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:
[email protected] 2 Department 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–
| · |p is normalized as |p|p = p . Let q be in Cp with |q – |p < p
and qx = exp(x log q) for
–qx
|x|p < . Then the q-analogue of x is defined to be [x]q = –q . The Bernoulli polynomials are given by the generating function
∞ t tn xt = Bn (x) e t e – n! n=
see [–] .
(.)
When x = , Bn = Bn () are called Bernoulli numbers. Carlitz [, , ] defined the degenerate Bernoulli polynomials as follows: t λ
x λ
( + λt) =
( + λt) –
∞ n=
βn (x|λ)
tn . n!
(.)
When x = , βn (|λ) = βn (λ) are called Carlitz’s degenerate Bernoulli numbers. From (.) we note that ∞ n=
lim βn (x|λ)
λ→
tn t x ( + λt) λ = lim n! λ→ ( + λt) λ –
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Lee and Kwon Advances in Difference Equations (2017) 2017:29
= =
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t ext et –
∞
Bn (x)
n=
tn . n!
(.)
Using the derivation given in (.), we have lim βn (x|λ) = Bn (x) (n ≥ ).
(.)
λ→
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
N
p – f (x) dμ (x) = lim N f (x) see [, , , , ] . N→∞ p Zp n=
(.)
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–
f (l),
(.)
l=
where fn (x) = f (x + n) (n ∈ N); see [, , , , ]. Thus, by (.) we get Zp
∞
e(x+y)t du (y) =
t xt tn Bn (x) . e = t e – n! 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) =
=
log( + λ) xt ( + λ) λ t λ ( + λ) λ –
t
∞
βn,λ (x)
n=
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)q
x
(see []).
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The degenerate q-Bernoulli polynomials are also defined by Kim as follows. ∞
tn βn,q,λ (x) = n! n=
Zp
( + λt)
[x+y]q λ
dμq (y) (see []).
(.)
The generating functions of Stirling numbers are given by
log( + t)
n
= n!
∞
S (l, n)
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 polynomials 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 characterize the modified degenerate q-Bernoulli polynomials used in the diagram
e[x+y]q t dμq (y) ∞ n = n= βn,q (x) tn! (q-Bernoulli polynomials) Zp
[x+y]q
λ dμq (y) Zp ( + λt) ∞ n = n= βn,q,λ (x) tn! (degenerate q-Bernoulli polynomials)
q–x e[x+y]q t dμq (y) tn = ∞ n= Bn,q (x) n! (modified q-Bernoulli polynomials) Zp
[x+y]q
q–y ( + λ) λ t dμq (y) tn = ∞ n= Bn,q,λ (x) n! (modified degenerate q-Bernoulli polynomials) Zp
A few studies have identified some of the properties of the degenerate q-Bernoulli polynomials and numbers. This paper defines the modified q-Bernoulli polynomials and numbers arising from the p-adic invariant integral on Zp and introduces additional characteristic properties of these polynomials and numbers, which are defined from the generating functions and p-adic invariant integral on Zp .
2 The modified degenerate q-Bernoulli polynomials and numbers – In the following discussions, we assume that λ, t ∈ Cp with < |λ| ≤ and |t|p < p p– . –
– p–
Then, as |λt|p < p p– , | log( + λt)|p = |λt|p , and hence | λ log( + λt)|p = |t|p < p makes sense to take the limit as λ → .
, it
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Following (.), we define the modified degenerate q-Bernoulli polynomials given by the generating function Zp
q–y ( + λ)
[x+y]q λ t
duq (y) =
∞
Bn,q,λ (x)
n=
tn . n!
(.)
When x = ,
Bn,q,λ () =
Bn,q,λ are called the modified degenerate q-Bernoulli numbers. Note that [x+y]q lim q–y ( + λ) λ t duq (y) λ→ Z p
=
=
Zp ∞
q–y e[x+y]q t duq (y)
Bn,q (x)
n=
tn , n!
