Recurrences and congruences for higher order geometric polynomials and related numbers
UDC 517.5We obtain new recurrence relations, an explicit formula, and convolution identities for higher order geometric polynomials. These relations generalize known results for geometric polynomials, and lead to congruences for higher order geometric polynomials, particularly for $p$-Bernoulli numb...
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| author | Kargın, L. Cenkci, M. Kargın, L. Cenkci, M. |
| author_facet | Kargın, L. Cenkci, M. Kargın, L. Cenkci, M. |
| author_sort | Kargın, L. |
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| datestamp_date | 2025-03-31T08:46:08Z |
| description | UDC 517.5We obtain new recurrence relations, an explicit formula, and convolution identities for higher order geometric polynomials. These relations generalize known results for geometric polynomials, and lead to congruences for higher order geometric polynomials, particularly for $p$-Bernoulli numbers. |
| doi_str_mv | 10.37863/umzh.v73i12.1080 |
| first_indexed | 2026-03-24T02:04:40Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v73i12.1080
UDC 517.5
L. Kargın, M. Cenkci (Akdeniz Univ., Antalya, Turkey)
RECURRENCES AND CONGRUENCES
FOR HIGHER ORDER GEOMETRIC POLYNOMIALS
AND RELATED NUMBERS
РЕКУРЕНТНI ТА КОНГРУЕНТНI СПIВВIДНОШЕННЯ
ДЛЯ ГЕОМЕТРИЧНИХ ПОЛIНОМIВ ВИЩОГО ПОРЯДКУ
I ВIДПОВIДНИХ ЧИСЕЛ
We obtain new recurrence relations, an explicit formula, and convolution identities for higher order geometric polynomials.
These relations generalize known results for geometric polynomials, and lead to congruences for higher order geometric
polynomials, in particular, for p-Bernoulli numbers.
Отримано новi рекурентнi спiввiдношення, точну формулу та тотожностi згортки для геометричних полiномiв
вищого порядку. Цi спiввiдношення узагальнюють вiдомi результати для геометричних полiномiв i дають можливiсть
отримати конгруентностi для геометричних полiномiв вищого порядку, зокрема для p-чисел Бернуллi.
1. Introduction. For a complex variable y, the geometric polynomials wn(y) of degree n are
defined by [31]
wn(y) =
n\sum
k=0
\Biggl\{
n
k
\Biggr\}
k!yk, (1.1)
where
\biggl\{
n
k
\biggr\}
is the Stirling number of the second kind [15]. These polynomials have been studied
from analytic, combinatoric, and number theoretic points of view. Analytically, they are used in
evaluating geometric series of the form [4]
\infty \sum
k=0
knyk
with \biggl(
y
d
dy
\biggr) n 1
1 - y
=
\infty \sum
k=0
knyk =
1
1 - y
wn
\biggl(
y
1 - y
\biggr)
for every | y| < 1 and every n \in \BbbZ , n \geq 0. Combinatorially, they are related to the total number of
preferential arrangements of n objects
wn(1) := wn =
n\sum
k=0
\Biggl\{
n
k
\Biggr\}
k!,
that is, the number of partitions of an n-element set into k nonempty distinguishable subsets
(c.f. [10]). Number theoretic studies on the geometric polynomials are mostly originated from their
exponential generating function
c\bigcirc L. KARGIN, M. CENKCI, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12 1619
1620 L. KARGIN, M. CENKCI
\infty \sum
n=0
wn(y)
tn
n!
=
1
1 - y(et - 1)
.
For example, setting y = - 1
2
gives
wn
\biggl(
- 1
2
\biggr)
=
2
n+ 1
\bigl(
1 - 2n+1
\bigr)
Bn+1 = - Tn
2n
,
where Bn are Bernoulli numbers and Tn are tangent numbers. Bernoulli numbers also occur in
integrals involving geometric polynomials, namely, we have [24]
1\int
0
wn( - y)dy = Bn, n > 0.
Moreover, we note that [21]
1\int
0
(1 - y)pwn( - y)dy =
1
p+ 1
Bn,p,
where Bn,p are p-Bernoulli numbers [30] (see Section 2 for definitions). The congruence identities
of geometric numbers is also one of the subjects studied. Gross [16] showed that
wn+4 = wn (\mathrm{m}\mathrm{o}\mathrm{d} 10),
which was generalized by Kauffman [19] later. Mező [27] also gave an elementary proof for Gross’
identity. Moreover, Diagana and Maı̈ga [11] used p-adic Laplace transform and p-adic integration
to give some congruences for geometric numbers. We refer to the papers [5 – 7, 12, 20, 29] and the
references therein for more on geometric numbers and polynomials.
In the literature, there are numerous studies for the generalization of geometric polynomials (see,
e.g., [13, 14, 22, 23]). One of the natural extension of geometric polynomials is the higher order
geometric polynomials [4]
w(r)
n (y) =
n\sum
k=0
\Biggl\{
n
k
\Biggr\}
(r)ky
k, r > 0, (1.2)
where (x)n is the Pochhammer symbol defined by (x)n = x(x+ 1) . . . (x+ n - 1) with (x)0 = 1.
It is evident that w(1)
n (y) = wn(y). The polynomials w
(r)
n (y) have the property [4]\biggl(
y
d
dy
\biggr) n 1
(1 - y)r+1
=
\infty \sum
k=0
\biggl(
k + r
k
\biggr)
knyk =
1
(1 - y)r+1
w(r+1)
n
\biggl(
y
1 - y
\biggr)
(1.3)
for any n, r = 0, 1, 2, . . . , and may be defined by means of the exponential generating function [4]
\infty \sum
n=0
w(r)
n (y)
tn
n!
=
\biggl(
1
1 - y(et - 1)
\biggr) r
.
On the other hand, the higher order geometric polynomials and exponential (or single variable
Bell) polynomials
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
RECURRENCES AND CONGRUENCES FOR HIGHER ORDER GEOMETRIC POLYNOMIALS . . . 1621
\varphi n(y) =
n\sum
k=0
\Biggl\{
n
k
\Biggr\}
yk
are connected by
w(r)
n (y) =
1
\Gamma (r)
\infty \int
0
\lambda r - 1\varphi n(y\lambda )e
- \lambda d\lambda (1.4)
(c.f. [4, 8]). According to this integral representation, several generating functions and recurrence
relations for higher order geometric polynomials were obtained in [8]. Namely, w(r)
n+m(y) admits a
recurrence relation according to the family
\Bigl\{
yjw
(r+j)
n (y)
\Bigr\}
as follows:
w
(r)
n+m(y) =
n\sum
k=0
m\sum
j=0
\Biggl(
n
k
\Biggr) \Biggl\{
m
j
\Biggr\}
(r)jj
n - kyjw
(r+j)
k (y). (1.5)
Setting y = 1 in (1.2), we have higher order geometric numbers w(r)
n . The higher order geometric
numbers and geometric numbers are connected with w
(1)
n = wn and the formula
w(r)
n =
1
r!2r
r\sum
k=0
\Biggl[
r + 1
k + 1
\Biggr]
wn+k, (1.6)
which was proved by a combinatorial method in [1] (Theorem 2). Here,
\biggl[
n
k
\biggr]
is the Stirling number
of the first kind [15]. Moreover, some congruence identities for the higher order geometric numbers
can also be found in the recent work [11].
