Jeong Gon Lee

Jongkyum Kwon

Springer - Publisher Connector

Lee and Kwon Advances in Diﬀerence Equations  (2017) 2017:29 
DOI 10.1186/s13662-017-1084-7
RESEARCH Open Access
The modiﬁed degenerate q-Bernoulli
polynomials arising from p-adic invariant
integral on Zp
Jeong Gon Lee1 and Jongkyum Kwon2*
*Correspondence:
mathkjk26@gnu.ac.kr
2Department of Mathematics
Education and RINS, Gyeongsang
National University, Jinju,
Gyeongsangnamdo, 52828,
Republic of Korea
Full list of author information is
available at the end of the article
Abstract
Dolgy et al. introduced the modiﬁed degenerate Bernoulli polynomials, which are
diﬀerent from Carlitz’s degenerate Bernoulli polynomials (see Dolgy et al. in Adv. Stud.
Contemp. Math. (Kyungshang) 26(1):1-9, 2016). In this paper, we study some explicit
identities and properties for the modiﬁed degenerate q-Bernoulli polynomials arising
from the p-adic invariant integral on Zp.
MSC: 11B68; 11S40; 11S80
Keywords: degenerate Bernoulli polynomials; modiﬁed degenerate q-Bernoulli
polynomials
1 Introduction
For a ﬁxed prime number p,Zp refers to the ring of p-adic integers,Qp to the ﬁeld of p-adic
rational numbers, and Cp to the completion of algebraic closure of Qp. The p-adic norm
| · |p is normalized as |p|p = p . Let q be in Cp with |q – |p < p–

p– and qx = exp(x logq) for
|x|p < . Then the q-analogue of x is deﬁned to be [x]q = –qx–q .
The Bernoulli polynomials are given by the generating function
( t
et – 
)
ext =
∞∑
n=
Bn(x)
tn
n!
(
see [–]
)
. (.)
When x = , Bn = Bn() are called Bernoulli numbers.
Carlitz [, , ] deﬁned the degenerate Bernoulli polynomials as follows:
t
( + λt) λ – 
( + λt) xλ =
∞∑
n=
βn(x|λ) t
n
n! . (.)
When x = , βn(|λ) = βn(λ) are called Carlitz’s degenerate Bernoulli numbers.
From (.) we note that
∞∑
n=
lim
λ→βn(x|λ)
tn
n! = limλ→
t
( + λt) λ – 
( + λt) xλ
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Lee and Kwon Advances in Diﬀerence Equations  (2017) 2017:29 Page 2 of 9
=
( t
et – 
)
ext
=
∞∑
n=
Bn(x)
tn
n! . (.)
Using the derivation given in (.), we have
lim
λ→βn(x|λ) = Bn(x) (n≥ ). (.)
Let f (x) be a uniformly diﬀerentiable function on Zp. Then the p-adic invariant integral
on Zp (also called the Volkenborn integral on Zp) is deﬁned by
∫
Zp
f (x)dμ(x) = limN→∞

pN
pN–∑
n=
f (x)
(
see [, , , , ]
)
. (.)
By using the formula deﬁned in (.) we note that
∫
Zp
f(x)du(x) –
∫
Zp
f (x)du(x) = f ′() (.)
and
∫
Zp
fn(x)du(x) –
∫
Zp
f (x)du(x) =
n–∑
l=
f ′(l), (.)
where fn(x) = f (x + n) (n ∈N); see [, , , , ].
Thus, by (.) we get
∫
Zp
e(x+y)t du(y) =
t
et – e
xt =
∞∑
n=
Bn(x)
tn
n! . (.)
The modiﬁed degenerate Bernoulli polynomials are recently revisited by Dolgy et al.,
and they are formulated with the p-adic invariant integral on Zp to be
∫
Zp
( + λ)(
x+y
λ
)t du(x) =
t
( + λ) tλ – 
(
log( + λ)
λ
)
( + λ) xtλ
=
∞∑
n=
βn,λ(x)
tn
n!
(
see []
)
, (.)
where λ ∈Cp with |λ|p < p–

p– .
When x = , we call βn,λ() = βn,λ the modiﬁed degenerate Bernoulli numbers.
Recently, Kim introduced p-adic q-integral on Zp is deﬁned by
Iq(f ) =
∫
Zp
f (x)dμq(x)
= lim
N→∞

