Degenerate Poly-Bernoulli Polynomials with Umbral Calculus Viewpoint Dae San Kim1,Taekyunkim2*, Hyuck in Kwon2 and Touﬁk Mansour3

Kim et al. Journal of Inequalities and Applications (2015)2015:228 DOI 10.1186/s13660-015-0748-7

R E S E A R C H Open Access Degenerate poly-Bernoulli polynomials with umbral calculus viewpoint Dae San Kim1,TaekyunKim2*, Hyuck In Kwon2 and Touﬁk Mansour3

*Correspondence: [email protected] 2Department of Mathematics, Abstract Kwangwoon University, Seoul, 139-701, S. Korea In this paper, we consider the degenerate poly-Bernoulli polynomials. We present Full list of author information is several explicit formulas and recurrence relations for these polynomials. Also, we available at the end of the article establish a connection between our polynomials and several known families of polynomials. MSC: 05A19; 05A40; 11B83 Keywords: degenerate poly-Bernoulli polynomials; umbral calculus

1 Introduction

The degenerate Bernoulli polynomials βn(λ, x)(λ =)wereintroducedbyCarlitz[ ]and rediscovered by Ustinov []underthenameKorobov polynomials of the second kind.They are given by the generating function

t tn ( + λt)x/λ = β (λ, x) . ( + λt)/λ – n n! n≥

When x =,βn(λ)=βn(λ, ) are called the degenerate Bernoulli numbers (see []). We

observe that limλ→ βn(λ, x)=Bn(x), where Bn(x)isthenth ordinary Bernoulli polynomial (see the references). (k) The poly-Bernoulli polynomials PBn (x)aredeﬁnedby

Li ( – e–t) tn k ext = PB(k)(x) , et – n n! n≥

n where Li (x)(k ∈ Z)istheclassicalpolylogarithm function given by Li (x)= x (see k k n≥ nk [–]). (k) For  = λ ∈ C and k ∈ Z,thedegenerate poly-Bernoulli polynomials Pβn (λ, x)arede- ﬁned by Kim and Kim to be

Li ( – e–t) tn k ( + λt)x/λ = Pβ(k)(λ, x) (see []). (.) ( + λt)/λ – n n! n≥

(k) (k) When x =,Pβn (λ)=Pβn (λ, ) are called degenerate poly-Bernoulli numbers.Weob- (k) (k) serve that limλ→ Pβn (λ, x)=PBn (x).

© 2015 Kim et al. 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, pro- vided 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. Kim et al. Journal of Inequalities and Applications (2015)2015:228 Page 2 of 13

The goal of this paper is to use umbral calculus to obtain several new and interesting identities of degenerate poly-Bernoulli polynomials. To do that we recall the umbral calculus as given in [, ]. We denote the algebra of polynomials in a single variable x over C by and the vector space of all linear functionals on by ∗. The action of a linear functional L on a polynomial p(x)isdenotedbyL | p(x). We deﬁne the vector space structure on ∗ by cL + c L | p(x) = cL | p(x) + c L | p(x),wherec, c ∈ C.Wedeﬁnethealgebra of formal power series in a single variable t to be

tk H = f (t)= a a ∈ C .(.) k k! k k≥

Apowerseriesf (t) ∈ H deﬁnes a linear functional on by setting n f (t) | x = an, for all n ≥ (see[, –]). (.)

By (.)and(.), we have k n t | x = n!δn,k, for all n, k ≥ , (.) | n tn | where δn,k is the Kronecker symbol. Let fL(t)= n≥ L x n! .From(.), we have fL(t) n n ∗ x = L | x .So,themapL → fL(t) is a vector space isomorphism from onto H.Thus, H is thought of as set of both formal power series and linear functionals. We call H the umbral algebra.Theumbral calculus is the study of umbral algebra. The order O(f (t)) of the non-zero power series f (t) ∈ H is the smallest integer k for which the coeﬃcient of tk does not vanish. Suppose that f (t), g(t) ∈ H such that O(f (t)) =  and

O(g(t)) = , then there exists a unique sequence sn(x) of polynomials such that k g(t) f (t) | sn(x) = n!δn,k,(.)

