Kim and Kim Advances in Difference Equations (2015) 2015:282 DOI 10.1186/s13662-015-0602-8

RESEARCH

Open Access

Some identities of Korobov-type polynomials associated with p-adic integrals on Zp Dae San Kim1 and Taekyun Kim2* *

Correspondence: [email protected] 2 Department of Mathematics, Kwangwoon University, Seoul, 139-701, Republic of Korea Full list of author information is available at the end of the article

Abstract In this paper, we consider Korobov-type polynomials derived from the bosonic and fermionic p-adic integrals on Zp , and we give some interesting and new identities of those polynomials and of their mixed-types. MSC: 11B68; 11B83; 11S80; 05A19 Keywords: Korobov-type polynomial; p-adic integral; λ-Changhee and Korobov mixed-type polynomial; Korobov and λ-Changhee mixed-type polynomial

1 Introduction Let p be a fixed odd prime number. Throughout this paper, Zp , Qp and Cp will denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of the algebraic closure of Qp . Let νp be the normalized exponential valuation of Cp with |p|p = p–νp (p) = p . Let UD(Zp ) be the space of uniformly differentiable functions on Zp . For f ∈ UD(Zp ), the bosonic p-adic integral on Zp is defined by  I (f ) =

Zp

f (x) dμ (x) pN –

= lim

N→∞



  f (x)μ x + pN Zp

x= N

p –   = lim N f (x). N→∞ p x=

(.)

Thus, by (.), we get

I (f ) = I (f ) + f  (),

where f (x) = f (x + ) (see [–]).

(.)

© 2015 Kim and Kim. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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The fermionic p-adic integral on Zp is defined by Kim as  I– (f ) =

Zp

f (x) dμ– (x) pN –



= lim

N→∞

  f (x)μ– x + pN Zp

x= pN –

= lim



N→∞

f (x)(–)x .

(.)

x=

Thus, from (.), we have I– (f ) = –I– (f ) + f () (see []).

(.)

From (.) and (.), we can derive the following equations:

I (fn ) – I (f ) =

n– 

f  (l),

I– (fn ) + (–)n– I– (f ) = 

l=

n– 

f (l),

(.)

l=

where fn (x) = f (x + n), f  (l) = dfdx(x) |x=l (see [, , ]). As is well known, the Bernoulli polynomials of order r (∈ N) are defined by the generating function 

t t e –

r ext =

∞ 

B(r) n (x)

n=

tn n!

(see [, ]).

(.)

(r) When x = , B(r) n = Bn () are called the Bernoulli numbers of order r. In particular, if r = , Bn (x) = B() n (x) are called the ordinary Bernoulli polynomials. The Euler polynomials of order r are also given by the generating function



 t e +

r ext =

∞ 

En(r) (x)

n=

tn n!

(see [, ]).

(.)

When x = , En(r) = En(r) () are called the Euler numbers of order r. In particular, if r = , then En (x) = En() (x) are called the ordinary Euler polynomials. The Daehee polynomials of order r are defined by the generating function 

log( + t) t

r ( + t)x =

∞ 

D(r) n (x)

n=

tn n!

(see []).

(.)

(r) When x = , D(r) n = Dn () are called the Daehee numbers of order r. In particular, if r = , () then Dn (x) = Dn (x) are called the ordinary Daehee polynomials. Now, we introduce the Changhee polynomials of order r given by the generating function



 t+

r ( + t)x =

∞  n=

Ch(r) n (x)

tn n!

(see []).

(.)

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(r) When x = , Ch(r) n = Chn () are called the Changhee numbers of order r. In particular, () if r = , then Chn (x) = Chn (x) are called the ordinary Changhee polynomials. Recently, Korobov introduced the special polynomials given by the generating function ∞

 tn λt x ( + t) = K (x | λ) n (t + )λ –  n! n=

(λ ∈ N) (see [–]).

(.)

Note that limλ→ Kn (x | λ) = bn (x), where bn (x) are the Bernoulli polynomials of the second kind defined by the generating function 

 ∞  tn t ( + t)x = bn (x) log( + t) n! n=

(see [, ]).

