negative integer. Then we have the following relation between the degenerate first kind Apostol-Euler numbers E_{n}^{(−k)}(λ|α) and the degenerate Simsek numbers y_{1}(n, k; λ|α):

k!y_{1}(n, k; λ|α) = 2^{k}E_{n}^{(−k)}(λ|α).  (56)

Example 23. According to Example 3, we have the following special cases of the degenerate first kind Apostol-Euler numbers E_{n}^{(−k)}(λ|α):

E_{0}^{(−k)}(λ|α) = 2^{−k}(λ + 1)^{k},
E_{n}^{(0)}(λ|α) = δ_{n,0},
E_{n}^{(−1)}(λ|α) = \frac{1}{2}(δ_{n,0} + λ(\frac{1}{α})_{n}α^{n}).

Substituting λ = 1 and letting α → 0 in Eq. (56), yields

E_{n}^{(−k)} = 2^{−k} ∑_{j=0}^{k} \binom{k}{j}j^{n},

the Euler numbers of the first kind of order −k, given by Simsek [28].

3 Conclusion

In this paper, we studied the degenerate Simsek numbers y_{1}(n, k; λ|α) and the (λ, α)-Stirling numbers of the second kind S_{2}(n, k; λ|α), based on generating functions. Some properties of S_{2}(n, k; λ|α) and y_{1}(n, k; λ|α), such as recurrence formulas, were presented. We also provided a relation between these numbers and with some kinds of special numbers. In particular, by specializing the parameters λ and α, we obtained the familiar Stirling numbers of the second kind, Simsek numbers [28], the λ-Stirling numbers of the second kind [19, 31], the degenerate Stirling numbers of the second kind [13, 14], the weighted Stirling numbers of the second kind [10], and the degenerate associated Stirling numbers of the second kind [9]. Finally, we introduced the degenerate first and second kind Apostol-Euler numbers of order k, and derived some related properties. A theorem which relates these numbers and y_{1}(n, k; λ|α) to each other was given.

4 Acknowledgments

The author would like to thank Anna Wysoczańska-Kula for her careful reading and helpful comments. The author sincerely appreciates the anonymous referee for his/her valuable comments and suggested references.

This research was partially supported by the Austrian FWF Project P29355-N35 and by the National Agency for Academic Exchange (NAWA) POLONIUM project PPN/BIL/ 2018/1/00197/U/00021.

References

[1] A. Bayad, Y. Simsek, and H. M. Srivastava, Some array type polynomials associated with special numbers and polynomials, Appl. Math. Comput. 244 (2014), 149–157.

[2] P. Blasiak, K. A. Penson, and A. I. Solomon, The boson normal ordering problem and generalized Bell numbers, Ann. Comb. 7 (2003), 127–139.

[3] L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers, Utilitas Math. 15 (1979), 51–88.

[4] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Springer, 1974.

[5] R. Golombek, Aufgabe 1088, Elem. Math. 49 (1994), 126–127.

[6] M. Goubi, A new class of generalized polynomials associated with Hermite-Bernoulli polynomials, J. Appl. Math. Inform. 38 (2020), 211–220.

[7] M. Goubi, An affirmative answer to two questions concerning special case of Simsek numbers and open problems, Appl. Anal. Discrete Math. 14 (2020), 94–105.

[8] M. Goubi, Generating functions for generalization Simsek numbers and their applications, to appear in Appl. Anal. Discrete Math., https://doi.org/10.2298/

AADM200522005G.

[9] F. T. Howard, Bell polynomials and degenerate Stirling numbers, Rend. Sem. Mat. Univ. Padova 61 (1979), 203–219.

[10] F. T. Howard, Degenerate weighted Stirling numbers, Discrete Math. 57 (1985), 45–58.

[11] L. C. Hsu and P. J.-S. Shiue, A unified approach to generalized Stirling numbers, Adv. in Appl. Math. 20 (1998), 366–384.

[12] H. K. Kim, Fully degenerate Bell polynomials associated with degenerate Poisson random variables, Open Math. 19 (2021), 284–296.