[pN ]q
pN–∑
x=
f (x)qx (see []).
(.)
Lee and Kwon Advances in Diﬀerence Equations  (2017) 2017:29 Page 3 of 9
The degenerate q-Bernoulli polynomials are also deﬁned by Kim as follows.
∞∑
n=
βn,q,λ(x)
tn
n! =
∫
Zp
( + λt)
[x+y]q
λ dμq(y) (see []). (.)
The generating functions of Stirling numbers are given by
(
log( + t)
)n = n!
∞∑
l=n
S(l,n)
tl
l! (n≥ ) (.)
and
(
et – 
)n = n!
∞∑
l=n
S(l,n)
tl
l! (n≥ ), (.)
where S(l,n) are the Stirling numbers of the ﬁrst kind, and S(l,n) are the Stirling numbers
of the second kind.
The following diagram illustrates the variations of several types of q-Bernoulli polyno-
mials and numbers. The deﬁnitions of the q-Bernoulli polynomials and the degenerate
q-Bernoulli polynomials applied in the given diagram are provided by Carlitz [, , ] and
Kim [], respectively. In this paper, we investigate some of the explicit identities to char-
acterize the modiﬁed degenerate q-Bernoulli polynomials used in the diagram
∫
Zp
e[x+y]qt dμq(y)
=
∑∞
n= βn,q(x) t
n
n!
(q-Bernoulli polynomials)
∫
Zp
q–xe[x+y]qt dμq(y)
=
∑∞
n= Bn,q(x) t
n
n!
(modiﬁed q-Bernoulli polynomials)
∫
Zp ( + λt)
[x+y]q
λ dμq(y)
=
∑∞
n= βn,q,λ(x) t
n
n!
(degenerate q-Bernoulli polynomials)
∫
Zp q
–y( + λ)
[x+y]q
λ
t dμq(y)
=
∑∞
n= Bn,q,λ(x) t
n
n!
(modiﬁed degenerate
q-Bernoulli polynomials)
A few studies have identiﬁed some of the properties of the degenerate q-Bernoulli poly-
nomials and numbers. This paper deﬁnes themodiﬁed q-Bernoulli polynomials and num-
bers arising from the p-adic invariant integral on Zp and introduces additional character-
istic properties of these polynomials and numbers, which are deﬁned from the generating
functions and p-adic invariant integral on Zp.
2 Themodiﬁed degenerate q-Bernoulli polynomials and numbers
In the following discussions, we assume that λ, t ∈ Cp with  < |λ| ≤  and |t|p < p–

