Abstract
In this paper, we consider the degenerate poly-Bernoulli polynomials. We present several explicit formulas and recurrence relations for these polynomials. Also, we establish a connection between our polynomials and several known families of polynomials.
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1 Introduction
The degenerate Bernoulli polynomials \(\beta_{n}(\lambda,x)\) (\(\lambda\neq0\)) were introduced by Carlitz [1] and rediscovered by Ustinov [2] under the name Korobov polynomials of the second kind. They are given by the generating function
When \(x=0\), \(\beta_{n}(\lambda)=\beta_{n}(\lambda,0)\) are called the degenerate Bernoulli numbers (see [3]). We observe that \(\lim_{\lambda\rightarrow0}\beta_{n}(\lambda,x)=B_{n}(x)\), where \(B_{n}(x)\) is the nth ordinary Bernoulli polynomial (see the references).
The poly-Bernoulli polynomials \(PB_{n}^{(k)}(x)\) are defined by
where \(Li_{k}(x)\) (\(k\in\mathbb{Z}\)) is the classical polylogarithm function given by \(Li_{k}(x)=\sum_{n\geq1}\frac{x^{n}}{n^{k}}\) (see [4–6]).
For \(0\neq\lambda\in\mathbb{C}\) and \(k\in\mathbb{Z}\), the degenerate poly-Bernoulli polynomials \(P\beta_{n}^{(k)}(\lambda,x)\) are defined by Kim and Kim to be
When \(x=0\), \(P\beta_{n}^{(k)}(\lambda)=P\beta_{n}^{(k)}(\lambda,0)\) are called degenerate poly-Bernoulli numbers. We observe that \(\lim_{\lambda\rightarrow0}P\beta_{n}^{(k)}(\lambda,x)=PB_{n}^{(k)}(x)\).
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 [7, 8]. We denote the algebra of polynomials in a single variable x over \(\mathbb {C}\) by Π and the vector space of all linear functionals on Π by \(\Pi^{*}\). The action of a linear functional L on a polynomial \(p(x)\) is denoted by \(\langle L\mid p(x)\rangle\). We define the vector space structure on \(\Pi^{*}\) by \(\langle cL+c'L'\mid p(x)\rangle=c\langle L\mid p(x)\rangle+c'\langle L'\mid p(x)\rangle\), where \(c,c'\in\mathbb{C}\). We define the algebra of formal power series in a single variable t to be
A power series \(f(t)\in\mathcal{H}\) defines a linear functional on Π by setting
where \(\delta_{n,k}\) is the Kronecker symbol. Let \(f_{L}(t)=\sum_{n\geq0}\langle L\mid x^{n}\rangle\frac{t^{n}}{n!}\). From (1.4), we have \(\langle f_{L}(t)\mid x^{n}\rangle=\langle L\mid x^{n}\rangle\). So, the map \(L\mapsto f_{L}(t)\) is a vector space isomorphism from \(\Pi ^{*}\) onto \(\mathcal{H}\). Thus, \(\mathcal{H}\) is thought of as set of both formal power series and linear functionals. We call \(\mathcal{H}\) the umbral algebra. The umbral calculus is the study of umbral algebra.
The order \(O(f(t))\) of the non-zero power series \(f(t)\in\mathcal {H}\) is the smallest integer k for which the coefficient of \(t^{k}\) does not vanish. Suppose that \(f(t),g(t)\in\mathcal{H}\) such that \(O(f(t))=1\) and \(O(g(t))=0\), then there exists a unique sequence \(s_{n}(x)\) of polynomials such that
where \(n,k\geq0\). The sequence \(s_{n}(x)\) is called the Sheffer sequence for \((g(t),f(t))\), which is denoted by \(s_{n}(x)\sim(g(t),f(t))\) (see [7, 8]). For \(f(t)\in\mathcal{H}\) and \(p(x)\in\Pi\), we have \(\langle e^{yt}\mid p(x)\rangle=p(y)\), \(\langle f(t)g(t)\mid p(x)\rangle =\langle g(t)\mid f(t)p(x)\rangle\), and
(see [7, 8]). From (1.6), we obtain \(\langle t^{k}\mid p(x)\rangle=p^{(k)}(0)\) and \(\langle1\mid p^{(k)}(x)\rangle =p^{(k)}(0)\), where \(p^{(k)}(0)\) denotes the kth derivative of \(p(x)\) with respect to x at \(x=0\). So, we get \(t^{k}p(x)=p^{(k)}(x)=\frac{d^{k}}{dx^{k}}p(x)\), for all \(k\geq0\). Let \(s_{n}(x)\sim(g(t),f(t))\), then we have
for all \(y\in\mathbb{C}\), where \(\bar{f}(t)\) is the compositional inverse of \(f(t)\) (see [7, 8]). For \(s_{n}(x)\sim(g(t),f(t))\) and \(r_{n}(x)\sim(h(t),\ell(t))\), let \(s_{n}(x)=\sum_{k=0}^{n} c_{n,k}r_{k}(x)\), then we have
From (1.1), we see that \(P\beta _{n}^{(k)}(\lambda,x)\) is the Sheffer sequence for the pair
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 polynomials, namely \(P\beta _{n}^{(k)}(\lambda,x)\). To do so, we recall that Stirling numbers \(S_{1}(n,k)\) of the first kind can be defined by means of exponential generating functions as \(\sum_{\ell\geq j}S_{1}(\ell,j)\frac{t^{\ell}}{\ell!}=\frac{1}{j!}\log^{j}(1+t)\) and can be defined by means of ordinary generating functions as
where \((x)_{n}=x(x-1)(x-2)\cdots(x-n+1)\) with \((x)_{0}=1\). For \(\lambda\neq 0\), we define \((x\mid \lambda)_{n}=\lambda^{n}(x/\lambda)_{n}\). Sometimes, for simplicity, we denote the function \(\frac{e^{t}-1}{Li_{k}(1-e^{-\frac {1}{\lambda}(e^{\lambda t}-1)})}\) by \(G_{k}(t)\).
