In this paper, we consider the degenerate poly-Bernoulli polynomials and
present new and explicit formulas for computing them in terms of the degenerate
Bernoulli polynomials and Stirling numbers of the second kind.Comment: 8 page

Dae San Kim

Taekyun Kim

Advances in Difference Equations

arXiv

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Kim and Kim Advances in Diﬀerence Equations  (2015) 2015:258 DOI 10.1186/s13662-015-0595-3RESEARCH Open AccessA note on degenerate poly-Bernoullinumbers and polynomialsDae San Kim1* and Taekyun Kim2*Correspondence:dskim@sogang.ac.kr1Department of Mathematics,Sogang University, Seoul, 121-742,Republic of KoreaFull list of author information isavailable at the end of the articleAbstractIn this paper, we consider the degenerate poly-Bernoulli polynomials and presentnew and explicit formulas for computing them in terms of the degenerate Bernoullipolynomials and Stirling numbers of the second kind.MSC: 11B68; 11B73; 11B83Keywords: degenerate poly-Bernoulli polynomial; degenerate Bernoulli polynomial;Stirling number of the second kind1 IntroductionFor λ ∈C, Carlitz considered the degenerate Bernoulli polynomials given by the generat-ing functiont( + λt) λ – ( + λt) xλ =∞∑n=βn(x | λ) tnn! (see [–]). (.)When x = , βn(λ) = βn( | λ) are called the degenerate Bernoulli numbers.Thus, by (.), we getβn(x | λ) =n∑l=(nl)βl(λ)(x | λ)n–l, (.)where (x | λ)n = x(x – λ)(x – λ) · · · (x – λ(n – )).The classical polylogarithm function Lik isLik(x) =∞∑n=xnnk (k ∈ Z; see [, –]). (.)From (.), we note that∞∑n=limλ→βn(x | λ)tnn!= limλ→t( + λt) λ – ( + λt) xλ© 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 anymedium, pro-vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, andindicate if changes were made.Kim and Kim Advances in Diﬀerence Equations  (2015) 2015:258 Page 2 of 8= tet – ext=∞∑n=Bn(x)tnn! , (.)where Bn(x) are called the Bernoulli polynomials (see [–]).Thus, by (.), we getlimλ→βn(x | λ) = Bn(x) (n≥ ). (.)In [, ], the poly-Bernoulli polynomials are given byLik( – e–t)et –  ext =∞∑n=B(k)n (x)tnn! . (.)For k = , we haveLi( – e–t)et –  ext = tet – ext=∞∑n=Bn(x)tnn! . (.)By (.) and (.), we get B()n (x) = Bn(x).The Stirling numbers of the second kind are given byxn =n∑l=S(n, l)(x)l (see [–]), (.)and the Stirling numbers of the ﬁrst kind are deﬁned by(x)n = x(x – ) · · · (x – n + ) =n∑l=S(n, l)xl (n≥ ). (.)The purpose of this paper is to construct the degenerate poly-Bernoulli polynomialsand present new and explicit formulas for computing them in terms of the degenerateBernoulli polynomials and Stirling numbers of the second kind.2 Degenerate poly-Bernoulli numbers and polynomialsFor λ ∈ C, k ∈ Z, we consider the degenerate poly-Bernoulli polynomials given by thegenerating functionLik( – e–t)( + λt) λ – ( + λt) xλ =∞∑n=β (k)n (x | λ)tnn! . (.)When x = , β (k)n (λ) = β (k)n ( | λ) are called the degenerate poly-Bernoulli numbers. Notethat β ()n (x | λ) = βn(x | λ) and limλ→ β (k)n (x | λ) = B(k)n (x).Kim and Kim Advances in Diﬀerence Equations  (2015) 2015:258 Page 3 of 8From (.), we can derive the following equation:∞∑n=β (k)n (x | λ)tnn! =(Lik( – e–t)( + λt) λ – )( + λt) xλ=( ∞∑l=β(k)l (λ)tll!)