(.)

In this paper, we define the higher-order Korobov polynomials given by the generating function 

λt (t + )λ – 

r ( + t)x =

∞ 

Kn(r) (x | λ)

n=

tn . n!

(.)

When x = , Kn(r) (λ) = Kn(r) ( | λ) are called the Korobov numbers of order r. In particular, if r = , then Kn (λ) = Kn() ( | λ) = Kn ( | λ) are called the ordinary Korobov numbers. Now, we consider the Korobov-type Changhee polynomials which are called the λ-Changhee polynomials as follows: 

 ∞  tn  x ( + t) = Chn (x | λ) . λ ( + t) +  n! n=

(.)

When x = , Chn (λ) = Chn ( | λ) are called λ-Changhee numbers. Note that limλ→ Chn (x | λ) = Chn (x), limλ→ Chn (x | λ) = (x)n , where (x)n = x(x – ) · · · (x – n + ) =

n 

S (n, l)xl

(see []).

l=

For r ∈ N, the λ-Changhee polynomials of order r are defined by the generating function 

 ( + t)λ + 

r ( + t)x =

∞  n=

Ch(r) n (x | λ)

tn . n!

(.)

The Stirling numbers of the second kind are defined by the generating function 

et – 

n

= n!

∞  l=n

S (l, n)

tl l!

(see [, ]).

(.)

The Korobov polynomials (of the first kind) were introduced in [] as the degenerate version of the Bernoulli polynomials of the second kind. In recent years, many researchers studied various kinds of degenerate versions of some familiar polynomials like

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Bernoulli polynomials, Euler polynomials and their variants by means of generating functions, p-adic integrals and umbral calculus (see [, , , ]). Here in this paper we introduce two Korobov-type polynomials obtained from the same function, namely the one by performing bosonic p-adic integrals on Zp and the other by carrying out fermionic p-adic integrals on Zp . In addition, we consider their higherorder versions and some mixed-types of them by considering multivariate p-adic integrals. In conclusion, we will obtain some connections between these new polynomials and Bernoulli polynomials, Euler polynomials, Daehee numbers and Bernoulli numbers of the second kind.

2 Korobov-type polynomials For λ ∈ N, by (.), we get  Zp

λ log( + t) ( + t)x ( + t)λ –     log( + t) λt ( + t)x = ( + t)λ –  t  ∞ ∞   tl tm = Kl (x | λ) Dm l! m! m= l=  ∞  n  n! tn Kl (x | λ)Dn–l = l!(n – l)! n! n= l=  n   ∞   tn n Kl (x | λ)Dn–l = . l n! n=

( + t)λy+x dμ (y) =

(.)

l=

From (.), we have   n     n λy + x dμ (y) = Kl (x | λ)Dn–l . n! n l Zp



(.)

l=

Therefore, by (.), we obtain the following theorem. Theorem . For n ≥ , we have   n   λy + x   n dμ (y) = Kl (x | λ)Dn–l . n l n! Zp



l=

Now, we observe that  Zp

(λy + x)n dμ (y) =

n  l=

=

n  l=

=

n  l=

 S (n, l)

Zp

S (n, l)λl

(λy + x)l dμ (y)

   x l y+ dμ (y) λ Zp

S (n, l)λl Bl

  x . λ

(.)

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Therefore by (.), we obtain the following corollary. Corollary . For n ≥ , we have n  l=

   n   x n = S (n, l)λ Bl Kl (x | λ)Dn–l . λ l l

l=

From (.), we have ∞ 

Kn (x | λ)

n=

tn λt = ( + t)x n! ( + t)λ –    t = ( + t)λy+x dμ (y) log( + t) Zp ∞  ∞   tl  tm = bl (λy + x)m dμ (y) l! m! m= Zp l=  n     n ∞  m   t n x l = . S (m, l)λ Bl bn–m λ n! m n= m=

(.)