where n, k ≥ . The sequence sn(x) is called the Sheﬀer sequence for (g(t), f (t)), which is yt denoted by sn(x) ∼ (g(t), f (t)) (see [, ]). For f (t) ∈ H and p(x) ∈ ,wehavee | p(x) = p(y), f (t)g(t) | p(x) = g(t) | f (t)p(x),and

tn xn f (t)= f (t) | xn , p(x)= tn | p(x) (.) n! n! n≥ n≥

(see [, ]). From (.), we obtain tk | p(x) = p(k)() and  | p(k)(x) = p(k)(), where p(k)() denotes the kth derivative of p(x)withrespecttox at x =.So,wegettkp(x)=p(k)(x)= k d p(x), for all k ≥ . Let s (x) ∼ (g(t), f (t)), then we have dxk n  ¯ tn eyf (t) = s (y) ,(.) ¯ n n! g(f (t)) n≥

∈ C ¯ ∼ for all y ,wheref (t) is the compositional inverse of f (t)(see[, ]). For sn(x) ∼ n (g(t), f (t)) and rn(x) (h(t), (t)), let sn(x)= k= cn,krk(x), then we have Kim et al. Journal of Inequalities and Applications (2015)2015:228 Page 3 of 13

¯  h(f (t)) ¯ k n cn,k = f (t) x (.) k! g(f¯(t))

(see [, ]). (k) From (.), we see that Pβn (λ, x) is the Sheﬀer sequence for the pair et –  g(t), f (t) = , eλt – .(.) –  (eλt–) λ Lik( – e λ )

In this paper, we will use umbral calculus in order to derive some properties, explicit formulas, recurrence relations, and identities as regards the degenerate poly-Bernoulli polynomials. Also, we establish a connection between our polynomials and several known families of polynomials.

2 Explicit formulas In this section we present several explicit formulas for the degenerate poly-Bernoulli poly- (k) nomials, namely Pβn (λ, x). To do so, we recall that Stirling numbers S(n, k) of the first t kind can be defined by means of exponential generating functions as ≥j S(, j) ! =  j j! log ( + t) and can be defined by means of ordinary generating functions as

n m t (x)n = S(n, m)x ∼ , e – ,(.) m=

n where (x)n = x(x–)(x–)···(x–n+)with(x) =.Forλ =,wedeﬁne( x | λ)n = λ (x/λ)n. t Sometimes, for simplicity, we denote the function e – by G (t). –  (eλt–) k Lik (–e λ ) First, we express the degenerate poly-Bernoulli polynomials in terms of degenerate poly- Bernoulli numbers.

Theorem . For all n ≥ , n n n Pβ(k)(λ, x)= S (, j)λ–jPβ(k) (λ)xj. n  n– j= =j ∼ n  ¯ – ¯ j | n j Proof By (.), for sn(x) (g(t), f (t)) we have sn(x)= j= j! g(f (t)) f (t) x x .Thus,in the case of degenerate poly-Bernoulli polynomials (see (.)), we have

 – g f¯(t) f¯(t)j | xn j!  Li ( – e–t)  j = k log( + λt) xn j! ( + λt)/λ – λ Li ( – e–t) logj( + λt) = λ–j k xn ( + λt)/λ – j! Li ( – e–t) λt = λ–j k S (, j) xn ( + λt)/λ –  ! ≥j n n Li ( – e–t) = S (, j)λ–j k xn–  ( + λt)/λ – =j Kim et al. Journal of Inequalities and Applications (2015)2015:228 Page 4 of 13

n n tm = S (, j)λ–j Pβ(k)(λ) xn–  m m! =j m≥ n n = S (, j)λ–jPβ(k) (λ),  n– =j

which completes the proof.

Note that Stirling numbers S(n, k) of the second kind can be deﬁned by the exponential generating functions as

xn (et –)k S (n, k) = . (.)  n! k! n≥k

Theorem . For all n ≥ , m–j n n m Pβ(k)(λ, x)= S (n, m)S (m – j, )λn––jPβ(k)(λ) xj. n j   j= m=j = | n n–m m ∼  λt Proof By (.), we have (x λ)n = m= S(n, m)λ x (, λ (e –)),andby(.), we have  G (t)Pβ(k)(λ, x) ∼ , eλt – ,(.) k n λ (k) n n–m m which implies Gk(t)Pβn (λ, x)= m= S(n, m)λ x .Thus,

n  λt Li ( – e– λ (e –)) Pβ(k)(λ, x)= S (n, m)λn–m k xm n  et – m=

n –v n–m Lik( – e ) m = S(n, m)λ x ( + λv)/λ – v=  (eλt–) m= λ n (  (eλt –)) = S (n, m)λn–mPβ(k)(λ) λ xm  ! m= ≥ n m λjtj = S (n, m)λn–m–Pβ(k)(λ) S (j, ) xm   j! m= = j≥ n m m m = S (n, m)S (j, )λn–m–+jPβ(k)(λ)xm–j j   m= = j=

n m m– m = S (n, m)S (m – j, )λn––jPβ(k)(λ)xj j   m= = j= m–j n n m = S (n, m)S (m – j, )λn––jPβ(k)(λ) xj,(.) j   j= m=j =

which completes the proof. Kim et al. Journal of Inequalities and Applications (2015)2015:228 Page 5 of 13