In this paper, dealing with two-variable geometric polynomials defined in [25] by
\infty \sum
n=0
w(r)
n (x; y)
tn
n!
=
\biggl(
1
1 - y(et - 1)
\biggr) r
ext, (1.7)
we obtain new recurrence relations, an explicit formula, and a result generalizing (c.f. [8])
n\sum
k=0
\Biggl(
n
k
\Biggr)
w
(r)
k (y)wn - k(y) =
w
(r)
n+1(y) + rw
(r)
n (y)
r(1 + y)
for higher order geometric polynomials. We particularly use the explicit formula to obtain an integral
representation similar to (1.4) involving r-Bell polynomials, which are defined in [26] as
\varphi n,r(y) =
n\sum
k=0
\Biggl\{
n+ r
k + r
\Biggr\}
r
yk, (1.8)
where
\biggl\{
n+ r
k + r
\biggr\}
r
are r-Stirling numbers of the second kind [3]. The resulting integral representation
enables us to utilize some properties of r-Bell polynomials for higher order geometric polynomials.
In particular, we evaluate the infinite sum
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
1622 L. KARGIN, M. CENKCI
\infty \sum
k=0
(k + r)n
\Biggl(
k + r - 1
k
\Biggr)
in terms of higher order geometric polynomials, obtain an ordinary generating function for higher
order geometric polynomials, introduce a new recurrence for w
(r)
n+m(y), and generalize (1.6). We
also give an integral representation relating the higher order geometric polynomials and p-Bernoulli
numbers, and express properties of p-Bernoulli numbers originating from those for the higher order
geometric polynomials. Besides, using some of theses results, we prove congruences for higher order
geometric polynomials and p-Bernoulli numbers. Particularly, we state a von Staudt – Clausen-type
congruence for p-Bernoulli numbers.
This paper is organized as follows. In Section 2, we summarize known results that we need
throughout the paper. We state and prove aforementioned results for higher order geometric polyno-
mials and p-Bernoulli numbers in Section 3. In Section 4, we deal with some congruences for higher
order geometric polynomials and p-Bernoulli numbers.
2. Preliminaries. The Stirling numbers of the first kind
\biggl[
n
k
\biggr]
can be defined by means of
x(x+ 1) . . . (x+ n - 1) =
n\sum
k=0
\Biggl[
n
k
\Biggr]
xk
or by the generating function
\bigl(
- \mathrm{l}\mathrm{o}\mathrm{g}(1 - x)
\bigr) k
= k!
\infty \sum
n=k
\Biggl[
n
k
\Biggr]
xk
k!
(c.f. [9, 15]). It follows from either of these definitions that\Biggl[
n
k
\Biggr]
= (n - 1)
\Biggl[
n - 1
k
\Biggr]
+
\Biggl[
n - 1
k - 1
\Biggr]
(2.1)
with \Biggl[
n
0
\Biggr]
= 0, if n > 0, and
\Biggl[
n
k
\Biggr]
= 0, if k > n or k < 0.
We note the following special values which will be used in the sequel:\Biggl[
n
1
\Biggr]
= 1,
\Biggl[
n
1
\Biggr]
= (n - 1)! if n > 0,
\Biggl[
n
n - 1
\Biggr]
=
\Biggl(
n
2
\Biggr)
,
\Biggl[
n
n - 2
\Biggr]
=
3n - 1
4
\Biggl(
n
3
\Biggr)
,
\Biggl[
n
n - 3
\Biggr]
=
\Biggl(
n
2
\Biggr) \Biggl(
n
4
\Biggr)
.
Many properties of
\biggl[
n
k
\biggr]
can be found in [9, p. 214 – 219]. In particular, we have
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
RECURRENCES AND CONGRUENCES FOR HIGHER ORDER GEOMETRIC POLYNOMIALS . . . 1623
k
\Biggl[
n
k
\Biggr]
=
n - 1\sum
i=k - 1
\Biggl(
n
i
\Biggr)
(n - i - 1)!
\Biggl[
i
k - 1
\Biggr]
.
This equality can be used to obtain some congruences for
\biggl[
n
k
\biggr]
. For example, if we take n = q,
where q is a prime number, then\Biggl[
q
k
\Biggr]
\equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} q), k = 2, 3, . . . , q - 1, (2.2)
since \Biggl(
q
i
\Biggr)
\equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} q), i = 1, 2, . . . , q - 1.
The Stirling numbers of the second kind
\Biggl\{
n
k
\Biggr\}
can be defined by means of
xn =
n\sum
k=0
\Biggl\{
n
k
\Biggr\}
x(x - 1) . . . (x - k + 1),
or by the generating function
(\mathrm{e}x - 1)k = k!
\infty \sum
n=k
\Biggl\{
n
k
\Biggr\}
xn
n!
(c.f. [9, 15]). It follows from the generating function that\Biggl\{
n
k
\Biggr\}
=
\Biggl\{
n - 1
k - 1
\Biggr\}
+ k
\Biggl\{
n - 1
k
\Biggr\}
with \Biggl\{
n
0
\Biggr\}
= 0, if n > 0,
\Biggl\{
n
k
\Biggr\}
= 0, if k > n or k < 0,
\Biggl\{
n
n
\Biggr\}
= 1,
\Biggl\{
n
1
\Biggr\}
= 1, if n > 0.
We note the following known identity for
\biggl\{
n
k
\biggr\}
for future reference:
\Biggl\{
n
k
\Biggr\}
=
1
k!
k\sum
j=1
( - 1)k - j
\Biggl(
k
j
\Biggr)
jn. (2.3)
Performing the product of two generating functions for
\biggl\{
n
k
\biggr\}
, we obtain the convolution for-
mula [18]
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
1624 L. KARGIN, M. CENKCI\Biggl(
k1 + k2
k1
\Biggr) \Biggl\{
n
k1 + k2
\Biggr\}
=
n\sum
m=0
\Biggl(
n
m
\Biggr) \Biggl\{
m
k1
\Biggr\} \Biggl\{
n - m
k2
\Biggr\}
.