l=

Therefore, by (.), we obtain the following corollary. Corollary . For n ≥ , we have Kn (x | λ) =

  n   m   n x . S (m, l)λl Bl bn–m λ m m= l=

By replacing t by et –  in (.), we get ∞ 

Km (x | λ)

m=

(et – )m λ(et – ) xt = λt e m! e – λt ( x )λt et –  eλ – t ∞    ∞  x tm  tl m λ Bm = λ m! l +  l! m= l=    ∞  n    n n–l x  tn λ Bn–l = . l λ l +  n! n= =

eλt

(.)

l=

On the other hand, ∞  m=

Km (x | λ)

∞ ∞    t m  tl e – = Km (x | λ) S (l, m) m! l! m= l=m  ∞  n  tn Km (x | λ)S (n, m) = . n! n= m=

Therefore, by (.) and (.), we obtain the following theorem.

(.)

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Theorem . For n ≥ , we have   n   n    n n–l x = Km (x | λ)S (n, m). λ Bn–l λ l +  m= l l=

It is easy to show that  Zp

 d a= d–

f (x) dμ (x) =

 f (a + dx) dμ (x) (d ∈ N).

Zp

(.)

By (.), we get  Zp

 d a= d–

( + t)λx dμ (x) =

 Zp

( + t)(a+dx)λ dμ (x)

 ( + t)aλ = d a= d–

 Zp

( + t)λdx dμ (x)

λd log( + t)  ( + t)aλ . d a= ( + t)dλ – 

(.)

λ log( + t) . ( + t)λ – 

(.)

d–

=

On the other hand,  Zp

( + t)λx dμ (x) =

Thus, by (.) and (.), we get  λt λdt ( + t)x = ( + t)aλ+x . λ ( + t) –  d a= ( + t)λd –  d–

(.)

Therefore, by (.) and (.), we obtain the following theorem. Theorem . For n ≥  and d ∈ N, we have  Kn (x | λ) = Kn (aλ + x | λd). d a= d–

From (.), we can derive the following equation: 

 Zp

( + t)(x+n)λ dμ (x) –

Zp

( + t)λx dμ (x) = λ log( + t)

n–  ( + t)λl

(n ∈ N). (.)

l=

Thus, by (.), we get  Zp

λ log( + t)  ( + t)λl . ( + t)λn –  n–

( + t)λx dμ (x) =

l=

(.)

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From (.), we have 

t log( + t)

 λnt ( + t)λl n ( + t)λn –  l=  n– ∞   tm . = Km (λl | λn) n m! m= n–

Zp

( + t)λx dμ (x) =

(.)

l=

On the other hand, t log( + t)

 ∞   tl tn ( + t) dμ (x) = bn (λx)l dμ (x) n! l! Zp n= l= Zp   ∞  m    tm m = (λx)l dμ (x) bm–l m! l Zp m= l=  m   l ∞    tm m n bm–l . S (l, n)λ Bn = l m! m= n= 



λx

∞ 

(.)

l=

Therefore, by (.) and (.), we obtain the following theorem. Theorem . For n ∈ N, m ≥ , we have n– m   l   m  bm–l Km (λl | λn) = S (l, n)λn Bn . l n n= l=

l=

Remark By (.), we easily get  Zp

  λ(x + n) m dμ (x) –

 Zp

(λx)m dμ (x) =

n–  m 

kS (m, k)λk lk– .

(.)

l= k=

Hence, by Theorem . and (.), we see m  l=

      m n–  m m m Kl (λn | λ)Dm–l Kl (λ)Dm–l kS (m, k)λk lk– . – = l l l=

l= k=

Now, we consider the multivariate p-adic integral on Zp given by 

 Zp

···

Zp

( + t)λ(x +···+xr )+x dμ (x ) · · · dμ (xr ).

(.)

By (.) and (.), we get 

 Zp

··· 

=  =

Zp

( + t)λ(x +···+xr )+x dμ (x ) · · · dμ (xr )

λ log( + t) ( + t)λ –  λt ( + t)λ – 

r ( + t)x r

 ( + t)x

log( + t) t

r .

(.)