( ∞∑m=(x | λ)m tmm!)=∞∑n=( n∑l=(nl)β(k)l (λ)(x | λ)n–l)tnn! . (.)Thus, by (.), we getβ (k)n (x | λ) =n∑l=(nl)β(k)l (λ)(x | λ)n–l. (.)Now, we observe thatLik( – e–t)( + λt) λ – ( + t) xλ=∞∑n=β (k)n (x | λ)tnn!= ( + t)xλ( + λt) λ – ∫ tey – ∫ yey – ∫ y· · · ey – ∫ y︸ ︷︷ ︸(k–) timesyey –  dy · · ·dy. (.)From (.), we have∞∑n=β ()n (x | λ)tnn!= ( + t)xλ( + λt) λ – ∫ tyey –  dy= ( + t)xλ( + λt) λ – ∞∑l=Bll!∫ tyl dy=( t( + λt) λ – ( + λt) xλ)( ∞∑l=Bll + tll!)=∞∑n={ n∑l=(nl) Bll + βn–l(x | λ)}tnn! , (.)where Bn = Bn() are the Bernoulli numbers.By comparing the coeﬃcients on both sides of (.), we obtain the following theorem.Kim and Kim Advances in Diﬀerence Equations  (2015) 2015:258 Page 4 of 8Theorem . For n≥ , we haveβ ()n (x | λ) =n∑l=(nl) Bll + βn–l(x | λ)= βn(x | λ) – nβn–(x | λ) +n∑l=(nl) Bll + βn–l(x | λ).Moreover,β (k)n (x | λ) =n∑l=(nl)β(k)l (λ)(x | λ)n–l.By (.), we easily get∞∑n=β (k)n (x | λ)tnn!= Lik( – e–t)( + λt) λ – ( + t) xλ= t( + λt) λ – ( + t) xλ Lik( – e–t)t . (.)We observe thatt Lik( – e–t)= t∞∑n=nk( – e–t)n= t∞∑n=(–)nnk n!∞∑l=nS(l,n)(–t)ll!= t∞∑l=l∑n=(–)n+lnk n!S(l,n)tll!=∞∑l=l+∑n=(–)n+l+nk n!S(l + ,n)l + tll! . (.)From (.) and (.), we have∞∑n=β (k)n (x | λ)tnn!=( ∞∑m=βm(x | λ) tmm!)( ∞∑l=( l+∑p=(–)p+l+pk p!S(l + ,p)l + )tll!)=∞∑n={ n∑l=(nl)( l+∑p=(–)p+l+p!pkS(l + ,p)l + )βn–l(x | λ)}tnn! . (.)By comparing the coeﬃcients on both sides of (.), we obtain the following theorem.Kim and Kim Advances in Diﬀerence Equations  (2015) 2015:258 Page 5 of 8Theorem . For n≥ , we haveβ (k)n (x | λ) =n∑l=(nl)( l+∑p=(–)p+l+p!pkS(l + ,p)l + )βn–l(x | λ).It is easy to show thatLik( – e–t)( + λt) λ – ( + λt) x+λ – Lik( – e–t)( + λt) λ – ( + λt) xλ= ( + λt) xλ Lik( – e–t)=( ∞∑l=(x | λ)l tll!)( ∞∑m=( – e–t)mmk)=( ∞∑l=(x | λ)l tll!)( ∞∑m=( – e–t)m+(m + )k)=( ∞∑l=(x | λ)l tll!)( ∞∑p=( p–∑m=(–)m+p+(m + )k (m + )!S(p,m + ))tpp!)=∞∑n={ n∑p=p–∑m=(–)m+p+(m + )k (m + )!S(p,m + )(np)(x | λ)n–p}tnn! . (.)On the other hand,Lik( – e–t)( + λt) λ – ( + λt) x+λ – Lik( – e–t)( + λt) λ – ( + λt) xλ=∞∑n={β (k)n (x +  | λ) – β (k)n (x | λ)} tnn! . (.)Therefore, by (.) and (.), we obtain the following theorem.Theorem . For n≥ , we haveβ (k)n (x +  | λ) – β (k)n (x | λ)=n∑p=( p–∑m=(–)m+k+(m + )k (m + )!S(k +m + ))(np)(x | λ)n–p.Now, we note thatLik( – e–t)( + λt) λ – ( + λt) xλ= Lik( – e–t)( + λt) dλ – d–∑a=( + λt) l+xλ=(Lik( – e–t)t) dd–∑a=dt( + λt) dλ – ( + λt) l+xλKim and Kim Advances in Diﬀerence Equations  (2015) 2015:258 Page 6 of 8=∞∑l=( l+∑p=(–)p+l+pk p!S(l + ,p)l + )tll!