Letting k = k1 + k2 and n = q, a prime number, we deduce that\Biggl\{
q
k
\Biggr\}
\equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} q), k = 2, 3, . . . , q - 1, (2.4)
since again \Biggl(
q
i
\Biggr)
\equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} q), i = 1, 2, . . . , q - 1,
and 1 < k < q.
Stirling numbers have been generalized in many ways. One of them is called r-Stirling numbers
(or weighted Stirling numbers). r-Stirling numbers of the second kind
\biggl\{
n
k
\biggr\}
r
can be defined by
means of the generating function (see [3])
(\mathrm{e}x - 1)k\mathrm{e}rx = k!
\infty \sum
n=k
\Biggl\{
n
k
\Biggr\}
r
xn
n!
. (2.5)
The Bernoulli numbers Bn are defined by the generating function
t
\mathrm{e}t - 1
=
\infty \sum
n=0
Bn
tn
n!
or by the equivalent recursion
B0 = 1 and
n - 1\sum
k=0
Bk
k!(n - k)!
= 0 for n \geq 2.
The first values are
B1 = - 1
2
, B2 =
1
6
, B4 = - 1
30
, B6 =
1
42
,
and B2k+1 = 0 for k \geq 1. The denominators of the Bernoulli numbers can be completely determined
due to von Staudt – Clausen theorem: for any integer n \geq 1, B2n can be written as
B2n = A2n -
\sum
q : (q - 1)| 2n
1
q
,
where A2n is an integer and the sum runs over all the prime numbers such that (q - 1)| 2n. It can be
stated equivalently as
qB2n \equiv
\left\{ 0 (\mathrm{m}\mathrm{o}\mathrm{d} q), if (q - 1) \nmid 2n,
- 1 (\mathrm{m}\mathrm{o}\mathrm{d} q), if (q - 1) | 2n.
(2.6)
We note that this classification is also valid for B1.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
RECURRENCES AND CONGRUENCES FOR HIGHER ORDER GEOMETRIC POLYNOMIALS . . . 1625
Many generalizations of Bernoulli numbers appear in the literature. One generalization is the
p-Bernoulli numbers Bn,p, which are due to Rahmani [30], defined by means of the generating
function
\infty \sum
n=0
Bn,p
xn
n!
= 2F1
\bigl(
1, 1; p+ 2, 1 - et
\bigr)
,
where 2F1(a, b; c; z) is the Gaussian hypergeometric function
2F1(a, b; c; z) =
\infty \sum
k=0
(a)k(b)k
(c)k
zk
k!
.
p-Bernoulli numbers are related to Bernoulli numbers in that Bn,0 = Bn and
p\sum
k=0
\Biggl[
p
k
\Biggr]
( - 1)kBn+k =
p!
p+ 1
Bn,p for n, p \geq 0, (2.7)
and satisfy an explicit formula of the form
Bn,p =
p+ 1
p!
n\sum
k=0
\Biggl\{
n+ p
k + p
\Biggr\}
p
( - 1)k(k + p)!
k + p+ 1
. (2.8)
3. Recurrence relations. From the generating function for higher order two-variable geometric
polynomials (1.7), we have
w(r)
n (x; y) =
n\sum
k=0
\Biggl(
n
k
\Biggr)
w
(r)
k (y)xn - k. (3.1)
Then it is obvious that
w(r)
n (0; y) = w(r)
n (y), w(1)
n (x; y) = wn(x; y),
w(r)
n (0; 1) = w(r)
n and w(1)
n (0; 1) = wn.
Setting x+ r instead of x in (1.7), we have
w(r)
n (x+ r; y) = ( - 1)nw(r)
n ( - x; - y - 1) for n \geq 0.
Then, for x = 0, we conclude that
w(r)
n (r; y) = ( - 1)nw(r)
n ( - y - 1), (3.2)
a relationship between two-variable and single variable higher order geometric polynomials.
Proposition 3.1. For n \geq 0 and r > 0, we have the following recurrence formulae:
n\sum
k=0
\Biggl(
n
k
\Biggr)
w
(r)
k (y)rn - k = ( - 1)nw(r)
n ( - y - 1) (3.3)
and
n\sum
k=0
\Biggl(
n
k
\Biggr)
w
(r+1)
k (y) =
1
ry
w
(r)
n+1(y). (3.4)
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
1626 L. KARGIN, M. CENKCI
Proof. Combining (3.1) and (3.2), we have (3.3). Furthermore, taking x = 1 in (3.1) and using
the recurrence relation presented in [25] (Theorem 3.4)
w
(r)
n+1(x; y) = w(r)
n (x; y) + ryw(r+1)
n (x+ 1; y)
for x = 1, we obtain (3.4).
Theorem 3.1. For every n \geq 0 and every r1, r2 > 0, we have the convolution identity
n\sum
k=0
\Biggl(
n
k
\Biggr)
w
(r1)
k (y)w
(r2)
n - k(y) =
w
(r1+r2 - 1)
n+1 (y) + (r1 + r2 - 1)w
(r1+r2 - 1)
n (y)
(r1 + r2 - 1)(1 + y)
. (3.5)
Proof. We first note that
yr
e(x+1)t\bigl(
1 - y(et - 1)
\bigr) r+1 =
d
dt
\biggl( \biggl(
1
1 - y(et - 1)
\biggr) r
ext
\biggr)
- xext\bigl(
1 - y(et - 1)
\bigr) r .
Let x = x1 + x2 - 1 and r = r1 + r2 - 1. Then by (1.7), product of two infinite series, and formal
differentiation under summation, we obtain
yr
e(x+1)t\bigl(
1 - y(et - 1)
\bigr) r+1 = y(r1 + r2 - 1)
e
\bigl(
x1+x2
\bigr)
t\bigl(
1 - y(et - 1)
\bigr) r1+r2
=
= y(r1 + r2 - 1)
\infty \sum
n=0
w(r1)
n
\bigl(
x1; y
\bigr) tn
n!
\infty \sum
n=0
w
\bigl(
r2
\bigr)
n
\bigl(
x2; y
\bigr) tn
n!
=
= y(r1 + r2 - 1)
\infty \sum
n=0
\Biggl[
n\sum
k=0
\Biggl(
n
k
\Biggr)
w
(r1)
k (x1; y)w
(r2)
n - k(x2; y)
\Biggr]
tn
n!
,
xext\bigl(
1 - y(et - 1)
\bigr) r = (x1 + x2 - 1)
\infty \sum
n=0
w(r1+r2 - 1)
n (x1 + x2 - 1; y)
tn
n!