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Thus, by (.), we get ∞ 

Kn(r) (x | λ)

n=

tn n!



r λt ( + t)x ( + t)λ –  r    t = ··· ( + t)λ(x +···+xr )+x dμ (x ) · · · dμ (xr ) log( + t) Zp Zp ∞  ∞   m     tl (r) t = λ(x + · · · + xr ) + x l dμ (x ) · · · dμ (xr ) bm ··· m! l! Zp m= l= Zp     ∞  n  l  n (r) t n x = b . λk S (l, k)B(r) k λ l n–l n! n= =

l= k=

By comparing the coefficients on both sides, we obtain the following theorem. Theorem . For n ≥ , we have Kn(r) (x | λ) =

n  l 

λk S (l, k)B(r) k

l= k=

   n (r) x b . λ l n–l

By replacing t by et –  in (.), we get 

λ(et – ) eλt – 

r ext =

∞ 

Km(r) (x | λ)

m=

=

∞  m=

=

Km(r) (x | λ)



  t m e – m! ∞ 

S (n, m)

n=m

∞ 

n 

n=

m=

tn n!

Km(r) (x | λ)S (n, m)

tn . n!

(.)

On the other hand, 

λ(et – ) eλt – 

r



λt eλt – 

r



 et –  r e = e t ∞  ∞   m   tl m (r) x t = λ Bm r!S (l + r, r) λ m! (l + r)! m= l=      n ∞  n n  t x (r) l . = l+r  λn–l S (l + r, r)Bn–l λ n! n= r xt

( λx )λt

l=

Therefore, by (.) and (.), we obtain the following theorem. Theorem . For n ≥ , we have n  m=

Km(r) (x | λ)S (n, m) =

n 

n

l l+r  λn–l S (l r l=

+ r, r)B(r) n–l

  x . λ

(.)

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From (.), we can derive the following equation:  Zp

∞

  tn x ( + t) . = Ch (x | λ) n ( + t)λ +  n! n=

( + t)λy+x dμ– (y) =

(.)

Thus, by (.), we get    λy + x dμ– (y) = Chn (x | λ) (n ≥ ). n n! Zp



(.)

We observe that   ( + t)x ( + t)λ +   ∞ ∞   tl tm = Chl (λ) (x)m l! m! m= l=    n ∞  n  n t = (x)m Chn–m (x) . m n! n= m=

∞ 

tn Chn (x | λ) = n! n=



(.)

Therefore, by (.) and (.), we obtain the following theorem. Theorem . For n ≥ , we have     n    n λy + x (x)m Chn–m (λ) . dμ– (y) = Chn (x | λ) = n! n! m n Zp m=



From (.), we have ∞ 

Chn (x | λ)

n=

tn = n! =

 Zp

( + t)λy+x dμ– (y)

∞   n=

Zp

(λy + x)n dμ– (y)

tn n!



   tn x l S (n, l)λ dμ– (y) y+ = λ n! Zp n= l=  n   n ∞   t x l = . S (n, l)λ El λ n! n= ∞  n 

l

(.)

l=

By replacing t by et –  in (.), we get ∞  n=

Chn (x | λ)

  t n  e –  = λt ext n! e +  x e( λ )λt eλt +    n ∞  x nt λ . = En λ n! n= =

(.)

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On the other hand, ∞ 

Chn (x | λ)

n=

∞ ∞    t n  tn e – = Chm (x | λ) S (n, m) n! n! n=m m=  ∞  n  tn . = Chm (x | λ)S (n, m) n! n= m=

(.)

Therefore, by (.), (.), and (.), we obtain the following theorem. Theorem . For n ≥ , we have En

  n  x = λ–n Chm (x | λ)S (n, m) λ m=

and Chn (x | λ) =

n 

S (n, l)λl El

l=

  x . λ

By replacing t by et –  in (.), we get En(r)

  n  x = λ–n Ch(r) m (x | λ)S (n, m). λ m=

(.)

