Some Formulae for Products of Geometric Polynomials with Applications

Levent Kargın Akseki Vocational School Alanya Alaaddin Keykubat University TR-07630 Antalya Turkey

               leventkargin48@gmail.com

Abstract

In this paper, we evaluate sums and integrals of products of two geometric polynomials and obtain new explicit formulas for geometric polynomials and numbers. As a consequence of these results, we give new explicit formulas for p-Bernoulli numbers, Apostol-Bernoulli functions, and integrals of products of two Apostol-Bernoulli functions.

1 Introduction

Let \left\{{n \atop k}\right\} be the Stirling numbers of the second kind [16]. Geometric polynomials are defined by [33]

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

They have the exponential generating function

\frac{1}{1 − y (e^{t} − 1)} = ∑_{n=0}^{∞} w_{n} (y) \frac{t^{n}}{n!},  (2)

and are related to the geometric series by [5]

(y \frac{d}{dy} )^{m} \frac{1}{1 − y} = ∑_{k=0}^{∞} k^{m}y^{k} = \frac{1}{1 − y} w_{m} (\frac{y}{1 − y} ) , |y| < 1.

In addition, the following recurrence relation holds for the geometric polynomials [13]:

w_{n+1} (y) = y \frac{d}{dy} (w_{n} (y) + yw_{n} (y)).  (3)

The nth geometric number (ordered Bell number, or Fubini number, or preferred arrangement number) [12, 17, 33], w_{n}, is defined by

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

and counts all the possible set partitions of an n element set such that the order of the blocks matters. Besides with this combinatorial property, these numbers are seen in the evaluation of the following series

∑_{k=0}^{∞} \frac{k^{n}}{2^{k}} = 2w_{n}.  (5)

In the literature, numerous identities concerned with these polynomials and numbers were obtained [5, 6, 7, 8, 14, 28] and some generalizations were given [5, 15, 20, 29].

The sums of products of various polynomials and numbers with or without binomial coefficients have been studied [2, 10, 21, 25, 26, 32, 34]. One of the famous results is [19, Eq. 50.11.2]

∑_{k=0}^{n} \binom{n}{k}B_{k} (x) B_{n−k} (y) = (1 − n) B_{n} (x + y − 1) + (x + y − 1) nB_{n−1} (x + y − 1) , (6)

where B_{n} (x) is the nth Bernoulli polynomial. In addition, the integrals of products of various polynomials and functions have been studied [3, 6, 11, 24, 27]. For example, the following interesting integral for a product of two Bernoulli polynomials appears in the book by Nörlund [30, p. 31]: For all k + m ≥ 2,

∫_{0}^{1} B_{k} (x) B_{m} (x) dx = \frac{k!m!}{(k + m)!}B_{k+m}.  (7)

Here, B_{n} is the nth Bernoulli number defined by B_{n} = B_{n} (0).

The main purpose of this paper is to obtain analogues of (6) and (7) for geometric polynomials, which generalize the binomial formulas [13]

∑_{k=0}^{n} \binom{n}{k}w_{k} = 2w_{n}, n > 0,  (8)

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

and the integral [22]

∫_{−1}^{0} w_{n} (y) dy = B_{n}, n > 0.  (10)

The paper is outlined as follows. We derive sums of products of two geometric polynomials and explicit formulas for these polynomials and numbers in Section 2. In Section 3, we give integrals of products of two geometric polynomials. As an application of this result, an explicit formula for p-Bernoulli numbers is obtained. Finally, in Section 4, we give a new explicit formula and integrals of products of Apostol-Bernoulli functions.

2 Sums of products of geometric polynomials

In this section, we define two variable geometric polynomials and obtain some basic properties which give us new formulas for w_{n} (y). Moreover, we consider the sums of products of two geometric polynomials.