,
and
d
dt
\biggl( \biggl(
1
1 - y(et - 1)
\biggr) r
ext
\biggr)
=
\infty \sum
n=0
w
(r1+r2 - 1)
n+1
\bigl(
x1 + x2 - 1; y
\bigr) tn
n!
.
Equating coefficients of
tn
n!
on both sides, we derive
n\sum
k=0
\Biggl(
n
k
\Biggr)
w
(r1)
k (x1; y)w
(r2)
n - k(x2; y) =
=
1
y(r1 + r2 - 1)
\Bigl[
w
(r1+r2 - 1)
n+1 (x1 + x2 - 1; y) - (x1 + x2 - 1)w(r1+r2 - 1)
n
\bigl(
x1 + x2 - 1; y
\bigr) \Bigr]
.
Setting x1 = r1, x2 = r2 and using (3.2), we obtain the convolution formula (3.5).
In the following theorem we give a new explicit expression for higher order geometric polyno-
mials and numbers.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
RECURRENCES AND CONGRUENCES FOR HIGHER ORDER GEOMETRIC POLYNOMIALS . . . 1627
Theorem 3.2. For n \geq 0,
w(r)
n (y) =
n\sum
k=0
\Biggl\{
n+ r
k + r
\Biggr\}
r
(r)k( - 1)n+k(y + 1)k. (3.6)
In particular,
w(r)
n =
n\sum
k=0
\Biggl\{
n+ r
k + r
\Biggr\}
r
(r)k( - 1)n+k2k.
Proof. Writing x = r in (1.7), employing the generalized binomial formula, and using the
generating function of r-Stirling numbers (2.5), we have
\infty \sum
n=0
w(r)
n (r; y)
tn
n!
=
\biggl(
1
1 - y(et - 1)
\biggr) r
ert =
\infty \sum
k=0
(r)ky
k
\bigl(
et - 1
\bigr) k
k!
ert =
=
\infty \sum
k=0
\infty \sum
n=k
\Biggl\{
n+ r
k + r
\Biggr\}
r
(r)ky
k t
n
n!
=
\infty \sum
n=0
\Biggl[
n\sum
k=0
\Biggl\{
n+ r
k + r
\Biggr\}
r
(r)ky
k
\Biggr]
tn
n!
.
Comparing the coefficients of
tn
n!
, we obtain
w(r)
n (r; y) =
n\sum
k=0
\Biggl\{
n+ r
k + r
\Biggr\}
r
(r)ky
k.
Using (3.2) and replacing y with - (y + 1), we reach the desired equation.
Now, with use of Theorem 3.2, we connect higher order geometric polynomials and r-Bell
polynomials in the following lemma which will be useful for the subsequent results.
Lemma 3.1. For every n \geq 0 and every r > 0, we have the integral representation
( - 1)nw(r)
n ( - y - 1) =
1
\Gamma (r)
\infty \int
0
\lambda r - 1\varphi n,r(y\lambda )e
- \lambda d\lambda . (3.7)
Proof. By (1.8) we have
\infty \int
0
\lambda r - 1\varphi n,r(y\lambda )e
- \lambda d\lambda =
n\sum
k=0
\Biggl\{
n+ r
k + r
\Biggr\}
r
yk
\infty \int
0
\lambda r+k - 1e - \lambda d\lambda =
n\sum
k=0
\Biggl\{
n+ r
k + r
\Biggr\}
r
\Gamma (r + k)yk.
Using (3.6) in the above yields the desired equation.
Higher order geometric polynomials are seen in the evaluation of the infinite series (1.3). If we
apply Lemma 3.1 to the Dobinski’s formula for r-Bell polynomials
\varphi n,r(y) =
1
ey
\infty \sum
n=0
(k + r)n
k!
xk,
we can evaluate a new infinite series in terms of higher order geometric polynomials.
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1628 L. KARGIN, M. CENKCI
Theorem 3.3. For every n \geq 0 and every r > 0, | y| < 1,
\infty \sum
k=0
(k + r)n
\biggl(
k + r - 1
k
\biggr)
yk =
( - 1)n
(1 - y)r
w(r)
n
\biggl(
1
y - 1
\biggr)
.
Next we introduce ordinary generating function for higher order geometric polynomials.
Theorem 3.4. For real y < - 1
2
, the higher order geometric polynomials have the generating
function
\infty \sum
n=0
w(r)
n (y)tn =
( - 1)r
(1 + rt)yr
2F1
\biggl(
rt+ 1
t
, r;
rt+ t+ 1
t
;
y + 1
y
\biggr)
.
Proof. We start by observing the ordinary generating function for r-Bell polynomials [26]
(Theorem 3.2)
\infty \sum
n=0
\varphi n,r(y)t
n =
- 1
rt - 1
1
ey
1F1
\biggl(
rt - 1
t
;
rt+ t - 1
t
; y
\biggr)
.
In light of the equation (3.7), this equation can be written as
\infty \sum
n=0
( - 1)nw(r)
n ( - y - 1)tn =
1
(1 - rt)\Gamma (r)
\infty \int
0
\lambda r - 1e - (y+1)\lambda
1F1
\biggl(
rt - 1
t
;
rt+ t - 1
t
; y\lambda
\biggr)
d\lambda =
=
1
(1 - rt)\Gamma (r)
\infty \sum
k=0
\biggl(
rt - 1
t
\biggr)
k\biggl(
rt+ t - 1
t
\biggr)
k
yk
k!
\infty \int
0
\lambda r+k - 1e - (y+1)\lambda d\lambda =
=
1
(1 - rt)(1 + y)r
\infty \sum
k=0
\biggl(
rt - 1
t
\biggr)
k
(r)k\biggl(
rt+ t - 1
t
\biggr)
k
k!
\biggl(
y
1 + y
\biggr) k
=
=
1
(1 - rt)(1 + y)r
2F1
\biggl(
rt - 1
t
, r;
rt+ t - 1
t
;
y
1 + y
\biggr)
.
We then replace - (y + 1) with y and - t with t to obtain the desired equation.
Now, we give an alternative representation for w
(r)
n+m(y), which also generalizes (1.6) in the
following theorem.
