1 Introduction
Throughout the paper assume that \(n,k\in\mathbb{Z}\) and \(0\neq q\in \mathbb{R}\). The poly-Cauchy polynomials with a
q
parameter of the first kind \(C_{n,q}^{(k)}(x)\) and of the second kind \(\widehat {C}_{n,q}^{(k)}(x)\) are, respectively, defined by
for all \(k\in\mathbb{Z}\), where
is the polylogarithm factorial function; see [1]. When \(x=0\), \(C_{n,q}^{(k)}=C_{n,q}^{(k)}(0)\), and \(\widehat {C}_{n,q}^{(k)}=\widehat{C}_{n,q}^{(k)}(0)\) are, respectively, called the poly-Cauchy numbers with a
q
parameter of the first kind and of the second kind. Note that \(\operatorname{Lif}_{1}(x)=\frac{e^{x}-1}{x}\).
$$\begin{aligned} &\operatorname{Lif}_{k}\bigl(\log(1+qt)/q\bigr) (1+qt)^{x/q}= \sum_{n\geq0}C_{n,q}^{(k)}(x) \frac{t^{n}}{n!},\\ &\operatorname{Lif}_{k}\bigl(-\log(1+qt)/q\bigr) (1+qt)^{-x/q}= \sum_{n\geq0}\widehat {C}_{n,q}^{(k)}(x) \frac{t^{n}}{n!}, \end{aligned}$$
$$\begin{aligned} \operatorname{Lif}_{k}(x)=\sum_{m\geq0} \frac{x^{m}}{m!(m+1)^{k}} \end{aligned}$$
(1.1)
Here the degenerate versions are introduced for the poly-Cauchy polynomials with a q parameter.
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Definition 1.1
The degenerate poly-Cauchy polynomials with a
q
parameter of the first kind
\(C_{n,q}^{(k)}(\lambda,x)\)
and of the second kind
\(\widehat {C}_{n,q}^{(k)}(\lambda,x)\) are, respectively, given by
$$\begin{aligned} &\operatorname{Lif}_{k} \biggl(\frac{(1+qt)^{\lambda}-1}{q\lambda} \biggr) (1+qt)^{\frac {x}{q}}=\sum_{n\geq0}C_{n,q}^{(k)}( \lambda,x)\frac{t^{n}}{n!}, \end{aligned}$$
(1.2)
$$\begin{aligned} &\operatorname{Lif}_{k} \biggl(-\frac{(1+qt)^{\lambda}-1}{q\lambda} \biggr) (1+t)^{-\frac {x}{q}}=\sum_{n\geq0}\widehat{C}_{n,q}^{(k)}( \lambda,x)\frac {t^{n}}{n!}. \end{aligned}$$
(1.3)
For \(q=1\), \(C_{n,1}^{(k)}(\lambda,x)=C_{n}^{(k)}(\lambda,x)\) and \(\widehat{C}_{n,1}^{(k)}(\lambda,x)=\widehat{C}_{n}^{(k)}(\lambda,x)\) are the degenerate poly-Cauchy polynomials of the first kind and of the second kind, respectively, which are studied in [2]. When \(x=0\), \(C_{n,q}^{(k)}(\lambda,0)\) and \(\widehat {C}_{n,q}^{(k)}(\lambda,0)\) are, respectively, called the degenerate poly-Cauchy numbers with a
q
parameter of the first kind and of the second kind.
In [3, 4], Carlitz introduced certain degenerate versions of Bernoulli and Euler polynomials. Almost half a century later these Carlitz degenerate Bernoulli polynomials were rediscovered under the name of Korobov polynomials of the second kind by Ustinov [5], while the degenerate version of the Bernoulli polynomials of the second kind were named the Korobov polynomials [6, 7]. It is remarkable that in recent years various degenerate versions of many important polynomials regained the attention of some researchers and many interesting results of them were obtained [2, 8‐13]. Thus these have become an active area of research.