Two variable geometric polynomials are defined by means of the following generating function:

∑_{n=0}^{∞} w_{n} (x; y) \frac{t^{n}}{n!} = \frac{e^{xt}}{1 − y (e^{t} − 1)}.  (11)

As some special cases of (11), we have

w_{n} (0; y) = w_{n} (y) and w_{n} (0; 1) = w_{n}.  (12)

We can rewrite (11) as

∑_{n=0}^{∞} w_{n} (x; y) \frac{t^{n}}{n!} = \frac{1}{1 − y (e^{t} − 1)}e^{xt}

= ∑_{n=0}^{∞} w_{n} (y) \frac{t^{n}}{n!} ∑_{n=0}^{∞} x^{n} \frac{t^{n}}{n!}

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

Comparing the coefficients of \frac{t^{n}}{n!} yields

w_{n} (x; y) = ∑_{k=0}^{n} \binom{n}{k}w_{k} (y) x^{n−k}.  (13)

Besides, from (11) we have

∑_{n=0}^{∞}(w_{n} (x + 1; y) − w_{n} (x; y))\frac{t^{n}}{n!} = \frac{e^{xt} (e^{t} − 1)}{1 − y (e^{t} − 1)}

= \frac{1}{y}(\frac{e^{xt}}{1 − y (e^{t} − 1)} − e^{xt})

= \frac{1}{y} ∑_{n=0}^{∞}(w_{n} (x; y) − x^{n})\frac{t^{n}}{n!}.

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

yw_{n} (x + 1; y) = (1 + y) w_{n} (x; y) − x^{n}.  (14)

Thus, setting x = 0 and x = −1 in (14), we find

yw_{n} (1; y) = (1 + y) w_{n} (y) , n > 0,  (15)
(1 + y) w_{n} (−1; y) = yw_{n} (y) + (−1)^{n} , n ≥ 0,  (16)

respectively. Combining these relations with (13) gives equations (8) and (9) which were obtained by using Euler-Seidel matrix method in [13].

Now, we want to give the generalization of the binomial formula (8). Derivative of (11) can be written as

\frac{∂}{∂t} (\frac{e^{xt}}{1 − y (e^{t} − 1)}) = \frac{xe^{xt}}{1 − y (e^{t} − 1)} + \frac{ye^{t}}{1 − y (e^{t} − 1)} \frac{e^{xt}}{1 − y (e^{t} − 1)}.

Taking x = x_{1} + x_{2} − 1 leads to

\frac{∂}{∂t} (\frac{e^{xt}}{1 − y (e^{t} − 1)}) = ∑_{n=0}^{∞} w_{n+1} (x_{1} + x_{2} − 1; y) \frac{t^{n}}{n!},

\frac{xe^{xt}}{1 − y (e^{t} − 1)} = (x_{1} + x_{2} − 1) ∑_{n=0}^{∞} w_{n} (x_{1} + x_{2} − 1; y) \frac{t^{n}}{n!},

and

\frac{ye^{t}}{1 − y (e^{t} − 1)} \frac{e^{xt}}{1 − y (e^{t} − 1)} = y (∑_{n=0}^{∞} w_{n} (x_{1}; y) \frac{t^{n}}{n!}) (∑_{n=0}^{∞} w_{n} (x_{2}; y) \frac{t^{n}}{n!})

= y ∑_{n=0}^{∞} ∑_{k=0}^{n} \binom{n}{k}w_{k} (x_{1}; y) w_{n−k} (x_{2}; y) \frac{t^{n}}{n!}.

By equating the coefficients of \frac{t^{n}}{n!}, we get

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

For x_{1} = x_{2} = 0 in the above equation, by using (16) gives the sums of products of the geometric polynomials.

Theorem 1. For n ≥ 0,

(y + 1) ∑_{k=0}^{n} \binom{n}{k}w_{k} (y) w_{n−k} (y) = w_{n+1} (y) + w_{n} (y) .  (17)

When y = 1 this becomes

2 ∑_{k=0}^{n} \binom{n}{k}w_{k}w_{n−k} = w_{n+1} + w_{n}.  (18)

We note that Boyadzhiev and Dil [9, Proposition 9] proved a more general form of Theorem 1 by different way.

Now, we investigate the sums of products of the geometric polynomials for different y values in the following theorem.