Theorem . For all n ≥ , n–j n–j– n n – n – Pβ(k)(λ, x)= λn–m–jS (n – j – , m)B(n)Pβ(k)(λ) xj. n j  m j= = m=

Proof Note that xn ∼ (, t). Thus, by (.) and transfer formula, we have λt n λt n G (t)Pβ(k)(λ, x)=x x–xn = x xn– k n eλt – eλt – λt n– n – = x B(n) xn– = x λB(n)xn–– ! ≥ = n– n – = λB(n)xn–. = (k) n– n– (n) – n– Therefore, Pβn (λ, x)= = λ B Gk(t) x ,which,by(.), completes the proof.

Theorem . For all n ≥ , n (k) m+ n (m +)! Pβ (λ, x)= (–) S ( +,m +) β (λ, x). n (m +)k( +)  n– = m=

Proof By (.), we have Li ( – e–t) Pβ(k)(λ, y)= k ( + λt)y/λ xn n ( + λt)/λ – Li ( – e–t) t = k ( + λt)y/λxn t ( + λt)/λ –

–t Lik( – e ) t n = β(λ, y) x t ! ≥ n –t m n  ( – e ) n– = β(λ, y) x t mk = m≥ n n–+ m –t m n (–) (e –) n– = β(λ, y) x .(.) mkt = m=

Thus, by (.), we obtain n n– m+ n–+ j (k) n (–) (m +)! (–) j– n– Pβ (λ, y)= β(λ, y) S (j, m +) t x n (m +)k  j! = m= j=m+

n n– n (–)m+(m +)! (–)n–+(n – )! = β(λ, y) S (n – +,m +) (m +)k  (n – +)! = m= n n– n+m– n (m +)! = (–) S (n – +,m +)β(λ, y), (m +)k(n – +)  = m=

which completes the proof. Kim et al. Journal of Inequalities and Applications (2015)2015:228 Page 6 of 13

Note that the above theorem has been obtained in Theorem . in [].

Theorem . For all n ≥ , n (k)  n + (k) Pβ (λ, x)= Pβ β (λ, x), n n + n – , m, – m + m n– = m=

where a = a! is the multinomial coeﬃcient. b,b,b b!b!b!

Proof By (.), we have n t –t (k) n e – Lik( – e ) n– Pβ (λ, y)= β(λ, y) x n t et – = n t m n e – (k) t n– = β(λ, y) Pβ x t m m! = m≥ n n– t n n – (k) e – n––m = β(λ, y)Pβ x . m m t = m= et– | n––m  n––m  Note that t x =  u du = n––m+ .Thus, n n– (k)  n n – (k) Pβ (λ, y)= Pβ β(λ, y) n n – – m + m m = m= n  n (k) = Pβ β (λ, y) – m + m m n– = m= n  n + (k) = Pβ β (λ, y), n + n – , m, – m + m n– = m=

which completes the proof.

j+ –t t y  t j Bjt ≥ Note that Li( – e )=  ey– dy = j≥ Bj j!  y dy = j≥ j!(j+) .Forgeneralk , the –t function Lik( – e ) has the integral representation t y y y –t   ···  y ··· Lik –e = y y y y dy dy dy dy,  e –  e – e –  e – (k–) times

which, by induction on k,implies

k– ··· Bj –t ··· j+ +jk–+ i Lik –e = t ··· .(.) ≥ ≥ ji!(j + + ji +) j  jk–  i=

Theorem . For all n ≥  and k ≥ , n k– B β(k) λ β λ ji P n ( , x)= (n) n–( , x) ··· . ··· ji!(j + + ji +) = j+ +jk–= i= Kim et al. Journal of Inequalities and Applications (2015)2015:228 Page 7 of 13

Proof By (.), we have

n –t (k) n Lik( – e ) n– Pβ (λ, x)= β(λ, x) x . n t =

Thus, by (.), we obtain n n! k– B β(k) λ β λ ji P n ( , x)= ( , x) ··· , ! ··· ji!(j + + ji +) = j+ +jk–=n– i=

which completes the proof.