p– .
Then, as |λt|p < p–

p– , | log( + λt)|p = |λt|p, and hence | λ log( + λt)|p = |t|p < p–

p– , it
makes sense to take the limit as λ → .
Lee and Kwon Advances in Diﬀerence Equations  (2017) 2017:29 Page 4 of 9
Following (.), we deﬁne themodiﬁed degenerate q-Bernoulli polynomials given by the
generating function
∫
Zp
q–y( + λ)
[x+y]q
λ
t duq(y) =
∞∑
n=
B˜n,q,λ(x)
tn
n! . (.)
When x = , B˜n,q,λ() = B˜n,q,λ are called the modiﬁed degenerate q-Bernoulli numbers.
Note that
lim
λ→
∫
Zp
q–y( + λ)
[x+y]q
λ
t duq(y)
=
∫
Zp
q–ye[x+y]qt duq(y)
=
∞∑
n=
Bn,q(x)
tn
n! , (.)
where Bn,q(x) are the modiﬁed Carlitz q-Bernoulli polynomials.
Now, we consider
∫
Zp
q–y( + λ)
[x+y]q
λ
t duq(y)
=
∫
Zp
q–ye
[x+y]q
λ
t log(+λ) duq(y)
=
∞∑
n=
(
log( + λ)
λ
)n ∫
Zp
q–y[x + y]nq duq(y)
tn
n!
=
∞∑
n=
(
log( + λ)
λ
)n
Bn,q(x)
tn
n! . (.)
By the deﬁnitions provided in (.), (.), and (.) we are able to derive the following
theorem.
Theorem . For n≥ , B˜n,q,λ(x) can be written as
B˜n,q,λ(x) =
(
log( + λ)
λ
)n
Bn,q(x). (.)
Note that (x)n =
∑n
l= S(n, l)xl (n ≥ ), where S are the Stirling numbers of the ﬁrst
kind.
Then, by using (.) we are able to state
∫
Zp
q–y( + λ)
[x+y]q
λ
t duq(y)
=
∞∑
n=
∫
Zp
q–y
( [x+y]q
λ
t
n
)
λn duq(y)
=
∞∑
n=
∫
Zp
q–yλn
n∑
l=
S(n, l)
( [x + y]q
λ
)l tl
n! duq(y)
Lee and Kwon Advances in Diﬀerence Equations  (2017) 2017:29 Page 5 of 9
=
∞∑
l=
∞∑
n=l
S(n, l)λn–l
tl
n!
∫
Zp
q–y[x + y]lq duq(y)
=
∞∑
l=
( ∞∑
n=l
S(n, l)λn–l
l!
n!Bl,q(x)
)
tl
l! . (.)
Given the descriptions in (.) and (.), we have another theorem.
Theorem . For n≥ , B˜n,q,λ(x) can be written as
B˜n,q,λ(x) =
∞∑
n=l
S(n, l)λn–l
l!
n!Bl,q(x). (.)
We observe that
∫
Zp
q–y( + λ)
[x+y]q
λ
t duq(y)
=
∫
Zp
q–y( + λ)
[x]q
λ
t( + λ)
[y]q
λ
qxt duq(y)
= ( + λ)
[x]q
λ
t
∫
Zp
q–y( + λ)
[y]q
λ
qxt duq(y)
=
( ∞∑
l=
(
log( + λ)
λ
)l
[x]lq
tl
l!
)( ∞∑
m=
B˜m,q,λ
qmxtm
m!
)
=
∞∑
n=
( n∑
m=
(n
m
)
B˜m,q,λ[x]n–mq qmx
(
log( + λ)
λ
)n–m) tn
n! . (.)
The third theorem is obtained by (.) and (.) as follows.
Theorem . For n≥ , B˜n,q,λ(x) can be written as
B˜n,q,λ(x) =
n∑
m=
(n
m
)
B˜m,q,λ[x]n–mq qmx
(
log( + λ)
λ
)n–m
. (.)
Remark .
lim
λ→ B˜m,q,λ(x) = limλ→
n∑
m=
(n
m
)
B˜m,q,λ[x]n–mq qmx
(
log( + λ)
λ
)n–m
=
n∑
m=
(n
m
)
B˜m,qqmx
= Bm,q(x). (.)
Note that
∫
Zp
q–y( + λ)
[x+y]q
λ
t duq(y)
= lim
N→∞

[dpN ]q
dpN–∑
y=
( + λ)
[x+y]q
λ
t
Lee and Kwon Advances in Diﬀerence Equations  (2017) 2017:29 Page 6 of 9
= lim
N→∞

[dpN ]q
d–∑
a=
pN–∑
y=
( + λ)
[x+a+dy]q
λ
t
= lim
N→∞

[d]q[pN ]qd
d–∑
a=
pN–∑
y=
( + λ)