(.)
where Bn,q (x) are the modified Carlitz q-Bernoulli polynomials. Now, we consider Zp
q–y ( + λ)
=
=
Zp
q–y e
duq (y)
[x+y]q λ t log(+λ)
duq (y)
∞ log( + λ) n λ
n=
=
[x+y]q λ t
Zp
∞ log( + λ) n λ
n=
q–y [x + y]nq duq (y)
Bn,q (x)
tn n!
tn . n!
(.)
By the definitions provided in (.), (.), and (.) we are able to derive the following theorem. Theorem . For n ≥ ,
Bn,q,λ (x) can be written as
Bn,q,λ (x) =
log( + λ) λ
n Bn,q (x).
(.)
Note that (x)n = nl= 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)
=
∞ n=
=
Zp
∞ n=
Zp
[x+y]q λ t
q–y
duq (y)
[x+y]q t n λ λ duq (y) n
q–y λn
n l=
S (n, l)
[x + y]q λ
l
tl duq (y) n!
Lee and Kwon Advances in Difference Equations (2017) 2017:29
=
∞ ∞
S (n, l)λn–l
l= n=l
=
∞
∞
l=
S (n, l)λ
tl n!
n–l
n=l
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q–y [x + y]lq duq (y)
Zp
tl l! Bl,q (x) . n! l!
(.)
Given the descriptions in (.) and (.), we have another theorem. Theorem . For n ≥ ,
Bn,q,λ (x) can be written as
Bn,q,λ (x) =
∞
S (n, l)λn–l
n=l
l! Bl,q (x). n!
(.)
We observe that [x+y]q q–y ( + λ) λ t duq (y) Zp
=
Zp
q–y ( + λ)
= ( + λ) =
[x]q λ t
[x]q λ t
( + λ)
[y]q x λ q t
Zp
q–y ( + λ)
∞ log( + λ) l
t [x]lq
[y]q x λ q t
l
duq (y) duq (y)
∞
qmx t m
Bm,q,λ m! m=
λ l! ∞ n n log( + λ) n–m t n n–m mx
= . Bm,q,λ [x]q q m λ n! n= m= l=
(.)
The third theorem is obtained by (.) and (.) as follows. Theorem . For n ≥ ,
Bn,q,λ (x) can be written as
Bn,q,λ (x) =
n–m n n n–m mx log( + λ)
Bm,q,λ [x]q q . m λ m=
(.)
Remark . n–m n n n–m mx log( + λ)
Bm,q,λ [x]q q λ→ λ m m= n n
= Bm,q qmx m m=
Bm,q,λ (x) = lim lim
λ→
= Bm,q (x). Note that [x+y]q q–y ( + λ) λ t duq (y) Zp
N
dp – [x+y]q ( + λ) λ t = lim N N→∞ [dp ]q y=
(.)
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N
d– p – [x+a+dy]q = lim ( + λ) λ t N N→∞ [dp ]q a= y= N
– d– p [d] [ x+a +y] t = lim ( + λ) λ q d qd N→∞ [d]q [pN ]qd a= y= N
p – d– [d] [ x+a +y] t = lim N ( + λ) λ q d qd q–dy qdy [d]q a= N→∞ [p ]qd y=
d– x+a –dy λ [d]q [ d +y]qd t = q ( + λ) duqd (y) [d]q a= Zp d– ∞ x + a [d]nq t n
Bn,qd ,λ [d]q a= n= d n! n ∞ d– t x+a n–
, [d]q Bn,qd ,λ = d n! n= a=
=
(.)
where d ∈ N. The following theorem is obtained from (.). Theorem . For n ≥ and d ∈ N,
Bn,q,λ (x) can be written as
Bn,q,λ (x) = [d]n– q
d–
Bn,qd ,λ
a=
x+a . d
(.)
tn . n!
(.)
Now, we observe that Zp
q–y e[x+y]q t duq (y) =
∞
Bn,q (x)
n=
t
We obtain Theorem . as follows by substituting t by log( + λ) λ in (.):
t
Zp
q–y e[x+y]q log(+λ) λ duq (y) =
=
Zp ∞ n= ∞
q–y ( + λ)
Bn,q (x)
[x+y]q λ t
duq (y)
t n log( + λ) λ n!
log( + λ) = Bn,q (x) λ n=
n
tn . n!