–t n+ Note that here we compute A = Lik( – e ) | x in several diﬀerent ways. As for the ﬁrst way, we have

t d t e–sLi ( – e–s) A Li e–s ds xn+ k– ds xn+ = k – = –s  ds  –e t Li ( – e–s) PB(k–) t = k– ds xn+ = m sm ds xn+ es – m!  m≥  PB(k–) = m tm+ | xn+ = PB(k–). (m +)! n m≥

As for the second way, we have

(et –)Li ( – e–t) Li ( – e–t) A = k xn+ = k et – xn+ et – et – Li ( – e–t) n n + Li ( – e–t) = k (x +)n+ – xn+ = k xm et – m et – m= n n + = PB(k). m m m=

As for the third way, by (.), we have

k– Bji A =(n +)! ··· . ··· ji!(j + + ji +) j+ +jk–=n i=

Hence, we can state the following result.

Theorem . For all n ≥ ,

n k– (k–) n + (k) Bji PBn = PBm =(n +)! ··· . m ··· ji!(j + + ji +) m= j+ +jk–=n i= Kim et al. Journal of Inequalities and Applications (2015)2015:228 Page 8 of 13

3 Recurrences In this section, we present several recurrences for the degenerate poly-Bernoulli polyno- (k) | ∼ eλt– mials, namely Pβn (λ, x). Note that, by (.)andthefactthat(x λ)n (, λ ), we obtain the following identity. ≥ (k) n n (k) | Proposition . For all n , Pβn (λ, x + y)= j= j Pβj (λ, x)(y λ)n–j.

It is well known that if sn(x) ∼ (g(t), f (t)), then we have f (t)sn(x)=nsn–(x). Thus, by (.), eλt– (k) (k) we obtain λ Pβn (λ, x)=nPβn–(λ, x), which implies the following result.

≥ (k) (k) (k) Proposition . For all n , Pβn (λ, x + λ)=Pβn (λ, x)+nλPβn–(λ, x).

Theorem . For all n ≥ ,

(k) (k) Pβn+(λ, x)–xPβn (λ, x – λ) m+ m+–i m + = λm+–i–S (m +–i, ) PB(k)B (x)–Pβ(k)(λ)B (x +–λ) . i  i i i= =

g (t)  ∼ Proof By applying the fact that sn+(x)=(x – g(t) ) f (t) sn(x) for all sn(x) (g(t), f (t)) and (.), we obtain g (t) g (t) Pβ(k) (λ, x)= x – e–λtPβ(k)(λ, x)=xPβ(k)(λ, x – λ)–e–λt Pβ(k)(λ, x), n+ g(t) n n g(t) n

where

g (t)  λt = log et – – log Li –e– λ (e –) g(t) k

t –  (eλt–) e  Lik–( – e λ )  λt = – eλte– λ (e –). t –  (eλt–) –  (eλt–) e – Lik( – e λ ) –e λ

–λt g (t) (k) Thus, the expression A = e g(t) Pβn (λ, x)isgivenby  te(–λ)t t G (t)– – G (t)– G (t)Pβ(k)(λ, x). t k  λt k– k n t e – e λ (e –) – (k) n n–m m Note that, by (.), we have Gk(x)Pβn (λ, x)= m= S(n, m)λ x . Therefore, n (–λ)t n–m  te – t – m A = S(n, m)λ Gk(t) – Gk–(t) x t et –  (eλt–) m= e λ – n (–λ)t S(n, m) n–m te – t – m+ = λ Gk(t) – Gk–(t) x .(.) m + et –  (eλt–) m= e λ –

We remark that the expression in the parentheses in (.) has order at least one. Now, let us simplify (.): Kim et al. Journal of Inequalities and Applications (2015)2015:228 Page 9 of 13

te(–λ)t G (t)–xm+ et – k te(–λ)t Li ( – e–s) = k xm+ t /λ eλt– e – ( + λs) – s= λ λt te(–λ)t m+ ( e – ) = Pβ(k)(λ) λ xm+ et – ! = te(–λ)t m+ m+ m + = λi–S (i, )Pβ(k)(λ)xm+–i et – i  = i= m+ m+–i m + te(–λ)t = λm+–i–S (m +–i, )Pβ(k)(λ) xi i  et – i= = m+ m+–i m + = λm+–i–S (m +–i, )Pβ(k)(λ)B (x +–λ)(.) i  i i= =

and

t – m+  λt Gk–(t) x e λ (e –) – t Li ( – e–s) = k– xm+ t s eλt– e – e – s= λ λt t m+ ( e – ) = PB(k) λ xm+ et – ! = t m+ m+ m + = λi–S (i, )PB(k)xm+–i et – i  = i= m+ m+–i m + t = λm+–i–S (m +–i, )PB(k) xi i  et – i= = m+ m+–i m + = λm+–i–S (m +–i, )PB(k)B (x). (.) i  i i= =