[13] T. Kim, A note on degenerate Stirling polynomials of second kind, Proc. Jangjeon Math. Soc. 20 (2017), 319–331.

[14] D. S. Kim and T. Kim, On degenerate Bell numbers and polynomials, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 111 (2017), 435–446.

[15] T. Kim and D. S. Kim, On some degenerate differential and degenerate difference operators, Russ. J. Math. Phys. 29 (2022), 37–46.

[16] T. Kim, D. S. Kim, H. Y. Kim, and J. Kwon, Degenerate Stirling polynomials of the second kind and some applications, Symmetry 11 (2019), 1046, https://doi.org/10.

3390/sym11081046.

[17] D. S. Kim, T. Kim, H. Y. Kim, and H. Lee, Two variable degenerate Bell polynomials associated with Poisson degenerate central moments, Proc. Jangjeon Math. Soc. 23 (2020), 587–596.

[18] I. Kucukoglu and Y. Simsek, Construction and computation of unified Stirling-type numbers emerging from p-adic integrals and symmetric polynomials, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 115 (2021), Article number 167,

https://doi.org/10.1007/s13398-021-01107-2.

[19] Q. M. Luo and H. M. Srivastava, Some generalizations of the Apostol-Genoucchi polynomials and the Stirling numbers of the second kind, Appl. Math. Comput. 217 (2011), 5702–5728.

[20] T. Mansour and M. Schork, The generalized Stirling and Bell numbers revisited, J. Integer Seq. 15 (2012), Article 12.8.3.

[21] T. Mansour and M. Schork, Commutation Relations, Normal Ordering, and Stirling Numbers, Chapman and Hall/CRC, 2015.

[22] L. Oussi, A (p, q)-deformed recurrence for the Bell numbers, J. Integer Seq. 23 (2020), Article 20.5.2.

[23] L. Oussi, (p, q)-analogues of the generalized Touchard polynomials and Stirling numbers,

Indag. Math. (N.S.) 33 (2022), 664–681, https://doi.org/10.1016/j.indag.2021.
12.009.

[24] H. Ozden and Y. Simsek, A new extension of q-Euler numbers and polynomials related to their interpolation functions, Appl. Math. Lett. 21 (2008), 934–939.

[25] H. Ozden and Y. Simsek, Modification and unification of the Apostol-type numbers and polynomials and their applications, Appl. Math. Comput. 235 (2014), 338–351.

[26] S. Roman, The Umbral Calculus, Academic Press, 1983.

[27] M. D. Schmidt, Combinatorial identities for generalized Stirling numbers expanding f-factorial functions and the f-harmonic numbers, J. Integer Seq. 21 (2018), Article 18.2.7.

[28] Y. Simsek, New families of special numbers for computing negative order Euler numbers and related numbers and polynomials, Appl. Anal. Discrete Math. 12 (2018), 1–35.

[29] Y. Simsek, Generating functions for generalized Stirling type numbers, Array type polynomials, Eulerian type polynomials and their applications, Fixed Point Theory Appl.

2013 (2013), Article number 87, https://doi.org/10.1186/1687-1812-2013-87.

[30] M. Z. Spivey, Combinatorial sums and finite differences, Discrete Math. 307 (2007), 3130–3146.

[31] H. M. Srivastava, Some generalizations and basic (or q-) extensions of the Bernoulli, Euler and Genoucchi polynomials, Appl. Math. Inf. Sci. 5 (2011), 390–444.

[32] A. Xu, On an open problem of Simsek concerning the computation of a family of special numbers, Appl. Anal. Discrete Math. 13 (2019), 61–72.

2020 Mathematics Subject Classification: Primary 11B73. Secondary 11B68, 11B75, 11B83. Keywords: degenerate Stirling number, degenerate Simsek number, generating function, Euler number.

(Concerned with sequences \underline{A000010}, \underline{A000110}, and \underline{A008277}.)

Received October 2 2021; revised versions received January 24 2022; February 2 2023; April 2 2023; May 18 2023. Published in Journal of Integer Sequences, May 18 2023.

Return to Journal of Integer Sequences home page.