On Noncentral Tanny-Dowling Polynomials and Generalizations of Some Formulas for Geometric Polynomials

Mahid M. Mangontarum and Norlailah M. Madid Department of Mathematics Mindanao State University–Main Campus Marawi City 9700 Philippines

                    mmangontarum@yahoo.com
               mangontarum.mahid@msumain.edu.ph
                  norlailahmadid07@gmail.com

Abstract In this paper, we establish some formulas for the noncentral Tanny-Dowling polynomials, such as sums of products and explicit formulas. Some special cases are also presented and discussed.

1 Introduction

The geometric polynomials [19], denoted by w_{n}(x), are defined by

w_{n}(x) = ∑_{k=0}^{n} k!\left\{{n \atop k}\right\}x^{k},  (1)

where \left\{{n \atop k}\right\} are the celebrated Stirling numbers of the second kind [7, 18]. These polynomials are known to satisfy the exponential generating function

∑_{n=0}^{∞} w_{n}(x)\frac{z^{n}}{n!} = \frac{1}{1 − x(e^{z} − 1)}  (2)

and the recurrence relation [9, Proposition 7]

w_{n+1}(x) = x\frac{d}{dx} (w_{n}(x) + xw_{n}(x)) .  (3)

The case when x = 1 yields

w_{n} := w_{n}(1) = ∑_{k=0}^{n} k!\left\{{n \atop k}\right\},  (4)

the geometric numbers (or ordered Bell numbers) whose values form the sequence \underline{A000670}. Recall that the numbers \left\{{n \atop k}\right\} count the number of partitions of a set X with n elements into k non-empty subsets. These numbers can also be interpreted as the number of ways to distribute n distinct objects into k identical boxes such that no box is empty. On the other hand, the numbers k!\left\{{n \atop k}\right\} can be combinatorially interpreted as the number of distinct ordered partitions of X with k blocks, or the numbers of ways to distribute n distinct objects into k distinct boxes. It follows immediately that the geometric numbers count the number of distinct ordered partitions of the n-set X.

The study of geometric polynomials and numbers has a long history. Aside from the work of Tanny [19], one may also see the papers written by Boyadzhiev [4], Dil and Kurt [9], Boyadzhiev and Dil [5], Kargın and Corcino [12], and the references therein for further readings. Benoumhani [3] studied two equivalent generalizations of w_{n}(x) given by

F_{m,1}(n; x) = ∑_{k=0}^{n} m^{k}k!W_{m}(n, k)x^{k}  (5)

and

F_{m,2}(n; x) = ∑_{k=0}^{n} k!W_{m}(n, k)x^{k},  (6)

where W_{m}(n, k) denote the Whitney numbers of the second kind of Dowling lattices [2]. These are called Tanny-Dowling polynomials and are known to satisfy the following exponential generating functions:

∑_{n=0}^{∞} F_{m,1}(n; x)\frac{z^{n}}{n!} = \frac{e^{z}}{1 − x(e^{mz} − 1)},  (7)

∑_{n=0}^{∞} F_{m,2}(n; x)\frac{z^{n}}{n!} = \frac{e^{z}}{1 − \frac{x}{m}(e^{mz} − 1)}.  (8)

More properties can be seen in [2, 3]. In a recent paper, Kargın [10] established a number of explicit formulas and formulas involving products of geometric polynomials, viz.

(x + 1) ∑_{k=0}^{n} \binom{n}{k}w_{k}(x)w_{n−k}(x) = w_{n+1}(x) + w_{n}(x),  (9)

∑_{k=0}^{n} \binom{n}{k}w_{k}(x_{1})w_{n−k}(x_{2}) = \frac{x_{2}w_{n}(x_{2}) − x_{1}w_{n}(x_{1})}{x_{2} − x_{1}},  (10)

w_{n}(x) = x ∑_{k=1}^{n} \left\{{n \atop k}\right\}(−1)^{n+k}k!(x + 1)^{k−1},  (11)

and

w_{n}(x) = ∑_{k=0}^{n} \left\{{n \atop k}\right\}k!x^{k} \frac{2^{n+1}(x + 1)x^{k} + (−1)^{k+1}}{(2x + 1)^{k+1}} .  (12)

These results were obtained by Kargın [10] with the aid of the two-variable geometric polynomials w_{k}(r; x) defined by

∑_{n=0}^{∞} w_{n}(r; x)\frac{z^{n}}{n!} = \frac{e^{rz}}{1 − x(e^{z} − 1)}.  (13)