First, we express the degenerate poly-Bernoulli polynomials in terms of degenerate poly-Bernoulli numbers.
Theorem 2.1
For all \(n\geq0\),
Proof
By (1.5), for \(s_{n}(x)\sim(g(t),f(t))\) we have \(s_{n}(x)=\sum_{j=0}^{n}\frac{1}{j!}\langle g(\bar{f}(t))^{-1}\bar{f}(t)^{j}\mid x^{n}\rangle x^{j}\). Thus, in the case of degenerate poly-Bernoulli polynomials (see (1.9)), we have
which completes the proof. □
Note that Stirling numbers \(S_{2}(n,k)\) of the second kind can be defined by the exponential generating functions as
Theorem 2.2
For all \(n\geq0\),
Proof
By (2.1), we have \((x\mid\lambda)_{n}=\sum_{m=0}^{n}S_{1}(n,m)\lambda^{n-m}x^{m}\sim(1,\frac {1}{\lambda}(e^{\lambda t}-1))\), and by (1.9), we have
which implies \(G_{k}(t)P\beta _{n}^{(k)}(\lambda,x)=\sum_{m=0}^{n}S_{1}(n,m)\lambda^{n-m}x^{m}\). Thus,
which completes the proof. □
Theorem 2.3
For all \(n\geq1\),
Proof
Note that \(x^{n}\sim(1,t)\). Thus, by (2.3) and transfer formula, we have
Therefore, \(P\beta _{n}^{(k)}(\lambda,x)=\sum_{\ell=0}^{n-1}\binom{n-1}{\ell}\lambda ^{\ell}B_{\ell}^{(n)}G_{k}(t)^{-1}x^{n-\ell}\), which, by (2.4), completes the proof. □
Theorem 2.4
For all \(n\geq0\),
Proof
By (2.3), we have
Thus, by (2.2), we obtain
which completes the proof. □
Note that the above theorem has been obtained in Theorem 2.2 in [5].
Theorem 2.5
For all \(n\geq0\),
where \(\binom{a}{b_{1},b_{2},b_{3}}=\frac{a!}{b_{1}!b_{2}!b_{3}!}\) is the multinomial coefficient.
Proof
By (2.5), we have
Note that \(\langle\frac{e^{t}-1}{t}\mid x^{n-\ell-m} \rangle =\int_{0}^{1} u^{n-\ell-m}\,du=\frac{1}{n-\ell-m+1}\). Thus,
which completes the proof. □
Note that \(Li_{2}(1-e^{-t})=\int_{0}^{t}\frac{y}{e^{y}-1}\,dy=\sum_{j\geq0}B_{j}\frac {1}{j!}\int_{0}^{t}y^{j}\,dy=\sum_{j\geq0}\frac{B_{j} t^{j+1}}{j!(j+1)}\). For general \(k\geq2\), the function \(Li_{k}(1-e^{-t})\) has the integral representation
which, by induction on k, implies
Theorem 2.6
For all \(n\geq0\) and \(k\geq2\),
Proof
By (2.5), we have
Thus, by (2.6), we obtain
which completes the proof. □
Note that here we compute \(A= \langle Li_{k}(1-e^{-t})\mid x^{n+1} \rangle\) in several different ways. As for the first way, we have
As for the second way, we have
As for the third way, by (2.6), we have
Hence, we can state the following result.
Theorem 2.7
For all \(n\geq0\),
3 Recurrences
In this section, we present several recurrences for the degenerate poly-Bernoulli polynomials, namely \(P\beta _{n}^{(k)}(\lambda,x)\). Note that, by (1.9) and the fact that \((x\mid\lambda)_{n}\sim (1,\frac{e^{\lambda t}-1}{\lambda})\), we obtain the following identity.