×d–∑a=∞∑m=βm( l + xd∣∣∣λd)dm– tmm!=d–∑a=( ∞∑n=( n∑l=l+∑p=(nl) (–)p+l+pk p!S(l + ,p)l +  βn–l( l + xd∣∣∣λd)dn–l–)tnn!)=∞∑n={ d–∑a=n∑l=l+∑p=(nl) (–)p+l+pk p!S(l + ,p)l +  βn–l( l + xd∣∣∣λd)dn–l–}tnn! , (.)where d is a ﬁxed positive integer.On the other hand,Lik( – e–t)( + λt) λ – ( + λt) xλ=∞∑n=β (k)n (x | λ)tnn! . (.)Therefore, by (.) and (.), we obtain the following theorem.Theorem . For n≥ , d ∈N and k ∈ Z, we haveβ (k)n (x | λ)=d–∑a=n∑l=l+∑p=(nl) (–)p+l+pk p!S(l + ,p)l +  βn–l( l + xd∣∣∣λd)dn–l–.From (.), we can derive the following equation:∞∑n=β (k)n (x + y | λ)tnn!= Lik( – e–t)( + λt) λ – ( + λt)x+yλ=(Lik( – e–t)( + λt) λ – ( + tλ) xλ)( + λt)yλ=( ∞∑l=β(k)l (x | λ)tll!)( ∞∑m=(y | λ)m tmm!)=∞∑n=( n∑l=(nl)β(k)l (x | λ)(y | λ)n–l)tnn! . (.)Therefore, by (.), we obtain the following theorem.Kim and Kim Advances in Diﬀerence Equations  (2015) 2015:258 Page 7 of 8Theorem . For n≥ , we haveβ (k)n (x + y | λ) =n∑l=(nl)β(k)l (x | λ)(y | λ)n–l.Remarkddxβ(k)n (x | λ)= ddxn∑l=(nl)β(k)n–l(λ)(x | λ)l=n∑l=(nl)β(k)n–l(λ)l–∑j=x – λjl–∏i=(x – λi)=n∑l=(nl)β(k)n–l(λ)l–∑j=l–∏i=i=j(x – λi).Competing interestsThe authors declare that they have no competing interests.Authors’ contributionsAll authors contributed equally to this work. All authors read and approved the ﬁnal manuscript.Author details1Department of Mathematics, Sogang University, Seoul, 121-742, Republic of Korea. 2Department of Mathematics,Kwangwoon University, Seoul, 139-701, Republic of Korea.AcknowledgementsThe authors would like to thank the referees for their valuable comments.Received: 18 March 2015 Accepted: 6 August 2015References1. Carlitz, L: Degenerate Stirling, Bernoulli and Eulerian numbers. Util. Math. 15, 51-88 (1979)2. Kim, DS, Kim, T, Dolgy, DV, Komatsu, T: Barnes-type degenerate Bernoulli polynomials. Adv. Stud. Contemp. Math.25(1), 121-146 (2015)3. Kim, T: Barnes’ type multiple degenerate Bernoulli and Euler polynomials. Appl. Math. Comput. 258, 556-564 (2015)4. Jolany, H, Mohebbi, H: Some results on Generalized multi poly-Bernoulli and Euler polynomials. Int. J. Math. Comb. 2,117-129 (2011)5. Kim, DS, Kim, T, Mansour, T, Dolgy, DV: On poly-Bernoulli polynomials of the second kind with umbral calculusviewpoint. Adv. Diﬀer. Equ. 2015(1), 27 (2015)6. Kim, DS, Kim, T: Some identities of degenerate Euler polynomials arising from p-adic fermionic integrals on Zp .Integral Transforms Spec. Funct. 