A natural generalization of F_{m,1}(n; x) and F_{m,2}(n; x) are the noncentral Tanny-Dolwing polynomials introduced by Mangontarum et al. [15] defined as

F̃_{m,a}(n; x) = ∑_{k=0}^{n} k!W̃_{m,a}(n, k)x^{k},  (14)

where W̃_{m,a}(n, k) are the noncentral Whitney numbers of the second kind. The polynomials F̃_{m,a}(n; x) satisfy the exponential generating function given by [15, Theorem 18]

∑_{n=k}^{∞} F̃_{m,a}(n; x)\frac{z^{n}}{n!} = \frac{me^{−az}}{m − x(e^{mz} − 1)}.  (15)

Looking at (15), it is readily observed that

F̃_{m,0}(n; x) = m^{n}w_{n} (\frac{x}{m}) ,

F̃_{m,−1}(n; x) = F_{m,2}(n; x),

and

F̃_{1,−r}(n; x) = w_{n}(r, x).

The numbers W̃_{m,a}(n, k) admit a variety of combinatorial properties which can be seen in [15]. One of these properties is the triangular recurrence relation [15, Proposition 8]

W̃_{m,a}(n + 1, k) = W̃_{m,a}(n, k − 1) + (mk − a)W̃_{m,a}(n, k)  (16)

with W̃_{m,a}(n, 0) = (−a)^{n} and W̃_{m,a}(n, k) = 1 when k = n. Using this recurrence relation, the following noncentral Tanny-Dowling polynomials can be derived for n = 0, 1, 2, 3, 4:

F̃_{m,a}(0; x) = 1
F̃_{m,a}(1; x) = x − a
F̃_{m,a}(2; x) = 2x^{2} + mx + a^{2}
F̃_{m,a}(3; x) = 6x^{3} + 2(3m − a)x^{2} + (m^{2} − ma_{a}^{2})x − a^{3}
F̃_{m,a}(4; x) = 24x^{4} + 6(6m − 2a)x^{3} + 2(7m^{2} − 6ma + 2a^{2})x^{2}
+ (m^{3} − 2m^{2}a + 2ma^{2} − 2a^{3})x + a^{4}.

These noncentral Whitney numbers of the second kind appear to be a common generalization of \left\{{n \atop k}\right\} and W_{m}(n, k), as well as other notable numbers reported by the respective authors in [1, 6, 13, 14, 16]. It is important to note that the noncentral Whitney numbers of the second kind are equivalent to the (r, β)-Stirling numbers by Corcino [8] and the r-Whitney numbers of the second kind by Mező [17]. On the other hand, the higher order generalized geometric polynomials, an even more generalized polynomial, were introduced in the paper of Kargın and Corcino [11]. However, the said polynomials and the noncentral Tanny-Dolwing polynomials were defined using different motivations. Moreover, the results obtained in this paper do not appear as particular cases of the ones seen in [11].

In the present paper, we establish some formulas for the noncentral Tanny-Dowling polynomials such as sums of products and explicit formulas. These formulas are shown to generalize the above-mentioned identities obtained by Kargın [10] for the geometric polynomials when the parameters are assigned with specific values. We also discuss some other identities resulting from the said formulas.

2 Formulas for sum of products

Now the exponential generating function in (15) can be rewritten as

∑_{n=0}^{∞} F̃_{m,a}(n; x)\frac{z^{n}}{n!} = \frac{1}{1 − \frac{x}{m}(e^{mz} − 1)} · e^{−az}.

Hence, by applying (2) and using Cauchy’s product for two series, we obtain

∑_{n=0}^{∞} F̃_{m,a}(n; x)\frac{z^{n}}{n!} = ∑_{n=0}^{∞} m^{n}w_{n} (\frac{x}{m}) \frac{z^{n}}{n!} ∑_{n=0}^{∞}(−a)^{n} \frac{z^{n}}{n!}

= ∑_{n=0}^{∞} (∑_{k=0}^{n} \binom{n}{k}w_{k} (\frac{x}{m}) m^{k}(−a)^{n−k}) \frac{z^{n}}{n!} .

Comparing the coefficients of \frac{z^{n}}{n!} yields the result in the next theorem.