Theorem 2. For n ≥ 0 and y_{1} ≠ y_{2},

∑_{k=0}^{n} \binom{n}{k}w_{k} (y_{1}) w_{n−k} (y_{2}) = \frac{y_{2}w_{n} (y_{2}) − y_{1}w_{n} (y_{1})}{y_{2} − y_{1}}.  (19)

Proof. We write

\frac{e^{x_{1}t}}{(1 − y_{1} (e^{t} − 1))} \frac{e^{x_{2}t}}{(1 − y_{2} (e^{t} − 1))}  (20)

= \frac{y_{2}}{y_{2} − y_{1}} \frac{e^{(x_{1}+x_{2})t}}{1 − y_{2} (e^{t} − 1)} − \frac{y_{1}}{y_{2} − y_{1}} \frac{e^{(x_{1}+x_{2})t}}{1 − y_{1} (e^{t} − 1)}.

Using the same method as in the proof of Theorem 1, we have

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

Setting x_{1} = x_{2} = 0 in the above equation gives the desired equation.

As we know, for y = 1, geometric polynomials reduce to geometric numbers. We now point out (see (24)) that geometric numbers also arise for other values of y. If we take y − 1 in place of y in (11), we have

w_{n} (x; y − 1) = (−1)^{n} w_{n} (1 − x; −y) .  (21)

Setting x = 0 in the above equation and using the relation (15), we have a reflection formula

w_{n} (y) = (−1)^{n} \frac{y}{y + 1}w_{n} (−y − 1) , n > 0.  (22)

Therefore, employing (1) gives a new explicit formula for geometric polynomials as in the following theorem.

Theorem 3. For n > 0, we obtain

w_{n} (y) = y ∑_{k=1}^{n} \left\{{n \atop k}\right\} (−1)^{n+k} k! (y + 1)^{k−1} .  (23)

Note that, when y = 1 (23) reduces to [18, Theorem 4.2]. Moreover, from (22), we get two conclusions

w_{2k} (\frac{−1}{2}) = 0 and w_{n} (−2) = (−1)^{n} 2w_{n}.  (24)

The first part of (24) was given in [9, Corollary 17]. If we take y_{1} = −2 and y_{2} = 1 in (19) and use the second part of (24), we obtain the alternating sums of products of geometric numbers.

Corollary 4. For n > 0, we have

∑_{k=0}^{n} \binom{n}{k} (−1)^{k} w_{k}w_{n−k} = \begin{cases}0, & if n is odd; \\ \frac{4}{3}w_{n}, & if n is even. \end{cases}

Finally, we obtain a new explicit formula for geometric polynomials and numbers in the following theorem.

Theorem 5. For y ≠ \frac{−1}{2} and n ≥ 0,

w_{n} (y) = ∑_{k=0}^{n} \left\{{n \atop k}\right\}k!y^{k} \frac{(2^{n+1} (y + 1) y^{k} + (−1)^{k+1})}{(2y + 1)^{k+1}}.  (25)

When y = 1 this becomes

w_{n} = ∑_{k=0}^{n} \left\{{n \atop k}\right\}k!\frac{(2^{n+2} + (−1)^{k+1})}{3^{k+1}}, n ≥ 0.  (26)

When y = −2 this becomes

w_{n} = ∑_{k=0}^{n} (−1)^{n−k} \left\{{n \atop k}\right\}k!\frac{2^{k−1}(2^{n+k+1} + 1)}{3^{k+1}}, n > 0.  (27)

Proof. If we take \frac{1}{y^{2}−1} in place of y in (2), we arrive at

\frac{1}{1 − \frac{1}{y^{2}−1} (e^{2t} − 1)} = \frac{y^{2} − 1}{2y^{2}}(\frac{1}{y − e^{t}} + \frac{1}{y + e^{t}} ).  (28)

Each of the functions in the above equation can be written as

\frac{1}{1 − \frac{1}{y^{2}−1} (e^{2t} − 1)} = ∑_{n=0}^{∞} 2^{n}w_{n} (\frac{1}{y^{2} − 1}) \frac{t^{n}}{n!},  (29)

\frac{1}{y − e^{t}} = \frac{y}{y − 1} ∑_{n=0}^{∞} w_{n} (\frac{1}{y − 1}) \frac{t^{n}}{n!},  (30)

\frac{1}{y + e^{t}} = \frac{y}{y + 1} ∑_{n=0}^{∞} w_{n} (\frac{−1}{y + 1}) \frac{t^{n}}{n!}.  (31)

By equating the coefficients of \frac{t^{n}}{n!}, we have

w_{n} (y) = 2^{n+1} (1 + y) w_{n} (\frac{y^{2}}{1 + 2y} ) − (1 + 2y) w_{n} (−y) .  (32)

Finally, using (1) in the right hand side of the above equation yields (25).