26(4), 295-302 (2015)7. Kim, DS, Kim, T: Higher-order Frobenius-Euler and poly-Bernoulli mixed-type polynomials. Adv. Diﬀer. Equ. 2013, 251(2013)8. Kim, DS, Kim, T: Hermite and poly-Bernoulli mixed-type polynomials. Adv. Diﬀer. Equ. 2013(343), 12 (2013)9. Kim, DS, Kim, T: A note on poly-Bernoulli and higher-order poly-Bernoulli polynomials. Russ. J. Math. Phys. 22(1),26-33 (2015)10. Kim, DS, Kim, T, Lee, SH: A note on poly-Bernoulli polynomials arising umbral calculus. Adv. Stud. Theor. Phys. 7(15),731-744 (2013)11. Kim, T, Jang, YS, Seo, JJ: Poly-Bernoulli polynomials and their applications. Int. J. Math. Anal. 8(30), 1495-1503 (2014)12. Acikgoz, M, Erdal, D, Araci, S: A new approach to q-Bernoulli numbers and q-Bernoulli polynomials related toq-Bernstein polynomials. Adv. Diﬀer. Equ. 9, Article ID 951764 (2010)13. Araci, S, Acikgoz, M, Jolany, H: On the families of q-Euler polynomials and their applications. J. Egypt. Math. Soc. 23(1),1-5 (2015)14. Bayad, A, Hamahata, Y: Multiple polylogarithms and multi-poly-Bernoulli polynomials. Funct. Approx. Comment.Math. 46, 45-61 (2012)15. Bayad, A, Gaboury, S: Generalized Dirichlet L-function of arbitrary order with applications. Adv. Stud. Contemp. Math.(Kyungshang) 23(4), 607-619 (2013)16. Gaboury, S, Tremblay, R, Fugère, B-J: Some explicit formulas for certain new classes of Bernoulli, Euler and Genocchipolynomials. Proc. Jangjeon Math. Soc. 17(1), 115-123 (2014)Kim and Kim Advances in Diﬀerence Equations  (2015) 2015:258 Page 8 of 817. Carlitz, L: A degenerate Staudt-Clausen theorem. Arch. Math. (Basel) 7, 28-33 (1956)18. Dere, R, Simsek, Y: Applications of umbral algebra to some special polynomials. Adv. Stud. Contemp. Math.(Kyungshang) 22(3), 433-438 (2012)19. Ding, D, Yang, J: Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials. Adv. Stud. Contemp.Math. (Kyungshang) 20(1), 7-21 (2010)20. Kim, T, Kwon, HK, Lee, SH, Seo, JJ: A note on poly-Bernoulli numbers and polynomials of the second kind. Adv. Diﬀer.Equ. 2014, 219 (2014)21. Kim, T: Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adicintegral on Zp . Russ. J. Math. Phys. 16(4), 484-491 (2009)22. Luo, Q-M, Guo, B-N, Qi, F: On evaluation of Riemann zeta function ζ (s). Adv. Stud. Contemp. Math. (Kyungshang) 7(2),135-144 (2003)23. Luo, Q-M, Qi, F: Relationships between generalized Bernoulli numbers and polynomials and generalized Eulernumbers and polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 7(1), 11-18 (2003)24. Park, J-W, Rim, S-H: On the modiﬁed q-Bernoulli polynomials with weight. Proc. Jangjeon Math. Soc. 17(2), 231-236(2014)25. S¸en, E: Theorems on Apostol-Euler polynomials of higher order arising from Euler basis. Adv. Stud. Contemp. Math.(Kyungshang) 23(2), 337-345 (2013)26. Srivastava, HM, Kim, T, Simsek, Y: q-Bernoulli numbers and polynomials associated with multiple q-zeta functions andbasic L-series. Russ. J. Math. Phys. 12(2), 241-268 (2005)27. Zhang, Z, Yang, J: On sums of products of the degenerate Bernoulli numbers. Integral Transforms Spec. Funct.20(9-10), 751-755 (2009)

A note on degenerate poly-Bernoulli numbers and polynomials

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