From (.), we can derive the following equation: ∞ 

t Ch(r) n (x | λ)

n!

n=



n

= 

 ( + t)λ +  

r ( + t)x r

ex log(+t) eλ log(+t) +    ∞  n  n (r) x λ log( + t) = En λ n! n=    ∞ ∞  tn (r) x λm = Em S (n, m) λ n! n=m m=    ∞  n  tn x (r) Em λm S (n, m) . = λ n! n= m= =

Therefore, by (.) and (.), we obtain the following theorem. Theorem . For n ≥ , we have Ch(r) n (x | λ) =

n  m=

(r) Em

  x m λ S (n, m) λ

and En(r)

  n x   (r) = n Ch (x | λ)S (n, m). λ λ m= m

(.)

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Let us observe the following multivariate fermionic p-adic integral on Zp : 

 Zp

··· 

= =

Zp

( + t)λ(x +···+xr )+x dμ– (x ) · · · dμ– (xr )

 ( + t)λ + 

∞ 

r ( + t)x tn . n!

Ch(r) n (x | λ)

n=

(.)

Thus, by (.), we get Ch(r) n (x | λ) n!   ··· = Zp

  λ(x + · · · + xr ) + x dμ– (x ) · · · dμ– (xr ) (n ≥ ). n Zp

(.)

Note that ∞ 

t Ch(r) n (x | λ)

n!

n=



n

=

 ( + t)λ + 

r x

( + t) =

 n ∞   n=

(x)m Ch(r) n–m (λ)

m=

  n n t . m n!

Thus, we get   n  n (r) (x)m Chn–m (λ) m m=

Ch(r) n (x | λ) =

(n ≥ ).

(.)

By (.) and (.), we easily get  Ch(r) n (x | λ) =



Zp n 

=

l= n 

=

···

Zp

(λx + · · · + λxr + x)n dμ– (x ) · · · dμ– (xr )

   x l S (n, l)λ ··· dμ– (x ) · · · dμ– (xr ) x + · · · + xr + λ Zp Zp 

l

S (n, l)λl El(r)

l=

  x . λ

(.)

Now, we consider the λ-Changhee and Korobov mixed-type polynomials which are given by the multivariate p-adic integral on Zp as follows: CK (r,s) n (x | λ)   ··· = Zp

=

n 

Zp

Ch(r) n (λx + · · · + λxs + x | λ) dμ (x ) · · · dμ (xs )

Ch(r) n–m (λ)

m=

where r, s ∈ N.

   n ··· (λx + · · · + λxs + x)m dμ (x ) · · · dμ (xs ), m Zp Zp

(.)

Kim and Kim Advances in Difference Equations (2015) 2015:282

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Now, we observe that 

 Zp

···

Zp



( + t)λx +···+λxs +x dμ (x ) · · · dμ (xs )

s   λt log( + t) s ( + t)x ( + t)λ –  t  n   n ∞   t (s) (s) n . = Kl (x | λ)Dn–l n! l n= =

(.)

l=

By (.), we get 

 Zp

···

=

Zp

n 

(λx + · · · + λxs + x)n dμ (x ) · · · dμ (xs )

Kl(s) (x | λ)D(s) n–l

l=

  n . l

(.)

From (.) and (.), we have CK (r,s) n (x | λ) =

n  m     m n (s) (s) Ch(r) n–m (λ)Kl (x | λ)Dm–l . l m m=

(.)

l=

The generating function of CK (r,s) n (x | λ) is given by ∞ 

CK (r,s) n (x | λ)

n=

=

∞   n=

 =

Zp

tn n!

 Zp

···

 

··· 

Zp

Zp Zp



r-times

Ch(r) n (λx + · · · + λxs + x | λ) dμ (x ) · · · dμ (xs )

tn n!

 ···

Zp



( + t)λy +···+λyr +λx +···+λxs +x

s-times

× dμ– (y ) · · · dμ– (yr ) dμ (x ) · · · dμ(xs )  r    λ log( + t) s = ( + t)x . ( + t)λ +  ( + t)λ – 

(.)

Theorem . For r, s ∈ N and n ≥ , we have CK (r,s) n (x | λ) =

n  m     m n (s) (s) Ch(r) n–m (λ)Kl (x | λ)Dm–l . l m m= l=

We consider the Korobov and λ-Changhee mixed-type polynomials, which are given by  KC (r,s) n (x | λ) =

Zp

 ···

where r, s ∈ N and n ≥ .