3 Integrals of products of geometric polynomials

In this section, we deal with an integral for a product of two geometric polynomials. First we need following lemmas.

Lemma 6. For all k ≥ 0 and n ≥ 1, we have

∫_{−1}^{0}y^{k}w_{n} (y) dy = \frac{(−1)^{k}}{k!} ∑_{j=0}^{k} \left[{k + 1 \atop j + 1}\right]B_{n+j},  (33)

where \left[{n \atop k}\right] is the Stirling number of the fist kind [16].

Proof. We prove (33) by induction on k. The case k = 0 of (33) is known from (10). If we integrate both sides of (3) with respect to y from −1 to 0 and apply integration by parts, we have

∫_{−1}^{0}w_{n+1} (y) dy = ∫_{−1}^{0}y \frac{d}{dy} (w_{n} (y) + yw_{n} (y))dy

= [y (w_{n} (y) + yw_{n} (y))]_{−1}^{0} − ∫_{−1}^{0}(w_{n} (y) + yw_{n} (y))dy.

So, using (10) yields the case k = 1 of (33) as

∫_{−1}^{0}yw_{n} (y) dy = − (B_{n+1} + B_{n}) .

Multiplying both sides of (3) by y and integrating it with respect to y from −1 to 0, we obtain

∫_{−1}^{0}yw_{n+1} (y) dy = ∫_{−1}^{0}y^{2} \frac{d}{dy} (w_{n} (y) + yw_{n} (y))dy.

Applying integration by parts and using (10) yield the case k = 2 of (33) as

2∫_{−1}^{0}y^{2}w_{n} (y) dy = B_{n+2} + 3B_{n+1} + 2B_{n}.

If we multiply both sides of (3) by y^{k} and integrating it with respect to y from −1 to 0, we obtain

∫_{−1}^{0}y^{k}w_{n+1} (y) dy = ∫_{−1}^{0}y^{k+1} \frac{d}{dy} (w_{n} (y) + yw_{n} (y))dy.

Applying integration by parts to the right hand side of the above equation and considering

∫_{−1}^{0}y^{k}w_{n} (y) dy = \frac{(−1)^{k}}{k!} ∑_{j=0}^{k} \left[{k + 1 \atop j + 1}\right]B_{n+j},

we have

∫_{−1}^{0}y^{k+1}F_{n+1} (y) dy = \frac{(−1)^{k+1}}{(k + 1)!} ∑_{j=0}^{k} \left[{k + 1 \atop j + 1}\right]B_{n+j+1} + \frac{(−1)^{k+1}}{(k + 1)!} ∑_{j=0}^{k} (k + 1) \left[{k + 1 \atop j + 1}\right]B_{n+j}.

Finally, the well-known relations

\left[{n + 1 \atop k}\right] = n\left[{n \atop k}\right] + \left[{n \atop k − 1}\right] and \left[{n \atop 1}\right] = (n − 1)!,

give that the statement is true for k + 1.

Lemma 7. For any non-negative integer m and j,

∑_{k=j}^{m} \left\{{m \atop k}\right\}\left[{k + 1 \atop j + 1}\right](−1)^{k} = (−1)^{m}\binom{m}{j}.  (34)

Proof. We rewrite this equation into matrix form by using the matrices

(S_{1})_{i,j} = (−1)^{i+j}\left[{i + 1 \atop j + 1}\right], (S_{2})_{i,j} = \left\{{i \atop j}\right\}, (B)_{i,j} = \binom{i}{j}.