Theorem 1. The noncentral Tanny-Dowling polynomials F̃_{m,a}(n; x) satisfy the following identity:

F̃_{m,a}(n; x) = ∑_{k=0}^{n} \binom{n}{k}m^{k}w_{k} (\frac{x}{m}) (−a)^{n−k}.  (17)

Alternative proof of Theorem 1. From [15, Theorem 10], the noncentral Whitney numbers of the second kind satisfy the following formula in terms of the Stirling numbers of the second kind:

W̃_{m,a}(n, k) = ∑_{j=0}^{n} \binom{n}{j}(−a)^{n−j}m^{j−k}\left\{{j \atop k}\right\}.

Multiplying both sides by k!x^{k} and summing over k gives the desired result.

Before proceeding, we see that when m = 1 and a = −r, (17) becomes

F̃_{1,−r}(n; x) = ∑_{k=0}^{n} \binom{n}{k}w_{k}(x)r^{n−k} := w_{n}(r; x),

which is precisely an identity obtained by Kargın [10, Equation (13)].

By applying the exponential generating function in (15),

∑_{n=0}^{∞} (F̃_{m,a−m}(n; x) − F̃_{m,a}(n; x)) \frac{z^{n}}{n!} = \frac{me^{−(a−m)z}}{m − x(e^{mz} − 1)} − \frac{me^{−az}}{m − x(e^{mz} − 1)}

= \frac{m}{x} (\frac{me^{−az}}{m − x(e^{mz} − 1)} − e^{−az})

= \frac{m}{x} ∑_{n=0}^{∞} F̃_{m,a}(n; x)\frac{z^{n}}{n!} − ∑_{n=0}^{∞} (−a)^{n} \frac{z^{n}}{n!}

= ∑_{n=0}^{∞} \frac{m}{x} (F̃_{m,a}(n; x) − (−a)^{n}) \frac{z^{n}}{n!} .

Comparing the coefficients of \frac{z^{n}}{n!} gives

F̃_{m,a−m}(n; x) − F̃_{m,a}(n; x) = \frac{m}{x} (F̃_{m,a}(n; x) − (−a)^{n}) .

The result in the next theorem follows by solving for xF̃_{m,a−m}(n; x).

Theorem 2. The noncentral Tanny-Dowling polynomials F̃_{m,a}(n; x) satisfy the following recurrence relation:

xF̃_{m,a−m}(n; x) = (m + x)F̃_{m,a}(n; x) − (−a)^{n}m.  (18)

Setting m = 1 and a = −r in (18) gives

xF̃_{1,−r−1}(n; x) = (1 + x)F̃_{1,−r}(n; x) − r^{n}

which is exactly the following identity [10, Equation (14)]:

xw_{n}(r + 1; x) = (1 + x)w_{n}(r; x) − r^{n}.

On the other hand, when a = 0 and a = m in (18), we get

xF̃_{m,−m}(n; x) = (m + x)m^{n}w_{n} (\frac{x}{m})  (19)

and

(m + x)F̃_{m,m}(n; x) = xm^{n}w_{n} (\frac{x}{m}) − (−m)^{n+1},  (20)

respectively. Substituting (17) to the right hand sides of these equations yields

x ∑_{k=0}^{n} \binom{n}{k}w_{k} (\frac{x}{m}) = (m + x)w_{n} (\frac{x}{m})  (21)

and

(m + x) ∑_{k=0}^{n} \binom{n}{k}w_{k} (\frac{x}{m}) (−1)^{n−k} = xw_{n} (\frac{x}{m}) − m(−1)^{n+1}.  (22)

These identities are generalizations of the results obtained by Dil and Kurt [9, Proposition 3] using the Euler-Seidel matrix method and by Kargın [10, Equations (8) and (9)]. That is, setting x = 1 and m = 1 gives

∑_{k=0}^{n} \binom{n}{k}w_{k} = 2w_{n}

and

2 ∑_{k=0}^{n} \binom{n}{k}(−1)^{k}w_{k} = (−1)^{n}w_{n} + 1.

The next theorem contains a formula for the sum of product of noncentral Tanny-Dowling polynomials for different values of a.

Theorem 3. The noncentral Tanny-Dowling polynomials satisfy the following relation:

x ∑_{k=0}^{n} \binom{n}{k}F̃_{m,a_{1}}(k; x)F̃_{m,a_{2}}(n − k; x) = F̃_{m,Ā}(n + 1; x) + ĀF̃_{m,Ā}(n; x),  (23)

where Ā = a_{1} + a_{2} + m for real numbers a_{1} and a_{2}.