Theorem 3.5. For all nonnegative integers n, m, r and p, we have
w
(r)
n+m(y) =
m\sum
k=0
\Biggl\{
m+ r
k + r
\Biggr\}
r
(r)k( - 1)m+k(y + 1)kw(r+k)
n (y) (3.8)
and
w(r+p)
n (y) =
1
(r)p(1 + y)p
p\sum
k=0
\Biggl[
p+ r
k + r
\Biggr]
r
w
(r)
n+k(y). (3.9)
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RECURRENCES AND CONGRUENCES FOR HIGHER ORDER GEOMETRIC POLYNOMIALS . . . 1629
Proof. We prove (3.8) first. Using the following property of r-Bell polynomials presented in
[28] (Eq. (8))
\varphi n+m,r(y) =
m\sum
k=0
\Biggl\{
m+ r
k + r
\Biggr\}
r
yk\varphi n,r+k(y)
in (3.7), we have
( - 1)n+mw
(r)
n+m( - y - 1) =
m\sum
k=0
\Biggl\{
m+ r
k + r
\Biggr\}
r
yk
\Gamma (k + r)
\Gamma (r)
1
\Gamma (k + r)
\infty \int
0
\lambda r+k - 1\varphi n,r+k(y\lambda )e
- \lambda d\lambda =
=
m\sum
k=0
\Biggl\{
m+ r
k + r
\Biggr\}
r
(r)ky
k( - 1)nw(r)
n ( - y - 1),
which is equal to (3.8).
To prove (3.9), we use the formula
yp\varphi n,r+p(y) =
p\sum
k=0
\Biggl[
p+ r
k + r
\Biggr]
r
( - 1)p - k\varphi n+k,r(y)
[28] (Eq. (11)) in (3.7).
Using (3.3) in (3.8), we obtain the following result similar which is slightly different from (1.5).
Corollary 3.1. We have
w
(r)
n+m(y) =
n\sum
k=0
m\sum
j=0
\Biggl\{
m+ r
j + r
\Biggr\}
r
\Biggl(
n
k
\Biggr)
(j + r)n - k( - 1)n+m+j(r)j(y + 1)jw
(r+j)
k ( - y - 1).
We note that it is also possible to derive this result by applying (1.4) and (3.7) in
\varphi n+m,r(y) =
n\sum
k=0
m\sum
j=0
\Biggl(
n
k
\Biggr) \Biggl\{
m+ r
j + r
\Biggr\}
r
(j + r)n - kyj\varphi k(y),
a formula given in [28] (Eq. (9)). Moreover, for r = 1, (3.9) can be written as
(1 + y)pw(p+1)
n (y) =
1
p!
p\sum
k=0
\Biggl[
p+ 1
k + 1
\Biggr]
wn+k(y), (3.10)
which is also polynomial extension of (1.6). Replacing y by - y and integrating both sides with
respect to y from 0 to 1, we have
1\int
0
(1 - y)pw(p+1)
n ( - y)dy =
1
p!
p\sum
k=0
\Biggl[
p+ 1
k + 1
\Biggr]
Bn+k.
Then using (2.7), we obtain the following integral representation for p-Bernoulli numbers.
Theorem 3.6. For n \geq 1 and p \geq 0,
1\int
0
(1 - y)pw(p+1)
n ( - y)dy = ( - 1)n - 1 p+ 1
p+ 2
Bn - 1,p+1. (3.11)
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1630 L. KARGIN, M. CENKCI
The explicit formula (2.8) for p-Bernoulli numbers can be also deduced using this integral repre-
sentation in (3.6).
The following theorem generalizes the identities (2.8) and (2.7).
Theorem 3.7. For n, p,m \geq 0, we have
Bn+m,p = (p+ 1)
m\sum
k=0
\Biggl\{
m+ p
k + p
\Biggr\}
p
( - 1)k(p+ 1)k
k + p+ 1
Bn,p+k. (3.12)
For n, r \geq 1 and p \geq 0, we get
Bn,p+r =
r(p+ r + 1)
(r + 1)(p+ r)(r)p
p\sum
k=0
\Biggl[
p+ r
k + r
\Biggr]
r
( - 1)kBn+k,r. (3.13)
Proof. Firstly, we replace y with - y in (3.9), multiply both sides by (1 - y)r - 1, and integrate
with respect to y from 0 to 1. The result is
1\int
0
(1 - y)p+r - 1w(r+p)
n ( - y)dy =
1
(p)r
p\sum
k=0
\Biggl[
p+ r
k + r
\Biggr]
r
1\int
0
(1 - y)r - 1w
(r)
n+k( - y)dy.
From (3.11) this equation turns into
Bn - 1,p+r =
r(p+ r + 1)
(r + 1)(p+ r)(r)p
p\sum
k=0
\Biggl[
p+ r
k + r
\Biggr]
r
( - 1)kBn+k - 1,r.
Replacing n with n+ 1 in the above equation completes the proof (3.13).
Applying the same method to the identity (3.8) gives (3.12).
4. Congruences. In this section, we first consider congruences modulo a prime number q for
higher order geometric polynomials. We start with two auxiliary results.
Lemma 4.1. Let q be an odd prime and y be an integer. Then we have
wq(y) \equiv y (\mathrm{m}\mathrm{o}\mathrm{d} q).
Proof. From (1.1), we obtain
wq(y) =
\Biggl\{
q
0
\Biggr\}
+
\Biggl\{
q
1
\Biggr\}
y +
\Biggl\{
q
q
\Biggr\}
q!yq +
q - 1\sum
k=2
\Biggl\{
q
k
\Biggr\}
k!yk.
Since \Biggl\{
q
0
\Biggr\}
= 0,
\Biggl\{
q
1
\Biggr\}
=
\Biggl\{
q
q
\Biggr\}
= 1
and by (2.4)
k!
\Biggl\{
q
k
\Biggr\}
\equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} q), k = 2, 3, . . . , q - 1,
we get the desired result
wq(y) \equiv y + q!yq \equiv y (\mathrm{m}\mathrm{o}\mathrm{d} q).
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RECURRENCES AND CONGRUENCES FOR HIGHER ORDER GEOMETRIC POLYNOMIALS . . . 1631
Lemma 4.2. Let q be a prime and y be an integer. Then, for all n \geq 1, we have
wq+n - 1(y) \equiv wn(y) (\mathrm{m}\mathrm{o}\mathrm{d} q).
Proof. If q = 2, then, by (1.1), we obtain
wn+1(y) - wn(y) =
n+1\sum
k=0
\Biggl\{
n+ 1
k
\Biggr\}
k!yk -
n\sum
k=0
\Biggl\{
n
k
\Biggr\}
k!yk
= (n+ 1)!yn+1 +
n\sum
k=2
\Biggl( \Biggl\{
n+ 1
k
\Biggr\}
-
\Biggl\{
n
k
\Biggr\} \Biggr)
k!yk \equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} 2),
since
\biggl\{
n
k
\biggr\}
= 0 and
\biggl\{
n
1
\biggr\}
= 1 for n > 0.