Proposition 3.1
For all \(n\geq0\), \(P\beta _{n}^{(k)}(\lambda,x+y)=\sum_{j=0}^{n}\binom{n}{j}P\beta _{j}^{(k)}(\lambda ,x)(y\mid\lambda)_{n-j}\).
It is well known that if \(s_{n}(x)\sim(g(t),f(t))\), then we have \(f(t)s_{n}(x)=ns_{n-1}(x)\). Thus, by (1.9), we obtain \(\frac{e^{\lambda t}-1}{\lambda} P\beta _{n}^{(k)}(\lambda,x)=nP\beta _{n-1}^{(k)}(\lambda,x)\), which implies the following result.
Proposition 3.2
For all \(n\geq0\), \(P\beta _{n}^{(k)}(\lambda,x+\lambda)=P\beta _{n}^{(k)}(\lambda,x)+n\lambda P\beta _{n-1}^{(k)}(\lambda,x)\).
Theorem 3.3
For all \(n\geq0\),
Proof
By applying the fact that \(s_{n+1}(x)=(x-\frac{g'(t)}{g(t)})\frac {1}{f'(t)} s_{n}(x)\) for all \(s_{n}(x)\sim(g(t),f(t))\) and (1.9), we obtain
where
Thus, the expression \(A=e^{-\lambda t}\frac{g'(t)}{g(t)}P\beta _{n}^{(k)}(\lambda,x)\) is given by
Note that, by (1.9), we have \(G_{k}(x)P\beta _{n}^{(k)}(\lambda,x)=\sum_{m=0}^{n}S_{1}(n,m)\lambda^{n-m}x^{m}\). Therefore,
We remark that the expression in the parentheses in (3.1) has order at least one. Now, let us simplify (3.1):
and
Hence, by (3.1)-(3.3), we complete the proof. □
In the next result we express \(\frac{d}{dx}P\beta _{n}^{(k)}(\lambda,x)\) in terms of \(P\beta _{n}^{(k)}(\lambda,x)\).
Proposition 3.4
For all \(n\geq0\), \(\frac{d}{dx}P\beta _{n}^{(k)}(\lambda,x)= n!\sum_{\ell=0}^{n-1}\frac{(-\lambda)^{n-\ell-1}}{\ell!(n-\ell)} P\beta _{\ell}^{(k)}(\lambda,x)\).
Proof
Note that \(\frac{d}{dx}s_{n}(x)=\sum_{\ell=0}^{n-1}\binom{n}{\ell}\langle \bar{f}(t)\mid x^{n-\ell}\rangle s_{\ell}(x)\) for all \(s_{n}(x)\sim (g(t),f(t))\). Thus, by (1.9), we have
which completes the proof. □
Theorem 3.5
For all \(n\geq1\),
Proof
By (1.9), we have
The term in (3.4) is given by
For the term in (3.5), we observe that \(\frac{d}{dt}\frac {Li_{k}(1-e^{-t})}{(1+\lambda t)^{1/\lambda}-1}=\frac{1}{t}(A-B)\), where
Note that the expression \(A-B\) has order of at least 1. Now, we are ready to compute the term in (3.5). By (1.9), we have
Thus, if we replace (3.4) by (3.6) and (3.5) by (3.7), we obtain
as claimed. □
4 Connections with families of polynomials
In this section, we present a few examples on the connections with families of polynomials. We start with the connection to Bernoulli polynomials \(B_{n}^{(s)}(x)\) of order s. Recall that the Bernoulli polynomials \(B_{n}^{(s)}(x)\) of order s are defined by the generating function \((\frac{t}{e^{t}-1} )^{s} e^{xt}=\sum_{n\geq0}B_{n}^{(s)}(x)\frac{t^{n}}{n!}\), equivalently,
(see [11–13]). In the next result, we express our polynomials \(P\beta _{n}^{(k)}(\lambda,x)\) in terms of Bernoulli polynomials of order s. To do that, we recall that the Bernoulli numbers \(b_{n}^{(s)}\) of the second kind of order s are defined as
Theorem 4.1
For all \(n\geq0\),
where \(c_{n,m}(\ell,r,j,i)=S_{1}(\ell ,m)S_{1}(j+s,j-i+s)S_{2}(j-i+s,s)b_{r}^{(s)}P\beta_{n-\ell-r-j}^{(k)}(\lambda)\) and \(\binom{a}{b_{1},\ldots,b_{m}}=\frac{a!}{b_{1}!\cdots b_{m}!}\) is the multinomial coefficient.