Hence, by (.)-(.), we complete the proof.

d (k) (k) Inthenextresultweexpress dx Pβn (λ, x)intermsofPβn (λ, x). ≥ d (k) n– (–λ)n–– (k) Proposition . For all n , dx Pβn (λ, x)=n! = !(n–) Pβ (λ, x). d n– n ¯ | n– ∼ Proof Note that dx sn(x)= = f (t) x s(x) for all sn(x) (g(t), f (t)). Thus, by (.), we have   xm f¯(t) | xn– = log( + λt) xn– = (–)m–λm(m –)! xn– λ λ m! m≥

=(–λ)n––(n – – )!,

which completes the proof. Kim et al. Journal of Inequalities and Applications (2015)2015:228 Page 10 of 13

Theorem . For all n ≥ ,

(k) (k) Pβn (λ, x)–xPβn–(λ, x – λ)  n n = Pβ(k–)(λ, x)B – Pβ(k)(λ, x +–λ)β (λ) . n m m n–m m n–m m=

Proof By (.), we have Li ( – e–t) Pβ(k)(λ, y)= k ( + λt)y/λ xn n ( + λt)/λ – Li ( – e–t) d = k ( + λt)y/λ xn– (.) ( + λt)/λ –dt d Li ( – e–t) + k ( + λt)y/λ xn– .(.) dt ( + λt)/λ –

The term in (.)isgivenby Li ( – e–t) y k ( + λt)(y–λ)/λ xn– = yPβ(k) (λ, y – λ). (.) ( + λt)/λ – n–

–t For the term in (.), we observe that d Lik (–e ) =  (A – B), where dt (+λt)/λ– t

t Li ( – e–t) t Li ( – e–t) A = k– , B = k ( + λt)/λ–. et –( + λt)/λ – ( + λt)/λ –( + λt)/λ –

Note that the expression A – B has order of at least . Now, we are ready to compute the term in (.). By (.), we have d Li ( – e–t) k ( + λt)y/λ xn– dt ( + λt)/λ –  = (A – B)( + λt)y/λ xn– t   = A( + λt)y/λ | xn – B( + λt)y/λ | xn n n  t tm = Pβ(k–)(λ, y) xn n et – m m! m≥  t tm – Pβ(k)(λ, y +–λ) xn n ( + λt)/λ – m m! m≥  n n t = Pβ(k–)(λ, y) xn–m n m m et – m=  n n t – Pβ(k)(λ, y +–λ) xn–m n m m ( + λt)/λ – m=  n n = Pβ(k–)(λ, y)B – Pβ(k)(λ, y +–λ)β (λ) .(.) n m m n–m m n–m m= Kim et al. Journal of Inequalities and Applications (2015)2015:228 Page 11 of 13

Thus, if we replace (.)by(.)and(.)by(.), we obtain

(k) (k) Pβn (λ, x)–xPβn–(λ, x – λ)  n n = Pβ(k–)(λ, x)B – Pβ(k)(λ, x +–λ)β (λ) , n m m n–m m n–m m=

as claimed.

4 Connections with families of polynomials In this section, we present a few examples on the connections with families of polynomials. (s) We start with the connection to Bernoulli polynomials Bn (x) of order s. Recall that the Bernoulli polynomials B(s)(x) of order s are deﬁned by the generating function ( t )sext = n et– (s) tn n≥ Bn (x) n! ,equivalently, et – s B(s)(x) ∼ , t (.) n t

(k) (see [–]). In the next result, we express our polynomials Pβn (λ, x)intermsof (s) Bernoulli polynomials of order s. To do that, we recall that the Bernoulli numbers bn of the second kind of order s are deﬁned as

ts tn = b(s) .(.) logs( + t) n n! n≥

Theorem . For all n ≥ , n n n– n––r j n (k) ,r,j ,n– –r–j +r+i–m (s) Pβn (λ, x)= j+s λ cn,m(, r, j, i) Bm (x), m= =m r= j= i= j