As was shown in the paper of Carlitz [3, 4], these degenerate versions have potential importance in number theory and combinatorics. For example, the authors have made some progress about symmetric identities involving the higher-order degenerate Euler and q-Euler polynomials by using the fermionic p-adic integrals. In a forthcoming paper, an investigation will be carried out as to some further results about the degenerate poly-Cauchy polynomials with a q parameter which are of arithmetic and combinatorial nature.
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The aim of this paper is to use umbral calculus techniques (see [14, 15]) in order to derive some properties, recurrence relations, and identities for the degenerate poly-Cauchy polynomials with a q parameter of the first kind and of the second kind.
From (1.2) and (1.3), one can see that \(C_{n,q}^{(k)}(\lambda,x)\) is the Sheffer sequence for the pair \(g(t)=\frac{1}{\operatorname{Lif}_{k} (\frac{e^{q\lambda t}-1}{q\lambda} )}\), \(f(t)=\frac{e^{qt}-1}{q}\), and that \(\widehat{C}_{n,q}^{(k)}(\lambda ,x)\) is the Sheffer sequence for the pair \(g(t)=\frac{1}{\operatorname{Lif}_{k} (-\frac{e^{-q\lambda t}-1}{q\lambda} )}\), \(f(t)=\frac {e^{-qt}-1}{q}\). Thus,
$$\begin{aligned} C_{n,q}^{(k)}(\lambda,x)\sim \biggl(\frac{1}{\operatorname{Lif}_{k} (\frac{e^{q\lambda t}-1}{q\lambda} )}, \frac{e^{qt}-1}{q} \biggr),\qquad \widehat {C}_{n,q}^{(k)}( \lambda,x)\sim \biggl(\frac{1}{\operatorname{Lif}_{k} (-\frac {e^{-q\lambda t}-1}{q\lambda} )},\frac{e^{-qt}-1}{q} \biggr). \end{aligned}$$
(1.4)
2 Explicit expressions
Let us start by presenting several explicit formulas for the degenerate poly-Cauchy polynomials with a q parameter, namely \(C_{n,q}^{(k)}(\lambda,x)\) and \(\widehat{C}_{n,q}^{(k)}(\lambda,x)\). To do so, recall here that Stirling numbers \(S_{1}(n,k)\) of the first kind can be defined by means of exponential generating functions as
the Stirling numbers \(S_{2}(n,k)\) of the second kind can be defined by the exponential generating functions as
and can be defined by means of ordinary generating functions as
where \((x)_{n}=x(x-1)(x-2)\cdots(x-n+1)\) with \((x)_{0}=1\).
$$\begin{aligned} \sum_{\ell\geq j}S_{1}(\ell,j) \frac{t^{\ell}}{\ell}=\frac{1}{j!}\log ^{j}(1+t), \end{aligned}$$
(2.1)
$$\begin{aligned} \sum_{n\geq k}S_{2}(n,k)\frac{x^{n}}{n!}= \frac{(e^{t}-1)^{k}}{k!}, \end{aligned}$$
(2.2)
$$\begin{aligned} (x|q)_{n}=q^{n}(x/q)_{n}=\sum _{m=0}^{n}S_{1}(n,m)q^{n-m}x^{m} \sim\biggl(1,\frac{e^{qt}-1}{q}\biggr), \end{aligned}$$
(2.3)
Theorem 2.1
For all
\(n\geq0\),
$$\begin{aligned} &C_{n,q}^{(k)}(\lambda,x) =\sum_{j=0}^{n} \Biggl(\sum_{\ell=j}^{n}\sum _{m=0}^{\ell-j}\frac{\binom{\ell }{j}}{(m+1)^{k}}S_{1}(n, \ell)S_{2}(\ell-j,m)q^{n-m-j} \lambda^{\ell-j-m} \Biggr)x^{j}, \\ &\widehat{C}_{n,q}^{(k)}(\lambda,x) =\sum _{j=0}^{n} \Biggl(\sum_{\ell=j}^{n} \sum_{m=0}^{\ell-j}(-1)^{m-j} \frac {\binom{\ell}{j}}{(m+1)^{k}}S_{1}(n,\ell)S_{2}(\ell-j,m)q^{n-m-j} \lambda^{\ell-j-m} \Biggr)x^{j}. \end{aligned}$$
Proof
By (1.4), one can see that
Thus, by (2.3) and (2.2), one obtains
which completes the proof of the first formula.