Proof. We start by taking the derivative of (15) with respect to z. That is,

\frac{∂}{∂z} (\frac{me^{−az}}{m − x(e^{mz} − 1)}) = \frac{me^{−az}}{m − x(e^{mz} − 1)} · \frac{xme^{mz}}{m − x(e^{mz} − 1)} − \frac{ame^{−az}}{m − x(e^{mz} − 1)}.

Replacing a with Ā = a_{1} + a_{2} + m yields

\frac{∂}{∂z} (\frac{me^{−Āz}}{m − x(e^{mz} − 1)}) = ∑_{n=k}^{∞} F̃_{m,Ā}(n + 1; x)\frac{z^{n}}{n!}

in the left-hand side while we get

\frac{me^{−Āz}}{m − x(e^{mz} − 1)} · \frac{xme^{mz}}{m − x(e^{mz} − 1)} = \frac{me^{−a_{1}z}}{m − x(e^{mz} − 1)} · \frac{me^{−a_{2}z}}{m − x(e^{mz} − 1)}

= x ∑_{n=k}^{∞} ∑_{k=0}^{n} \binom{n}{k}F̃_{m,a_{1}}(k; x)F̃_{m,a_{2}}(n − k; x)\frac{z^{n}}{n!}

and

\frac{Āme^{−Āz}}{m − x(e^{mz} − 1)} = Ā · ∑_{n=k}^{∞} F̃_{m,Ā}(n; x)\frac{z^{n}}{n!}

in the right-hand side. Combining the above equations and comparing the coefficients of \frac{z^{n}}{n!} gives the desired result.

When a_{1} = a_{2} = 0 in (23),

xm^{n} ∑_{k=0}^{n} \binom{n}{k}w_{k} (\frac{x}{m}) w_{n−k} (\frac{x}{m}) = F̃_{m,m}(n + 1; x) + mF̃_{m,m}(n; x).

Applying (20) to the right-hand side of this equation gives

xm^{n} ∑_{k=0}^{n} \binom{n}{k}w_{k} (\frac{x}{m}) w_{n−k} (\frac{x}{m}) = \frac{xm^{n+1}w_{n+1} (\frac{x}{m}) − (−m)^{n+2}}{m + x} +m\frac{xm^{n}w_{n} (\frac{x}{m}) − (−m)^{n+1}}{m + x}

which can be simplified into the following identity:

\frac{m + x}{m} ∑_{k=0}^{n} \binom{n}{k}w_{k} (\frac{x}{m}) w_{n−k} (\frac{x}{m}) = w_{n+1} (\frac{x}{m}) + w_{n} (\frac{x}{m}) .  (24)

Obviously, this identity boils down to the result obtained by Kargın [10] in (9) when m = 1.

Theorem 4. For x_{1} ≠ x_{2}, the following formula holds:

∑_{k=0}^{n} \binom{n}{k}F̃_{m,a_{1}}(k; x_{1})F̃_{m,a_{2}}(n − k; x_{2}) = \frac{x_{2}F̃_{m,a_{1}+a_{2}}(n; x_{2}) − x_{1}F̃_{m,a_{1}+a_{2}}(n; x_{1})}{x_{2} − x_{1}}.  (25)

Proof. Note that we can write

\frac{me^{−a_{1}z}}{m − x_{1}(e^{mz} − 1)} · \frac{me^{−a_{2}z}}{m − x_{2}(e^{mz} − 1)} = \frac{1}{x_{2} − x_{1}} (\frac{x_{2}me^{−(a_{1}+a_{2})z}}{m − x_{2}(e^{mz} − 1)} − \frac{x_{1}me^{−(a_{1}+a_{1})z}}{m − x_{1}(e^{mz} − 1)}) .

Following the same method used in the previous theorem leads us to the desired result.

This theorem contains a formula for the sums of products of noncentral Tanny-Dowling polynomials for different values of x. When a_{1} = a_{2} = 0, (25) reduces to

∑_{k=0}^{n} \binom{n}{k}w_{k} (\frac{x_{1}}{m} ) w_{n−k} (\frac{x_{2}}{m} ) = \frac{x_{2}w_{n} (\frac{x_{2}}{m} ) − x_{1}w_{n} (\frac{x_{1}}{m} )}{x_{2} − x_{1}}.  (26)

It is clear to see that when m = 1, we recover the sum of products of geometric polynomials in (10).