(.)
For r ∈ N, we define the modified degenerate q-Bernoulli polynomials of order r as follows: [x +x +···+xr +x]q t λ ··· q–(x +x +···+xr ) ( + λ) duq (x ) duq (x ) · · · duq (xr ) Zp
=
Zp
∞ n=
B(r) n,q,λ (x)
tn . n!
(.)
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(r) When x = ,
B(r) n,q,λ () = Bn,q,λ are called the modified degenerate q-Bernoulli numbers of order r. We observe that [x +x +···+xr +x]q t λ ··· q–(x +x +···+xr ) ( + λ) duq (x ) duq (x ) · · · duq (xr ) Zp
Zp
=
Zp
···
Zp
q–(x +x +···+xr )
∞ log( + λ) n λ
n=
tn duq (x ) duq (x ) · · · duq (xr ) n! ··· q–(x +x +···+xr ) [x + x + · · · + xr + x]nq
× [x + x + · · · + xr + x]nq =
∞ log( + λ) n λ
n=
Zp
Zp
× duq (x ) duq (x ) · · · duq (xr ) =
∞ log( + λ) n λ
n=
B(r) n,q (x)
tn n!
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)
=
=
Zp
···
Zp
q–(x +x +···+xr )
duq (x ) duq (x ) · · · duq (xr )
∞ [x +x +···+xr +x]q t l λ λ duq (x ) duq (x ) · · · duq (xr ) l l=
∞ l S (l, n) l= n=
[x +x +···+xr +x]q t λ
l!
λl–n t n
Zp
···
Zp
q–(x +x +···+xr )
× [x + x + · · · + xr + x]nq duq (x ) duq (x ) · · · duq (xr ) =
∞ l S (l, n) l= n=
=
l!
λl–n t n B(r) n,q (x)
∞ ∞ S (l, n) n=
l=n
l!
λ
l–n
n!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) =
∞ S (l, n) l=n
l!
λl–n n!B(r) n,q (x).
(.)
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Now, we observe that, for d ∈ N,
Zp
···
Zp
q–(x +x +···+xr ) ( + λ) N
N
r
[x +x +···+xr +x]q t λ
duq (x ) duq (x ) · · · duq (xr )
dp – dp – [x +x +···+xr +x]q t λ · · · ( + λ) = lim N r N→∞ [dp ]q x = x = N
N
r
dp – d– dp – d– [a +···+ar +x+dx +dx +···+dxr ]q t λ = lim · · · · · · ( + λ) N r N→∞ [dp ]q a = a = x = x = r
N
N
r
– p – d– d– p a +···+ar +x +x +x +···+xr ]qd t λ [d]q [ d = lim · · · · · · ( + λ) r N→∞ [d]rq [pN ] d q a = a = x = x = r
pN –
N
p – d– d– a +···+ar +x +x +x +···+x ] [d] t r qd q λ[ d = · · · lim · · · ( + λ) [d]rq a = a = N→∞ [pN ]rqd x = x = r
=
[d]rq
d–
···
a =
r
d– ar = Zp
···
Zp
q–d(x +x +···+xr )
[ a +···+ar +x +x +x +···+x ]
[d] t
r qd q d × ( + λ) λ duqd (x ) duqd (x ) · · · duqd (xr ) n ∞ d– d– a + · · · + ar + x t (r) n–r
[d]q . ··· Bn,qd ,λ = d n! n= a = a =
(.)
r
Finally, by comparing the coefficients on both sides of (.) we get the following theorem. Theorem . For n ≥ and d ∈ N,
B(r) n,q (x) can be written as n–r
B(r) n,q,λ (x) = [d]q
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 1 Division of Mathematics and informational Statistics, Nanoscale Science and Technology Institute, Wonkwang University, Iksan, 54538, Republic of Korea. 2 Department 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 References 1. Dolgy, DV, Kim, T, Kwon, H-I, Seo, J-J: On the modified degenerate Bernoulli polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 26(1), 1-9 (2016) 2. Al-Salam, WA: q-Bernoulli numbers and polynomials. Math. Nachr. 17, 239-260 (1959)
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