Now, suppose that q is an odd prime and let n \geq q - 1. Then again by (1.1) we write
wq+n - 1(y) - wn(y) =
q+n - 1\sum
k=0
\Biggl\{
q + n - 1
k
\Biggr\}
k!yk -
n\sum
k=0
\Biggl\{
n
k
\Biggr\}
k!yk =
=
q - 1\sum
k=0
\Biggl( \Biggl\{
q + n - 1
k
\Biggr\}
-
\Biggl\{
n
k
\Biggr\} \Biggr)
k!yk+
+
q+n - 1\sum
k=q
\Biggl\{
q + n - 1
k
\Biggr\}
k!yk -
n\sum
k=q
\Biggl\{
n
k
\Biggr\}
k!yk \equiv
\equiv
q - 1\sum
k=0
\Biggl( \Biggl\{
q + n - 1
k
\Biggr\}
-
\Biggl\{
n
k
\Biggr\} \Biggr)
k!yk (\mathrm{m}\mathrm{o}\mathrm{d} q).
By using (2.3), we obtain
wq+n - 1(y) - wn(y) \equiv
q - 1\sum
k=0
k!yk
1
k!
k\sum
j=1
( - 1)k - j
\Biggl(
k
j
\Biggr)
jn
\bigl(
jq - 1 - 1
\bigr)
\equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} q),
since (j, q) = 1 and jq - 1 - 1 \equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} q).
If 1 \leq n < q - 1, then we write
wq+n - 1(y) - wn(y) =
q+n - 1\sum
k=0
\Biggl\{
q + n - 1
k
\Biggr\}
k!yk -
n\sum
k=0
\Biggl\{
n
k
\Biggr\}
k!yk =
=
q - 1\sum
k=0
\Biggl( \Biggl\{
q + n - 1
k
\Biggr\}
-
\Biggl\{
n
k
\Biggr\} \Biggr)
k!yk+
+
q+n - 1\sum
k=q
\Biggl\{
q + n - 1
k
\Biggr\}
k!yk -
q - 1\sum
k=n
\Biggl\{
n
k
\Biggr\}
k!yk +
q - 1\sum
k=n+1
\Biggl\{
n
k
\Biggr\}
k!yk \equiv
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1632 L. KARGIN, M. CENKCI
\equiv
q - 1\sum
k=0
k\sum
j=1
( - 1)k - j
\Biggl(
k
j
\Biggr)
jn(jq - 1 - 1) \equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} q),
since
\biggl\{
n
k
\biggr\}
= 0 when k > n.
Therefore, for n \geq 1, wq+n - 1(y) \equiv wn(y) (\mathrm{m}\mathrm{o}\mathrm{d} q).
We note that a more general result can be found in [2] for Fubini numbers.
Theorem 4.1. Let q be an odd prime. If 1+y is not a multiple of q, then w
(q)
q (y) \equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} q).
Proof. We set p = q - 1 and n = q in (3.10) to obtain
(1 + y)q - 1(q - 1)!w(q)
q (y) =
q\sum
k=1
\Biggl[
q
k
\Biggr]
wq+k - 1(y) =
=
\Biggl[
q
1
\Biggr]
wq(y) +
\Biggl[
q
q
\Biggr]
w1(y) +
q - 1\sum
k=2
\Biggl[
q
k
\Biggr]
wq+k - 1(y) =
= (q - 1)!wq(y) + y +
q - 1\sum
k=2
\Biggl[
q
k
\Biggr]
wq+k - 1(y).
By Lemmas 4.1 and 4.2, we find that
(1 + y)q - 1(q - 1)!w(q)
q (y) \equiv ( - 1)y + y +
q - 1\sum
k=2
\Biggl[
q
k
\Biggr]
wk(y) \equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} q),
since by (2.2)
\biggl[
q
k
\biggr]
\equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} q) for 2 \leq k \leq q - 1 and wk(y) is an integer when y is an integer.
The result now follows from Fermat’s and Wilson’s theorems.
It is obvious from (1.2) that if y is an integer which is a multiple of q, then w
(r)
n (y) \equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} q),
since
\biggl\{
n
k
\biggr\}
(r)k is an integer. We note that Theorem 4.1 is a special case which can be drawn from
the following result.
Theorem 4.2. If y is an integer that is not a multiple of q, then w
(r)
n (y) \equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} q) for n \geq 1
and r \equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} q).
Proof. Let r = tq for some integer t. By (1.2), we have
w(r)
n (y) =
n\sum
k=0
k!
\biggl(
tq + k - 1
k
\biggr) \Biggl\{
n
k
\Biggr\}
yk.
Since
k!
\Biggl(
tq + k - 1
k
\Biggr)
= (tq + k - 1)(tq + k - 2) . . . (tq + 1)(tq) \equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} q),
we have the result.
Theorem 4.3. If y is an integer such that y and 1+ y are not multiples of an odd prime q, then
w
(r)
q - 1(y) \equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} q) for r \equiv 1 (\mathrm{m}\mathrm{o}\mathrm{d} q).
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RECURRENCES AND CONGRUENCES FOR HIGHER ORDER GEOMETRIC POLYNOMIALS . . . 1633
Proof. Let r = 1 + tq for some integer t. By (1.2), we have
w
(r)
q - 1(y) =
q - 1\sum
k=0
k!
\Biggl(
tq + k
k
\Biggr) \Biggl\{
q - 1
k
\Biggr\}
yk.
Since \Biggl(
tq + k
k
\Biggr)
=
(tq + k)(tq + k - 1) . . . (tq + 1)
k!
\equiv k(k - 1) . . . 1
k!
= 1(\mathrm{m}\mathrm{o}\mathrm{d} q),
we deduce that
w
(r)
q - 1(y) \equiv
q - 1\sum
k=0
k!
\Biggl\{
q - 1
k
\Biggr\}
yk (\mathrm{m}\mathrm{o}\mathrm{d} q).
It follows from (2.3) that
k!
\Biggl\{
q - 1
k
\Biggr\}
\equiv ( - 1)k - 1 (\mathrm{m}\mathrm{o}\mathrm{d} q)
for 1 \leq k \leq q - 1. Since
\biggl\{
q - 1
0
\biggr\}
= 0, then we have
w
(r)
q - 1(y) \equiv
q - 1\sum
k=0
( - 1)k - 1yk = 1 -
q - 1\sum
k=0
( - 1)kyk = 1 - 1 + yq
1 + y
(\mathrm{m}\mathrm{o}\mathrm{d} q),
which implies
(1 + y)w
(r)
q - 1(y) \equiv 1 + y - 1 + yq \equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} q),
and the result.