λ
[d]q[ x+ad +y]qd t
= [d]q
d–∑
a=
lim
N→∞

[pN ]qd
pN–∑
y=
( + λ)

λ
[d]q[ x+ad +y]qd tq–dyqdy
= [d]q
d–∑
a=
(∫
Zp
q–dy( + λ)

λ
[d]q[ x+ad +y]qd t duqd (y)
)
= [d]q
d–∑
a=
∞∑
n=
B˜n,qd ,λ
(x + a
d
) [d]nqtn
n!
=
∞∑
n=
(
[d]n–q
d–∑
a=
B˜n,qd ,λ
(x + a
d
)) tn
n! , (.)
where d ∈N.
The following theorem is obtained from (.).
Theorem . For n≥  and d ∈N, B˜n,q,λ(x) can be written as
B˜n,q,λ(x) = [d]n–q
d–∑
a=
B˜n,qd ,λ
(x + a
d
)
. (.)
Now, we observe that
∫
Zp
q–ye[x+y]qt duq(y) =
∞∑
n=
Bn,q(x)
tn
n! . (.)
We obtain Theorem . as follows by substituting t by log( + λ) tλ in (.):
∫
Zp
q–ye[x+y]q log(+λ)
t
λ duq(y) =
∫
Zp
q–y( + λ)
[x+y]q
λ
t duq(y)
=
∞∑
n=
Bn,q(x)

n!
(
log( + λ) tλ
)n
=
∞∑
n=
Bn,q(x)
(
log( + λ)
λ
)n tn
n! . (.)
For r ∈ N, we deﬁne the modiﬁed degenerate q-Bernoulli polynomials of order r as fol-
lows:
∫
Zp
· · ·
∫
Zp
q–(x+x+···+xr )( + λ)
[x+x+···+xr+x]q
λ
t duq(x)duq(x) · · · duq(xr)
=
∞∑
n=
B˜(r)n,q,λ(x)
tn
n! . (.)
Lee and Kwon Advances in Diﬀerence Equations  (2017) 2017:29 Page 7 of 9
When x = , B˜(r)n,q,λ() = B˜
(r)
n,q,λ are called the modiﬁed degenerate q-Bernoulli numbers of
order r.
We observe that
∫
Zp
· · ·
∫
Zp
q–(x+x+···+xr)( + λ)
[x+x+···+xr+x]q
λ
t duq(x)duq(x) · · · duq(xr)
=
∫
Zp
· · ·
∫
Zp
q–(x+x+···+xr)
∞∑
n=
(
log( + λ)
λ
)n
× [x + x + · · · + xr + x]nq
tn
n! duq(x)duq(x) · · · duq(xr)
=
∞∑
n=
(
log( + λ)
λ
)n ∫
Zp
· · ·
∫
Zp
q–(x+x+···+xr )[x + x + · · · + xr + x]nq
× duq(x)duq(x) · · · duq(xr) t
n
n!
=
∞∑
n=
((
log( + λ)
λ
)n
B(r)n,q(x)
) tn
n! . (.)
Therefore, we are able to derive the following theorem.
Theorem . For n≥ , B˜(r)n,q,λ(x) can be written as
B˜(r)n,q,λ(x) =
(
log( + λ)
λ
)n
B(r)n,q(x). (.)
Now, we consider
∫
Zp
· · ·
∫
Zp
q–(x+x+···+xr )( + λ)
[x+x+···+xr+x]q
λ
t duq(x)duq(x) · · · duq(xr)
=
∫
Zp
· · ·
∫
Zp
q–(x+x+···+xr)
∞∑
l=
( [x+x+···+xr+x]q
λ
t
l
)
λl duq(x)duq(x) · · · duq(xr)
=
∞∑
l=
l∑
n=
S(l,n)
l! λ
l–ntn
∫
Zp
· · ·
∫
Zp
q–(x+x+···+xr)
× [x + x + · · · + xr + x]nq duq(x)duq(x) · · · duq(xr)
=
∞∑
l=
l∑
n=
S(l,n)
l! λ
l–ntnB(r)n,q(x)
=
∞∑
n=
( ∞∑
l=n
S(l,n)
l! λ
l–nn!B(r)n,q(x)
)
tn
n! . (.)
Now, (.) yields the following theorem.
Theorem . For n≥ , B˜(r)n,q,λ(x) can be written as
B˜(r)n,q,λ(x) =
∞∑
l=n
S(l,n)
l! λ
l–nn!B(r)n,q(x). (.)
Lee and Kwon Advances in Diﬀerence Equations  (2017) 2017:29 Page 8 of 9
Now, we observe that, for d ∈N,
∫
Zp
· · ·
∫
Zp
q–(x+x+···+xr )( + λ)
[x+x+···+xr+x]q
λ
t duq(x)duq(x) · · · duq(xr)
= lim
N→∞