These can be considered as infinite matrices so that the statement we are going to prove takes the form

S_{2}S_{1} = B^{−1},

where the element-wise inverse of the matrix B is (B)_{i,k}^{−1} = (−1)^{i+k}\binom{i}{k}. The above equation is equivalent to

S_{1} = B^{−1}S_{2}^{−1} = (S_{2}B)^{−1} .

The matrix on the right hand side is easily decipherable. Element-wise, it is

((S_{2}B)^{−1})_{i,j} = ∑_{k=0}^{i} \left\{{i \atop k}\right\}\binom{k}{j}.

The latter sum simply equals to

∑_{k=0}^{i} \left\{{i \atop k}\right\}\binom{k}{j} = \left\{{i + 1 \atop j + 1}\right\}

[16, Eq. 6.15]. Hence, our original statement equals to the matrix equation

(S_{1})_{i,j}^{−1} = \left\{{i + 1 \atop j + 1}\right\}.

This is nothing but the reformulation of the fact that the second and signed first kind Stirling matrices are inverses of each other.

Now, we are ready to give the integrals of products of geometric polynomials. Using (1), we have

∫_{−1}^{0}w_{m} (y) w_{n} (y) dy = ∫_{−1}^{0} ∑_{k=0}^{m} \left\{{m \atop k}\right\}k!y^{k}w_{n} (y) dy.

Then, interchanging the sum and integral in the above equation and using (33) yield

∫_{−1}^{0}w_{m} (y) w_{n} (y) dy = ∑_{j=0}^{m} ∑_{k=j}^{m} \left\{{m \atop k}\right\}\left[{k + 1 \atop j + 1}\right] (−1)^{k} B_{n+j}.

Finally, using Lemma 7 gives the following theorem.

Theorem 8. For all m ≥ 0 and n ≥ 1, we have

∫_{−1}^{0}w_{m} (y) w_{n} (y) dy = (−1)^{m} ∑_{j=0}^{m} \binom{m}{j}B_{n+j}.  (35)

Using the representation (1) in (35) and integrating termwise, one obtains

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

This double sum identity extends to the explicit formula [16, p. 560]

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

In order to give an application of Lemma 6, we now emphasize the summation in the right hand of (33). Rahmani [31] defined p-Bernoulli numbers as

∑_{n=0}^{∞} B_{n,p} \frac{t^{n}}{n!} = _{2}F_{1} (1, 1; p + 2; 1 − e^{t}) ,

where _{2}F_{1} (a, b; c; z) denotes the Gaussian hypergeometric function [1]. These numbers can be written in terms of Stirling numbers of the first kind as follows:

∑_{j=0}^{p} (−1)^{j} \left[{p \atop j}\right]B_{n+j} = \frac{p!}{p + 1}B_{n,p}, n, p ≥ 0.

From the above equation, we have

∑_{j=0}^{p} (−1)^{j+1} \left[{p + 1 \atop j + 1}\right]B_{n+j} = \frac{(p + 1)!}{p + 2}B_{n−1,p+1}, n ≥ 1, p ≥ 0.  (36)

Moreover, when n is odd or even, we have

(−1)^{j+1} B_{n+j} = B_{n+j} or (−1)^{j+1} B_{n+j} = −B_{n+j}, n > 1,

respectively. Therefore, we obtain

∑_{j=0}^{p} \left[{p + 1 \atop j + 1}\right]B_{n+j} = \begin{cases}\frac{(p+1)!}{p+2}B_{n−1,p+1}, & if n is odd; \\ −\frac{(p+1)!}{p+2}B_{n−1,p+1}, & if n is even. \end{cases}

Using the above equation, (33) can be written as

∫_{−1}^{0}y^{p}w_{n} (y) dy = \begin{cases}(−1)^{p} \frac{p+1}{p+2}B_{n−1,p+1}, & if n is odd; \\ (−1)^{p+1} \frac{p+1}{p+2}B_{n−1,p+1}, & if n is even, \end{cases}

where n > 1, p ≥ 0. On the other hand, using (1) in the left part of (3), a new explicit formula for p-Bernoulli numbers is obtained.