These results and their proofs are direct generalizations of the corresponding congruences for
higher order geometric numbers given in [11] (Corollary 4.2).
We conclude the study of congruences for higher order geometric polynomials by a similar result.
Theorem 4.4. If y is an integer that is not a multiple of an odd prime q, then w
(r)
q+1(y) \equiv 0
(\mathrm{m}\mathrm{o}\mathrm{d} q) for r \equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} q) and w
(r)
q+1(y) \equiv - y (\mathrm{m}\mathrm{o}\mathrm{d} q) for r \equiv - 1 (\mathrm{m}\mathrm{o}\mathrm{d} q).
Proof. For a prime q and nonnegative integer m, we have\Biggl\{
q +m
k
\Biggr\}
\equiv
\Biggl\{
m+ 1
k
\Biggr\}
+
\Biggl\{
m
k - q
\Biggr\}
(\mathrm{m}\mathrm{o}\mathrm{d} q).
This result was given by Howard in [17], and can be easily verified by induction on m. It then
follows that
\biggl\{
q + 1
k
\biggr\}
\equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} q) for k = 3, 4, . . . , q and
\biggl\{
q + 1
2
\biggr\}
\equiv 1 (\mathrm{m}\mathrm{o}\mathrm{d} q).
Now, we write (1.2) as
w
(r)
q+1(y) =
q+1\sum
k=0
k!
\Biggl(
r + k - 1
k
\Biggr) \Biggl\{
q + 1
k
\Biggr\}
yk = ry + r(r + 1)
\Biggl\{
q + 1
2
\Biggr\}
y2+
+(q + 1)!
\Biggl(
r + q
q
\Biggr)
yq+1 +
q\sum
k=3
k!
\Biggl(
r + k - 1
k
\Biggr) \Biggl\{
q + 1
k
\Biggr\}
yk \equiv ry + r(r + 1)y2 (\mathrm{m}\mathrm{o}\mathrm{d} q),
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1634 L. KARGIN, M. CENKCI
from which the results follow.
In the rest of this section we consider congruences for p-Bernoulli numbers. In particular, the
following theorem states a von Staudt – Clausen-type result for p-Bernoulli numbers.
Theorem 4.5. Let n be a positive integer. Then we have 4B2n,2 \equiv - 1 (\mathrm{m}\mathrm{o}\mathrm{d} 2) and if
(q - 1) | 2n for an odd prime q, then qB2n,q \equiv - 1
2
(\mathrm{m}\mathrm{o}\mathrm{d} q).
Proof. First, we take p = 2 in (2.6). This gives
2
3
B2n,2 = B2n+2,
or, equivalently,
4B2n,2 = 3 \cdot 2B2n+2.
The result then follows from the von Staudt – Clausen theorem.
Next, let q be an odd prime. Then we replace n by 2n and p by q in (2.6) and obtain
q!
q + 1
B2n,q =
q\sum
k=0
\Biggl[
q
k
\Biggr]
( - 1)kB2n+k =
=
\Biggl[
q
0
\Biggr]
B2n -
\Biggl[
q
1
\Biggr]
B2n+1 +
\Biggl[
q
q - 1
\Biggr]
B2n+q - 1 -
\Biggl[
q
q
\Biggr]
B2n+q +
q - 2\sum
k=2
\Biggl[
q
k
\Biggr]
( - 1)kB2n+k.
Since \Biggl[
q
0
\Biggr]
= 0,
\Biggl[
q
1
\Biggr]
= (q - 1)!,
\Biggl[
q
q - 1
\Biggr]
=
q(q - 1)
2
,
\Biggl[
q
q
\Biggr]
= 1
and B2n+1 = 0, n \geq 1, the above equality turns into
q!
q + 1
B2n,q =
q - 1
2
qB2n+q - 1 +
q - 2\sum
k=2
1
q
\Biggl[
q
k
\Biggr]
( - 1)kqB2n+k.
If (q - 1) | 2n, then (q - 1) | 2n + q - 1, so qB2n+q - 1 \equiv - 1 (\mathrm{m}\mathrm{o}\mathrm{d} q) by (2.6). We also have
(q - 1) \nmid 2n+k for k = 2, 3, . . . , q - 2, so qB2n+k \equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} q) again by (2.6). Noting that
\biggl[
q
k
\biggr]
\equiv 0
(\mathrm{m}\mathrm{o}\mathrm{d} q) for 2 \leq k \leq q - 2, we observe that the sum vanishes modulo q. Thus,
q!
q + 1
B2n,q \equiv - q - 1
2
(\mathrm{m}\mathrm{o}\mathrm{d} q),
or, equivalently,
qB2n,q \equiv - 1
2
(\mathrm{m}\mathrm{o}\mathrm{d} q),
by Wilson’s theorem.
In the following theorem, we give a congruence for Bq,q, where q > 3 is a prime.
Theorem 4.6. For a prime q > 3, we have
qBq,q \equiv
1
12
- Bq+1 (\mathrm{m}\mathrm{o}\mathrm{d} q).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
RECURRENCES AND CONGRUENCES FOR HIGHER ORDER GEOMETRIC POLYNOMIALS . . . 1635
Proof. Let q > 3 be a prime. Writing n = p = q in (2.7) gives
q!
q + 1
Bq,q =
\Biggl[
q
0
\Biggr]
Bq -
\Biggl[
q
1
\Biggr]
Bq+1 +
\Biggl[
q
q - 1
\Biggr]
B2q - 1 -
\Biggl[
q
q
\Biggr]
B2q -
-
\Biggl[
q
q - 2
\Biggr]
B2q - 2 +
q - 3\sum
k=2
\Biggl[
q
k
\Biggr]
( - 1)kBq+k =
= - (q - 1)!Bq+1 - B2q -
\Biggl[
q
q - 2
\Biggr]
B2q - 2 +
q - 3\sum
k=2
\Biggl[
q
k
\Biggr]
( - 1)kBq+k,
or, equivalently,
q!Bq,q = - (q - 1)!(q + 1)Bq+1 - (q + 1)B2q -
- (q + 1)(3q - 1)(q - 1)(q - 2)
24
qB2q - 2 + (q + 1)
q - 3\sum
k=2
1
q
\Biggl[
q
k
\Biggr]
( - 1)kqBq+k,
since \Biggl[
q
q - 2
\Biggr]
=
(3q - 1)q(q - 1)(q - 2)
24
.
