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

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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 cal- culus 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 func- tional 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.
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