Proof
Let \(h_{s}(t)= (\frac{(1+\lambda t)^{1/\lambda}-1}{t} )^{s}\) and \(P\beta _{n}^{(k)}(\lambda,x)=\sum_{m=0}^{n}c_{n,m}B_{m}^{(s)}(x)\). By (1.8), (1.9), and (4.1), we have
which, by (4.2), implies
One can show that
Thus, by (1.9), we have
as required. □
Similar techniques as in the proof of the previous theorem, we can express our polynomials \(P\beta _{n}^{(k)}(\lambda,x)\) in terms of other families. Below we present three examples, where we leave the proofs to the interested reader.
The first example is to express our polynomials \(P\beta _{n}^{(k)}(\lambda ,x)\) in terms of Frobenius-Euler polynomials. Note that the Frobenius-Euler polynomials \(H_{n}^{(s)}(x\mid\mu)\) of order s are defined by the generating function \((\frac{1-\mu}{e^{t}-\mu} )^{s} e^{xt}=\sum_{n\geq 0}H_{n}^{(s)}(x\mid\mu)\frac{t^{n}}{n!}\) (\(\mu\neq1\)), or equivalently, \(H_{n}^{(s)}(x\mid\mu)\sim ( (\frac{e^{t}-\mu}{1-\mu} )^{s},t )\) (see [10, 14]).
Theorem 4.2
For all \(n\geq0\),
where \(c_{n,m}(\ell,r,i)=S_{1}(\ell,m)(i\mid\lambda)_{n-\ell-r}P\beta _{r}^{(k)}(\lambda)\).
If we express our polynomials \(P\beta _{n}^{(k)}(\lambda,x)\) in terms of falling polynomials \((x\mid\lambda)_{n}\), then we get the following result.
Theorem 4.3
For all \(n\geq0\), \(P\beta _{n}^{(k)}(\lambda,x)=\sum_{m=0}^{n}\binom{n}{m}P\beta _{n-m}^{(k)}(\lambda)(x\mid\lambda)_{m}\).
Our last example is to express our polynomials \(P\beta _{n}^{(k)}(\lambda,x)\) in terms of degenerate Bernoulli polynomials \(\beta _{n}^{(s)}(\lambda,x)\) of order s. Note that the degenerate Bernoulli polynomials \(\beta_{n}^{(s)}(\lambda,x)\) of order s are given by
Theorem 4.4
For all \(n\geq0\),
where \(c_{n,m}(j,i)=S_{1}(j+s,j-i+s)S_{2}(j-i+s,s)P\beta _{n-m-j}^{(k)}(\lambda)\).
References
Carlitz, L: Degenerate Stirling, Bernoulli and Eulerian numbers. Util. Math. 15, 51-88 (1979)
Ustinov, AV: Korobov polynomials and umbral analysis. Chebyshevskiĭ Sb. 4, 137-152 (2003) (in Russian)
Kim, DS, Kim, T, Dolgy, DV: A note on degenerate Bernoulli numbers and polynomials associated with p-adic invariant integral on \(\Bbb{Z}_{p}\). Appl. Math. Comput. 295, 198-204 (2015)
Jolany, H, Faramarzi, H: Generalization on poly-Eulerian numbers and polynomials. Sci. Magna 6, 9-18 (2010)
Kim, DS, Kim, T: A note on degenerate poly-Bernoulli numbers and polynomials (2015). arXiv:1503.08418
Kim, DS, Kim, T: A note on poly-Bernoulli and higher-order poly-Bernoulli polynomials. Russ. J. Math. Phys. 22, 26-33 (2015)
Roman, S: More on the umbral calculus, with emphasis on the q-umbral calculus. J. Math. Anal. Appl. 107, 222-254 (1985)
Roman, S: The Umbral Calculus. Dover, New York (2005)
Kim, DS, Kim, T: q-Bernoulli polynomials and q-umbral calculus. Sci. China Math. 57, 1867-1874 (2014)
Kim, T: Identities involving Laguerre polynomials derived from umbral calculus. Russ. J. Math. Phys. 21, 36-45 (2014)
Bayad, A, Kim, T: Identities involving values of Bernstein, q-Bernoulli, and q-Euler polynomials. Russ. J. Math. Phys. 18(2), 133-143 (2011)
Ding, D, Yang, J: Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials. Adv. Stud. Contemp. Math. 20(1), 7-21 (2010)
Kim, T: A note on q-Bernstein polynomials. Russ. J. Math. Phys. 18, 73-82 (2011)
Araci, S, Acikgoz, M: A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials. Adv. Stud. Contemp. Math. 22(3), 399-406 (2012)
Acknowledgements
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MOE) (No. 2012R1A1A2003786).
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Kim, D.S., Kim, T., Kwon, H.I. et al. Degenerate poly-Bernoulli polynomials with umbral calculus viewpoint. J Inequal Appl 2015, 228 (2015). https://doi.org/10.1186/s13660-015-0748-7
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DOI: https://doi.org/10.1186/s13660-015-0748-7