$$\begin{aligned} \frac{1}{\operatorname{Lif}_{k} (\frac{e^{q\lambda t}-1}{q\lambda} )}C_{n,q}^{(k)}(\lambda,x) \sim\biggl(1, \frac{e^{qt}-1}{q}\biggr). \end{aligned}$$
(2.4)
$$\begin{aligned} C_{n,q}^{(k)}(\lambda,x)&=\operatorname{Lif}_{k} \biggl(\frac{e^{q\lambda t}-1}{q\lambda } \biggr) (x|q)_{n}=\sum _{m=0}^{n}S_{1}(n,m)q^{n-m} \operatorname{Lif}_{k} \biggl(\frac{e^{q\lambda t}-1}{q\lambda} \biggr)x^{m} \\ &=\sum_{m=0}^{n}\sum _{\ell=0}^{m}S_{1}(n,m)q^{n-m} \frac{(e^{q\lambda t}-1)^{\ell}}{\ell!(\ell+1)^{k}\lambda^{\ell}q^{\ell}}x^{m} \\ &=\sum_{m=0}^{n}\sum _{\ell=0}^{m}\sum_{j=\ell}^{m}S_{1}(n,m)S_{2}(j, \ell )q^{n-m}\frac{\lambda^{j}q^{j}}{j!(\ell+1)^{k}\lambda^{\ell}q^{\ell}}t^{j}x^{m} \\ &=\sum_{m=0}^{n}\sum _{\ell=0}^{m}\sum_{j=\ell}^{m} \binom {m}{j}S_{1}(n,m)S_{2}(j,\ell)q^{n-m} \frac{\lambda^{j}q^{j}}{(\ell+1)^{k}\lambda ^{\ell}q^{\ell}}x^{m-j} \\ &=\sum_{\ell=0}^{n}\sum _{m=0}^{\ell}\sum_{j=0}^{\ell-m} \binom{\ell }{j}S_{1}(n,\ell)S_{2}(\ell-j,m)q^{n-m-j} \lambda^{\ell-j-m}\frac{x^{j}}{(m+1)^{k}} \\ &=\sum_{j=0}^{n} \Biggl(\sum _{\ell=j}^{n}\sum_{m=0}^{\ell-j} \frac{\binom{\ell }{j}}{(m+1)^{k}}S_{1}(n,\ell)S_{2}(\ell-j,m)q^{n-m-j} \lambda^{\ell-j-m} \Biggr)x^{j}, \end{aligned}$$
The second formula follows by similar arguments from the facts that
and \((-x|q)_{n}=\sum_{m=0}^{n}(-1)^{m}S_{1}(n,m)q^{n-m}x^{m}\sim(1,\frac{e^{-qt}-1}{q})\). □
$$\begin{aligned} \frac{1}{\operatorname{Lif}_{k} (-\frac{e^{-q\lambda t}-1}{q\lambda} )} \widehat {C}_{n,q}^{(k)}(\lambda,x)\sim \biggl(1,\frac{e^{-qt}-1}{q}\biggr) \end{aligned}$$
(2.5)
Theorem 2.2
For all
\(n\geq0\),
$$\begin{aligned} &C_{n,q}^{(k)}(\lambda,x) =\sum_{j=0}^{n} \Biggl(\sum_{\ell=j}^{n}\sum _{m=0}^{\ell-j}\binom{\ell }{j}S_{1}(n, \ell)S_{2}(\ell-j,m)q^{n-m-j} C_{m,q}^{(k)}( \lambda,0) \Biggr)x^{j}, \\ &\widehat{C}_{n,q}^{(k)}(\lambda,x) =\sum _{j=0}^{n} \Biggl(\sum_{\ell=j}^{n} \sum_{m=0}^{\ell-j}(-1)^{j} \binom{\ell }{j}S_{1}(n,\ell)S_{2}(\ell-j,m)q^{n-m-j} \widehat{C}_{m,q}^{(k)}(\lambda,0) \Biggr)x^{j}. \end{aligned}$$
Proof
By (2.4) and (2.3), one has \(C_{n,q}^{(k)}(\lambda,x)=\sum_{\ell=0}^{n}S_{1} (n,\ell)q^{n-\ell}\operatorname{Lif}_{k} (\frac{e^{q\lambda t}-1}{q\lambda} )x^{\ell}\). By (1.1), one obtains
Thus, by (2.2), one gets
which completes the proof of the first formula.