Now (q - 1) \nmid (q + k) for k = 2, 3, . . . , q - 3. So by the von Staudt – Clausen theorem qBq+k \equiv 0
(\mathrm{m}\mathrm{o}\mathrm{d} q). Moreover,
\biggl[
q
k
\biggr]
\equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} q) for k = 2, 3, . . . , q - 3, hence the sum above vanishes modulo
q. The von Staudt – Clausen theorem also implies that qBq+1 \equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} q), qB2q \equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} q) and
qB2q - 2 \equiv - 1 (\mathrm{m}\mathrm{o}\mathrm{d} q). All these and the Wilson’s theorem give
qBq,q \equiv
1
12
- Bq+1 +B2q (\mathrm{m}\mathrm{o}\mathrm{d} q).
The result readily follows by employing Adam’s theorem which states that q | n implies Bn \equiv 0
(\mathrm{m}\mathrm{o}\mathrm{d} q) for primes (q - 1) \nmid n.
Finally, we give a congruence for Bq,q+1, where q > 3 is a prime.
Theorem 4.7. For a prime q > 3, we have
qBq,q+1 \equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} q).
Proof. Letting n = p = q and r = 1 in (3.13) and using Bn,1 = - 2Bn+1, we get
(q + 1)!Bq,q+1 = (q + 2)
q\sum
k=0
\Biggl[
q + 1
k + 1
\Biggr]
( - 1)k - 1Bq+k+1 =
= q(q + 2)
q\sum
k=0
\Biggl[
q
k
\Biggr]
( - 1)kBq+k + (q + 2)
q\sum
k=0
\Biggl[
q
k
\Biggr]
( - 1)k - 1Bq+k+1.
Equation (2.7) enable us to write the first sum as
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
1636 L. KARGIN, M. CENKCI
q(q + 2)
q\sum
k=0
\Biggl[
q
k
\Biggr]
( - 1)kBq+k =
(q + 2)q!
(q + 1)
qBq,q.
By Theorem 4.6, we conclude that
q(q + 2)
q\sum
k=0
\Biggl[
q
k
\Biggr]
( - 1)kBq+k \equiv (q + 2)q!
(q + 1)
\biggl(
1
12
- Bq+1
\biggr)
\equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} q),
by the von Staudt – Clausen theorem.
Now, we seperate the terms of the second sum as
(q + 2)
q\sum
k=0
\Biggl[
q
k
\Biggr]
( - 1)k - 1Bq+k+1 = - (q + 2)
\Biggl[
q
q - 1
\Biggr]
B2q - (q + 2)
\Biggl[
q
q - 3
\Biggr]
B2q - 2+
+
q + 2
q
q - 4\sum
k=2
\Biggl[
q
k
\Biggr]
( - 1)k - 1qBq+k+1 =
= - (q + 2)
q - 1
2
qB2q - q + 2
\Biggl(
q
4
\Biggr)
q - 1
2
qB2q - 2 ++
q + 2
q
q - 4\sum
k=2
\Biggl[
q
k
\Biggr]
( - 1)k - 1qBq+k+1,
since
\biggl[
q
q - 1
\biggr]
=
\biggl(
q
2
\biggr)
and
\biggl[
q
q - 3
\biggr]
=
\biggl(
q
2
\biggr) \biggl(
q
4
\biggr)
. For k = 2, 3, . . . , q - 4, (q - 1) \nmid (q + k + 1),
so by the von Staudt – Clausen theorem qBq+k+1 \equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} q). Moreover,
\biggl[
q
k
\biggr]
\equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} q) in the
same range, so we conclude that the sum above vanishes modulo q. The result now follows by noting\biggl(
q
4
\biggr)
\equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} q), qB2q - 2 \equiv - 1 (\mathrm{m}\mathrm{o}\mathrm{d} q) and qB2q \equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} q).
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Received 26.09.19
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
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| id | umjimathkievua-article-1080 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:04:40Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c2/e63923e65cfdaedee1fa5d26e8c075c2.pdf |
| spelling | umjimathkievua-article-10802025-03-31T08:46:08Z Recurrences and congruences for higher order geometric polynomials and related numbers Recurrences and congruences for higher order geometric polynomials and related numbers Kargın, L. Cenkci, M. Kargın, L. Cenkci, M. Higher order geometric polynomials p-Bernoulli numbers congruences Higher order geometric polynomials p-Bernoulli numbers congruences UDC 517.5We obtain new recurrence relations, an explicit formula, and convolution identities for higher order geometric polynomials. These relations generalize known results for geometric polynomials, and lead to congruences for higher order geometric polynomials, particularly for $p$-Bernoulli numbers. УДК 517.5 Рекурентні та конгруентні співвідношення для геометричних поліномів вищого порядку і відповідних чисел Отримано новi рекурентнi спiввiдношення, точну формулу та тотожностi згортки для геометричних полiномiв вищого порядку. Цi спiввiдношення узагальнюють вiдомi результати для геометричних полiномiв i дають можливiсть отримати конгруентностi для геометричних полiномiв вищого порядку, зокрема для $p$-чисел Бернуллi. Institute of Mathematics, NAS of Ukraine 2021-12-17 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1080 10.37863/umzh.v73i12.1080 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 12 (2021); 1619 - 1637 Український математичний журнал; Том 73 № 12 (2021); 1619 - 1637 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1080/9159 Copyright (c) 2021 Levent KARGIN, Mehmet Cenkci |
| spellingShingle | Kargın, L. Cenkci, M. Kargın, L. Cenkci, M. Recurrences and congruences for higher order geometric polynomials and related numbers |
| title | Recurrences and congruences for higher order geometric polynomials and related numbers |
| title_alt | Recurrences and congruences for higher order geometric polynomials and related numbers |
| title_full | Recurrences and congruences for higher order geometric polynomials and related numbers |
| title_fullStr | Recurrences and congruences for higher order geometric polynomials and related numbers |
| title_full_unstemmed | Recurrences and congruences for higher order geometric polynomials and related numbers |
| title_short | Recurrences and congruences for higher order geometric polynomials and related numbers |
| title_sort | recurrences and congruences for higher order geometric polynomials and related numbers |
| topic_facet | Higher order geometric polynomials p-Bernoulli numbers congruences Higher order geometric polynomials p-Bernoulli numbers congruences |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1080 |
| work_keys_str_mv | AT kargınl recurrencesandcongruencesforhigherordergeometricpolynomialsandrelatednumbers AT cenkcim recurrencesandcongruencesforhigherordergeometricpolynomialsandrelatednumbers AT kargınl recurrencesandcongruencesforhigherordergeometricpolynomialsandrelatednumbers AT cenkcim recurrencesandcongruencesforhigherordergeometricpolynomialsandrelatednumbers |