3 Explicit formulas

In Theorem 1, we obtained an explicit formula that expresses the noncentral Tanny-Dowling polynomials in terms of the geometric polynomials. Now, with g_{n} = \frac{1}{a^{n}} F̃_{m,a}(n; x) and f_{j} = (\frac{m}{a})^{j} w_{j} (\frac{x}{m}), the binomial inversion formula

f_{n} = ∑_{j=0}^{n} \binom{n}{j}g_{j} ⟺ g_{n} = ∑_{j=0}^{n}(−1)^{n−j}\binom{n}{j}f_{j}  (27)

allows us to express the geometric polynomials w_{n} (\frac{x}{m}) in terms of the noncentral Tanny-Dowling polynomials as follows:

w_{n} (\frac{x}{m}) = \frac{1}{m^{n}} ∑_{j=0}^{n} \binom{n}{j}a^{n−j}F̃_{m,a}(j; x).  (28)

In this section, we will derive more explicit formulas for both polynomials.

Using x − m in place of x in (15) gives

∑_{n=k}^{∞} F̃_{m,a}(n; x − m)\frac{z^{n}}{n!} = \frac{me^{−(−a−m)(−z)}}{m + x(e^{−mz} − 1)}

= ∑_{n=k}^{∞} F̃_{m,−a−m}(n; −x)\frac{(−z)^{n}}{n!} .

By comparing the coefficients of \frac{z^{n}}{n!} , we get

F̃_{m,a}(n; x − m) = (−1)^{n}F̃_{m,−a−m}(n; −x).  (29)

Applying (18) to the right-hand side gives

F̃_{m,a}(n; x − m) = (−1)^{n} (\frac{(m − x)F̃_{m,−a}(n; −x) − a^{n}m}{−x}) .

Replacing −x and −a with x and a, respectively, and solving for F̃_{m,a}(n; x) yields

F̃_{m,a}(n; x) = \frac{(−1)^{n}xF̃_{m,−a}(n; −x − m) + (−a)^{n}m}{m + x}.

By (14), we get the next theorem.

Theorem 5. The noncentral Tanny-Dowling polynomials satisfy the following explicit formula:

F̃_{m,a}(n; x) = x ∑_{k=0}^{n}(−1)^{n+k}k!W̃_{m,−a}(n, k)(m + x)^{k−1} + \frac{(−a)^{n}m}{m + x}.  (30)

Setting a = 0 in W̃_{m,a}(n, k) allows us to express the noncentral Whitney numbers of the second kind in terms of \left\{{n \atop k}\right\}. More precisely, when a = 0 in [15, Proposition 7], we can see that

W̃_{m,0}(n, k) = m^{n−k}\left\{{n \atop k}\right\}.

Thus, (30) becomes

w_{n} (\frac{x}{m}) = x ∑_{k=0}^{n} \frac{(−1)^{n+k}k!}{m^{k}} \left\{{n \atop k}\right\}(m + x)^{k−1}  (31)

when a = 0. Furthermore, when m = 1, we recover the explicit formula in (11). The expression m^{n−k}\left\{{n \atop k}\right\} is actually called translated Whitney numbers of the second kind and is denoted by \left\{{n \atop k}\right\}^{(m)}. These numbers satisfy the recurrence relation given by [1, Theorem 8]

\left\{{n \atop k}\right\}^{(m)} = \left\{{n − 1 \atop k − 1}\right\}^{(m)} + mk\left\{{n − 1 \atop k}\right\}^{(m)}

and the explicit formula [16, Proposition 2]

\left\{{n \atop k}\right\}^{(m)} = \frac{1}{m^{k}k!} ∑_{j=0}^{k}(−1)^{k−j}\binom{k}{j}(mj)^{n}.

More properties of these numbers can be seen in [14]. With these, we may also write

w_{n} (\frac{x}{m}) = \frac{x}{m^{n}} ∑_{k=0}^{n}(−1)^{n+k}k!\left\{{n \atop k}\right\}^{(m)}(m + x)^{k−1},  (32)

an explicit formula for the geometric polynomials w_{n} (\frac{x}{m}) in terms of the translated Whitney numbers of the second kind.

Now it can be shown that

\frac{y^{2} − 1}{2y} (\frac{e^{−a(2z)}}{y − e^{mz}} + \frac{e^{−a(2z)}}{y + e^{mz}} ) = \frac{e^{−a(2z)}}{1 − (\frac{1}{y^{2}−1}) (e^{m(2z)−1})}.