Theorem 9. For n > 1 and p > 0,

B_{2n−1,p} = \frac{p + 1}{p} ∑_{k=0}^{2n−1} \left\{{2n − 1 \atop k + 1}\right\}\frac{(−1)^{k+1} (k + 1)!}{k + p + 1}

and

B_{2n,p} = \frac{p + 1}{p} ∑_{k=0}^{2n} \left\{{2n + 1 \atop k + 1}\right\}\frac{(−1)^{k} (k + 1)!}{k + p + 1}.

4 Applications

Apostol-Bernoulli functions B_{n} (λ) have the following explicit expression:

B_{n} (λ) = \frac{n}{λ − 1} ∑_{k=0}^{n−1} \left\{{n − 1 \atop k}\right\}k! (\frac{λ}{1 − λ})^{k} , λ ∈ ℂ\\{1\}.  (37)

Thus, for λ ≠ 1,

B_{0} (λ) = 0, B_{1} (λ) = \frac{1}{λ − 1}, B_{2} (λ) = \frac{−2λ}{(λ − 1)^{2}} , …

The functions B_{n} (λ) are rational functions in the variable λ. Apostol [4] introduced these functions in order to evaluate the Lerch transcendent (also known as the Lerch zeta function) for negative integer values of s. Also, these functions were studied and generalized recently in a number of papers, under the name Apostol-Bernoulli numbers.

Comparing (37) and (1), Boyadzhiev [7] showed that Apostol-Bernoulli functions can be expressed by geometric polynomials as

B_{n+1} (λ) = \frac{n + 1}{λ − 1}w_{n} (\frac{λ}{1 − λ}) , λ ∈ ℂ\\{1\}.  (38)

We can use this relation to obtain some new properties of B_{n} (λ). For example, setting y = \frac{−λ}{λ−1} in (23), we have [18, Theorem 4.3]

\frac{B_{n+1} (λ)}{(n + 1)} = (−1)^{n} λ ∑_{k=0}^{n} \left\{{n \atop k}\right\}k! (\frac{1}{λ − 1})^{k+1} , λ ≠ 1, n ≥ 0.

Similarly, from Theorem 1, we get the sums of products of Apostol-Bernoulli functions as given in [23, Corollary 1.3]. Moreover, using equation (25) of Theorem 5 gives a new explicit formula for Apostol-Bernoulli functions.

Corollary 10. For λ ≠ ±1 and n ≥ 0,

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

To give a different application of the relation (38), we first deal with Lemma 6. Replacing y with \frac{λ}{1−λ} in (33), we have

∫_{−∞}^{0} \frac{λ^{k}}{(λ − 1)^{k+1}} B_{n+1} (λ) dλ = \frac{n + 1}{k!} ∑_{j=0}^{k} \left[{k + 1 \atop j + 1}\right]B_{n+j},

where k ≥ 0 and n ≥ 1. Then, from Theorem 8, we obtain the integrals of products of Apostol-Bernoulli functions as given in the following corollary.

Corollary 11. For all m ≥ 0 and n ≥ 1, we have

∫_{−∞}^{0} B_{m} (λ) B_{n} (λ) dλ = (−1)^{m} (m + 1) (n + 1) ∑_{j=0}^{m} \binom{m}{j}B_{n+j}.

5 Acknowledgments

The author would like to thank to Professor Istvan Mező for his valuable suggestions regarding improvement of the paper.

References

[1] G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University, 1999.

[2] T. Agoh and K. Dilcher, Convolution identities and lacunary recurrences for Bernoulli numbers, J. Number Theory 124 (2007), 105–122.

[3] T. Agoh and K. Dilcher, Integrals of products of Bernoulli polynomials, J. Math. Anal. Appl. 381 (2011), 10-16.

[4] T. M. Apostol, On the Lerch zeta function, Pacific J. Math. 1 (1951), 161–167.

[5] K. N. Boyadzhiev, A series transformation formula and related polynomials, Int. J. Math. Math. Sci. 23 (2005), 3849–3866.

[6] K. N. Boyadzhiev, Exponential polynomials, Stirling numbers and evaluation of some gamma integrals, Abstr. Appl. Anal. 18 (2009), 1–18.

[7] K. N. Boyadzhiev, Apostol-Bernoulli functions, derivative polynomials and Eulerian polynomials, Adv. Appl. Discrete Math. 1 (2008), 109–122.