[dpN ]rq
dpN–∑
x=
· · ·
dpN–∑
xr=
( + λ)
[x+x+···+xr+x]q
λ
t
= lim
N→∞

[dpN ]rq
d–∑
a=
· · ·
d–∑
ar=
dpN–∑
x=
· · ·
dpN–∑
xr=
( + λ)
[a+···+ar+x+dx+dx+···+dxr ]q
λ
t
= lim
N→∞

[d]rq[pN ]rqd
d–∑
a=
· · ·
d–∑
ar=
pN–∑
x=
· · ·
pN–∑
xr=
( + λ)

λ
[d]q[ a+···+ar+xd +x+x+···+xr]qd t
= [d]rq
d–∑
a=
· · ·
d–∑
ar=
lim
N→∞

[pN ]rqd
pN–∑
x=
· · ·
pN–∑
xr=
( + λ)

λ
[ a+···+ar+xd +x+x+···+xr]qd [d]qt
= [d]rq
d–∑
a=
· · ·
d–∑
ar=
∫
Zp
· · ·
∫
Zp
q–d(x+x+···+xr)
× ( + λ) λ [
a+···+ar+x
d +x+x+···+xr ]qd [d]qt duqd (x)duqd (x) · · · duqd (xr)
=
∞∑
n=
(
[d]n–rq
d–∑
a=
· · ·
d–∑
ar=
B˜(r)n,qd ,λ
(a + · · · + ar + x
d
)) tn
n! . (.)
Finally, by comparing the coeﬃcients on both sides of (.) we get the following theo-
rem.
Theorem . For n≥  and d ∈N, B˜(r)n,q(x) can be written as
B˜(r)n,q,λ(x) = [d]n–rq
d–∑
a=
· · ·
d–∑
ar=
B˜(r)n,qd ,λ
(a + · · · + ar + x
d
)
. (.)
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed equally to this work. Both authors read and approved the ﬁnal manuscript.
Author details
1Division of Mathematics and informational Statistics, Nanoscale Science and Technology Institute, Wonkwang University,
Iksan, 54538, Republic of Korea. 2Department of Mathematics Education and RINS, Gyeongsang National University, Jinju,
Gyeongsangnamdo, 52828, Republic of Korea.
Acknowledgements
The authors would like to express their sincere gratitude to the Editor, who gave us valuable comments to improve this
paper.
Received: 19 November 2016 Accepted: 14 January 2017
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The modified degenerate q-Bernoulli polynomials arising from p-adic invariant integral on 
                
                    
                                    
                        
                            Z
                            p
                        
                    
                
                $\mathbb{Z}_{p}$
            
        

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The modified degenerate q-Bernoulli polynomials arising from p-adic invariant integral on Z p $\mathbb{Z}_{p}$

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The modified degenerate q-Bernoulli polynomials arising from p-adic invariant integral on Z p Zp\mathbb{Z}_{p}Zp​

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The modified degenerate q-Bernoulli polynomials arising from p-adic invariant integral on Z p $\mathbb{Z}_{p}$