Notice that the right-hand side is

\frac{e^{−a(2z)}}{1 − (\frac{1}{y^{2}−1}) (e^{m(2z)−1})} = \frac{me^{−a(2z)}}{m − (\frac{m}{y^{2}−1}) (e^{m(2z)−1})}

= ∑_{n=0}^{∞} 2^{n}F̃_{m,a} (n; \frac{m}{y^{2} − 1}) \frac{z^{n}}{n!} .

Also, in the left-hand side, we have

\frac{e^{−a(2z)}}{y − e^{mz}} = \frac{1}{y − 1} ∑_{n=0}^{∞} F̃_{m,2a} (n; \frac{m}{y − 1}) \frac{z^{n}}{n!}

and

\frac{e^{−a(2z)}}{y + e^{mz}} = \frac{1}{y + 1} ∑_{n=0}^{∞} F̃_{m,2a} (n; \frac{−m}{y + 1}) \frac{z^{n}}{n!} .

Combining these equations and comparing the coefficients of \frac{z^{n}}{n!} results to

2^{n+1}F̃_{m,a} (n; \frac{m}{y^{2} − 1}) = \frac{y + 1}{y}F̃_{m,2a} (n; \frac{m}{y − 1}) + \frac{y − 1}{y}F̃_{m,2a} (n; \frac{−m}{y + 1}) .

Note that if we set x = \frac{m}{y−1}, then y = \frac{m+x}{x}. Hence, skipping the tedious computations allow us to write

(m + 2x)F̃_{m,2a}(n; x) = 2^{n+1}(m + x)F̃_{m,a} (n; \frac{x^{2}}{m + 2x}) − mF̃_{m,2a} (n; \frac{−mx}{m + 2x}) .

The next theorem is obtained by applying (14).

Theorem 6. The noncentral Tanny-Dowling polynomials satisfy the following explicit formula:

F̃_{m,2a}(n; x) = ∑_{k=0}^{n} k!x^{k} (\frac{2^{n+1}(m + x)x^{k}W̃_{m,a}(n, k) + (−m)^{k+1}W̃_{m,2a}(n, k)}{(m + 2x)^{k+1}}) .  (33)

Since it is already known that W̃_{m,0}(n, k) = \left\{{n \atop k}\right\}^{(m)}, then the right-hand side can be expressed in terms of the translated Whitney numbers of the second kind when a = 0. That is,

w_{n} (\frac{x}{m}) = \frac{1}{m^{n}} ∑_{k=0}^{n} k!x^{k}\left\{{n \atop k}\right\}^{(m)} (\frac{2^{n+1}(m + x)x^{k} + (−m)^{k+1}}{(m + 2x)^{k+1}} ) .  (34)

Lastly, we recover the explicit formula in (12) when m = 1.

Finally, we will conclude this paper by mentioning an explicit formula for F̃_{m,a}(n; x) established in [15, Theorem 19] that is given by

F̃_{m,a}(n; x) = \frac{m}{m + x} ∑_{k=0}^{∞} (\frac{x}{m + x})^{k} (mk − a)^{n}.  (35)

This explicit formula entails interesting particular cases. For instance, when a = 0,

w_{n} (\frac{x}{m}) = \frac{m}{m + x} ∑_{k=0}^{∞} (\frac{x}{m + x})^{k} k^{n}.  (36)

When m = 1 and then x = 1, we get formulas for the ordinary geometric polynomials and numbers. That is,

w_{n}(x) = \frac{1}{x + 1} ∑_{k=0}^{∞} (\frac{x}{x + 1})^{k} k^{n}  (37)

and

w_{n} = ∑_{k=0}^{∞} \frac{k^{n}}{2^{k+1}} .  (38)

4 Acknowledgment

The authors are thankful to the editor-in-chief and the anonymous referee for the helpful comments and suggestions which helped improve the paper.

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2010 Mathematics Subject Classification: Primary 11B83; Secondary 11B73. Keywords: Geometric polynomial, noncentral Tanny-Dowling polynomial, Stirling number of the second kind, Whitney number of the second kind.

(Concerned with sequences \underline{A000670}, \underline{A008277}, \underline{A000110}, and \underline{A003575}.)

Received April 8 2020; revised versions received June 5 2020; September 23 2020; October 25 2020. Published in Journal of Integer Sequences, January 26 2021.

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