[8] K. N. Boyadzhiev, Close encounters with the Stirling numbers of the second kind, Math. Mag. 85 (2012), 252–266.

[9] K. N. Boyadzhiev and A. Dil, Geometric polynomials: properties and applications to series with zeta values, Anal. Math. 42 (2016), 203–224.

[10] W. Chu and R. R. Zhou, Convolutions of Bernoulli and Euler polynomials, Sarajevo J. Math. 6 (2010), 147-163.

[11] M. C. Dağlı and M. Can, On reciprocity formula of character Dedekind sums and the integral of products of Bernoulli polynomials, J. Number Theory 156 (2015), 105–124.

[12] M. E. Dasef and S. M. Kautz, Some sums of some importance, College Math. J. 28 (1997), 52–55.

[13] A. Dil and V. Kurt, Investigating geometric and exponential polynomials with Euler-Seidel matrices, J. Integer Sequences 14 (2011), Article 11.4.6.

[14] A. Dil and V. Kurt, Polynomials related to harmonic numbers and evaluation of harmonic number series I, Integers 12 (2012), #A38.

[15] A. Dil and V. Kurt, Polynomials related to harmonic numbers and evaluation of harmonic number series II, Appl. Anal. Discrete Math. 5 (2011), 212–229.

[16] R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley Publ. Com., 1994.

[17] O. A. Gross, Preferential arrangements, Amer. Math. Monthly 69 (1962), 4–8.

[18] B. N. Guo and F. Qi, Some identities and an explicit formula for Bernoulli and Stirling numbers, J. Comput. Appl. Math. 255 (2014), 568–579.

[19] E. R. Hansen, A Table of Series and Products, Prentice-Hall Inc., 1975.

[20] L. Kargin and R. B. Corcino, Generalization of Mellin derivative and its applications, Integral Transforms Spec. Funct. 27 (2016), 620–631.

[21] K. Kamano, Sums of products of Bernoulli numbers, including poly-Bernoulli numbers, J. Integer Sequences 13 (2010), Article 10.5.2.

[22] B. C. Kellner, Identities between polynomials related to Stirling and harmonic numbers, Integers 14 (2014), #A54.

[23] M. S. Kim and S. Hu, Sums of products of Apostol–Bernoulli numbers, Ramanujan J. 28 (2012), 113–123.

[24] T. Kim, Some properties on the integral of the product of several Euler polynomials, Quaest. Math. 38 (2015), 553-562.

[25] T. Komatsu, Sums of products of Cauchy numbers, including poly-Cauchy numbers, J. Discrete Math. 2013 (2013), Article 373927.

[26] T. Komatsu, Convolution identities for Cauchy numbers, Acta Math. Hungar. 144 (2014), 76–91.

[27] J. Liu, H. Pan and Y. Zhang, On the integral of the product of the Appell polynomials, Integral Transforms Spec. Funct. 25 (2014), 680–685.

[28] I. Mező, Periodicity of the last digits of some combinatorial sequences, J. Integer Sequences 17 (2014), Article 14.1.1.

[29] I. Mező and G. Nyul, r-Fubini numbers and r-Eulerian numbers, manuscript.

[30] N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, 1924.

[31] M. Rahmani, On p-Bernoulli numbers and polynomials, J. Number Theory 157 (2015), 350–366.

[32] J. Singh, Defining Sums of products of power sums, J. Integer Sequences 19 (2016), Article 16.1.2.

[33] S. M. Tanny, On some numbers related to the Bell numbers, Canad. Math. Bull. 17 (1974), 733–738.

[34] F. Z. Zhao, Sums of products of Cauchy numbers, Discrete Math. 309 (2009), 3830– 3842.

2010 Mathematics Subject Classification: Primary 11B83; Secondary 11B68, 11B75. Keywords: geometric number, geometric polynomial, Apostol-Bernoulli function, p-Bernoulli number.

(Concerned with sequence \underline{A000670}.)

Received September 21 2016; revised version received February 2 2017; February 6 2017. Published in Journal of Integer Sequences, February 9 2017.

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