(s) (k) a where cn,m(, r, j, i)=S(, m)S(j + s, j – i + s)S(j – i + s, s)br Pβ (λ) and = n––r–j b,...,bm a! is the multinomial coeﬃcient. b!···bm! (+λt)/λ– s (k) n (s) Proof Let hs(t)=( t ) and Pβn (λ, x)= m= cn,mBm (x). By (.), (.), and (.), we have

m m!λ cn,m –t /λ s s Li ( – e ) ( + λt) – λt m = k log( + λt) xn , ( + λt)/λ – t log( + λt)

which, by (.), implies

n n Li ( – e–t) λt s λmc = λS (, m) k h (t) xn– n,m  ( + λt)/λ – s log( + λt) =m n n Li ( – e–t) λt s = λS (, m) k h (t) xn–  ( + λt)/λ – s log( + λt) =m Kim et al. Journal of Inequalities and Applications (2015)2015:228 Page 12 of 13

n n– n n – Li ( – e–t) = λ+rS (, m)b(s) k h (t) xn––r r  r ( + λt)/λ – s =m r= n n– n n – Li ( – e–t) = λ+rS (, m)b(s) k h (t)xn––r . r  r ( + λt)/λ – s =m r=

One can show that  s e λ log(+λt) – h (t)= s t

j λi = s! S (j + s, j – i + s)S (j – i + s, s) tj.   (j + s)! j≥ i=

Thus, by (.), we have

j n n– n– –r n n – c = s! λ+r–mS (, m)b(s)S (j + s, j – i + s) n,m r  r  =m r= j= i= λi Li ( – e–t) × S (j – i + s, s) (n – – r) k xn––r–j  (j + s)! j ( + λt)/λ –

n n– n––r j n ,r,j ,n– –r–j +r+i–m = j+s λ S(, m)S(j + s, j – i + s) =m r= j= i= j × (s) (k) S(j – i + s, s)br Pβn––r–j(λ) ,

as required.

Similar techniques as in the proof of the previous theorem, we can express our polyno- (k) mials Pβn (λ, x) in terms of other families. Below we present three examples, where we leave the proofs to the interested reader. (k) The ﬁrst example is to express our polynomials Pβn (λ, x) in terms of Frobenius-Euler (s) | μ polynomials. Note that the Frobenius-Euler polynomials Hn (x )oforders are deﬁned –μ s xt (s) | tn (s) | by the generating function ( et–μ ) e = n≥ Hn (x μ) n! (μ = ), or equivalently, Hn (x ∼ et–μ s μ) (( –μ ) , t)(see[, ]).

Theorem . For all n ≥ , n n n– s n n – s λ–m(–μ)s–i Pβ(k)(λ, x)= c (, r, i) H(s)(x | μ), n r i ( – μ)s n,m m m= =m r= i=

(k) where cn,m(, r, i)=S(, m)(i | λ)n––rPβr (λ).

(k) If we express our polynomials Pβn (λ, x)intermsoffalling polynomials (x | λ)n,thenwe get the following result. ≥ (k) n n (k) | Theorem . For all n , Pβn (λ, x)= m= m Pβn–m(λ)(x λ)m. Kim et al. Journal of Inequalities and Applications (2015)2015:228 Page 13 of 13

(k) Our last example is to express our polynomials Pβn (λ, x)intermsofdegenerate (s) Bernoulli polynomials βn (λ, x)oforders. Note that the degenerate Bernoulli polynomials (s) βn (λ, x)oforders are given by t s tn ( + λt)x/λ = β(s)(λ, x) . ( + λt)/λ – n n! n≥

Theorem . For all n ≥ , n n–m j n–m n j Pβ(k)(λ, x)= λic (j, i) β(s)(λ, x), n m j+s n,m m m= j= i= s

(k) where cn,m(j, i)=S(j + s, j – i + s)S(j – i + s, s)Pβn–m–j(λ).

Competing interests The authors declare that they have no competing interests.

Authors’ contributions All authors contributed equally to the writing of this paper. All authors read and approved the ﬁnal manuscript.

Author details 1Department of Mathematics, Sogang University, Seoul, 121-742, S. Korea. 2Department of Mathematics, Kwangwoon University, Seoul, 139-701, S. Korea. 3Department of Mathematics, University of Haifa, Haifa, 3498838, Israel.

Acknowledgements This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MOE) (No. 2012R1A1A2003786).

Received: 7 April 2015 Accepted: 2 July 2015

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