Zp

Kn(r) (λx + · · · + λxs + x | λ) dμ– (x ) · · · dμ– (xs ),

(.)

Kim and Kim Advances in Difference Equations (2015) 2015:282

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Then, by (.), we get KC (r,s) n (x | λ)   n    n = ··· (λx + · · · + λxs + x)n–m dμ– (x ) · · · dμ– (xs ) Km(r) (λ) m Zp Zp m= =

n    n Km(r) (λ)Ch(s) n–m (x | λ). m m=

(.)

The generating function of KC (r,s) n (x | λ) is given by ∞ 

KC (r,s) n (x | λ)

n=

=

n=

=  =



∞  



tn n!

Zp

···

Zp

λt ( + t)λ –  λt ( + t)λ – 

Kn(r) (λx + · · · + λxs + x | λ) dμ– (x ) · · · dμ– (xs )

r 



Zp

r 

tn n!

···

Zp

( + t)λx +···+λxs +x dμ– (x ) · · · dμ– (xs )

 ( + t)λ + 

s ( + t)x .

(.)

Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to this work. All authors read and approved the final manuscript. Author details 1 Department of Mathematics, Sogang University, Seoul, 121-742, Republic of Korea. 2 Department of Mathematics, Kwangwoon University, Seoul, 139-701, Republic of Korea. Acknowledgements The authors would like to thank the referees and editor for their valuable comments. The work reported in this paper was conducted during the sabbatical year of Kwangwoon University in 2014. Received: 1 February 2015 Accepted: 10 August 2015 References 1. Kim, T: Symmetry of power sum polynomials and multivariate fermionic p-adic invariant integral on Zp . Russ. J. Math. Phys. 16(1), 93-96 (2014) 2. Jang, L-C, Kim, W-J, Simsek, Y: A study on the p-adic integral representation on Zp associated with Bernstein and Bernoulli polynomials. Adv. Differ. Equ. 2010, Article ID 163217 (2010) 3. Araci, S, Acikgoz, M, Sen, E: On the extended Kim’s p-adic q-deformed fermionic integrals in the p-adic integer ring. J. Number Theory 133(10), 3348-3361 (2013) 4. Lim, D, Do, Y: Some identities of Barnes-type special polynomials. Adv. Differ. Equ. 2015, 42 (2015) 5. Kim, T: New approach to q-Euler polynomials of higher order. Russ. J. Math. Phys. 17(2), 218-225 (2010) 6. Dolgy, DV, Kim, DS, Kim, T, Mansour, T: Barnes-type degenerate Euler polynomials. Appl. Math. Comput. 261, 388-396 (2015) 7. Roman, S: The Umbral Calculus. Pure and Applied Mathematics, vol. 111. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York (1984) 8. Dick, J, Pillichshammer, F, Waterhouse, BJ: The construction of good extensible Korobov rules. Computing 79(1), 79-91 (2007) 9. Karatsuba, A: Korobov trigonometric sums. Chebyshevski˘ı Sb. 7, 101-109 (2006) 10. Korobov, NM: On some properties of special polynomials. In: Proceedings of the IV International Conference “Modern Problems of Number Theory and Its Applications” (Russian), Tula, 2001, vol. 1, pp. 40-49 (2001) 11. Pylypiv, VM, Maliarchuk, AR: On some properties of Korobov polynomials. Carpath. Math. Publ. 6(1), 130-133 (2014) 12. Ustinov, AV: Korobov polynomials and umbral analysis. Chebyshevski˘ı Sb. 4, 137-152 (2003) 13. Park, JW: On the twisted Daehee polynomials with q-parameter. Adv. Differ. Equ. 2014, 304 (2014) 14. Kim, DS, Kim, T, Dolgy, DV, Komatsu, T: Barnes-type degenerate Bernoulli polynomials. Adv. Stud. Contemp. Math. 25(1), 121-146 (2015)