$$\begin{aligned} C_{n,q}^{(k)}(\lambda,x) &=\sum_{\ell=0}^{n}S_{1}(n, \ell)q^{n-\ell}\operatorname{Lif}_{k} \biggl(\frac{(1+qs)^{\lambda}-1}{q\lambda} \biggr)\bigg|_{s=\frac{e^{qt}-1}{q}}x^{\ell}\\ &=\sum_{\ell=0}^{n}\sum _{m=0}^{\ell}S_{1}(n,\ell)q^{n-\ell }C_{m,q}^{(k)}( \lambda,0)\frac{ (\frac{e^{qt}-1}{q} )^{m}}{m!}x^{\ell}. \end{aligned}$$
$$\begin{aligned} C_{n,q}^{(k)}(\lambda,x) &=\sum_{\ell=0}^{n} \sum_{m=0}^{\ell}\sum _{j=m}^{\ell}S_{1}(n,\ell )S_{2}(j,m)q^{n-\ell}C_{m,q}^{(k)}(\lambda,0) \frac{q^{j-m}}{j!}t^{j}x^{\ell}\\ &=\sum_{\ell=0}^{n}\sum _{m=0}^{\ell}\sum_{j=m}^{\ell}\binom{\ell}{j} S_{1}(n,\ell)S_{2}(j,m)q^{n-\ell+j-m}C_{m,q}^{(k)}( \lambda,0)x^{\ell-j} \\ &=\sum_{\ell=0}^{n}\sum _{m=0}^{\ell}\sum_{j=0}^{\ell-m} \binom{\ell}{j} S_{1}(n,\ell)S_{2}(\ell-j,m)q^{n-j-m}C_{m,q}^{(k)}( \lambda,0)x^{j} \\ &=\sum_{j=0}^{n} \Biggl(\sum _{\ell=j}^{n}\sum_{m=0}^{\ell-j} \binom{\ell}{j} S_{1}(n,\ell)S_{2}(\ell-j,m)q^{n-m-j}C_{m,q}^{(k)}( \lambda,0) \Biggr)x^{j}, \end{aligned}$$
For the second formula, one uses (2.5) to obtain
Along the lines of the proof of the first formula, one derives
as required. □
$$\widehat{C}_{n,q}^{(k)}(\lambda,x)=\sum _{\ell=0}^{n}(-1)^{\ell}S_{1}(n,\ell )q^{n-\ell}\operatorname{Lif}_{k} \biggl(-\frac{e^{-q\lambda t}-1}{q\lambda} \biggr)x^{\ell}. $$
$$\begin{aligned} \widehat{C}_{n,q}^{(k)}(\lambda,x) &=\sum _{\ell=0}^{n}\sum_{m=0}^{\ell}\sum_{j=m}^{\ell}(-1)^{\ell+j} S_{1}(n,\ell )S_{2}(j,m)q^{n-\ell} \widehat{C}_{m,q}^{(k)}(\lambda,0)\frac {q^{j-m}}{j!}t^{j}x^{\ell}\\ &=\sum_{\ell=0}^{n}\sum _{m=0}^{\ell}\sum_{j=m}^{\ell}(-1)^{\ell+j} \binom{\ell }{j} S_{1}(n,\ell)S_{2}(j,m)q^{n-\ell+j-m} \widehat{C}_{m,q}^{(k)}(\lambda ,0)x^{\ell-j} \\ &=\sum_{\ell=0}^{n}\sum _{m=0}^{\ell}\sum_{j=0}^{\ell-m}(-1)^{j} \binom{\ell }{j} S_{1}(n,\ell)S_{2}(\ell-j,m)q^{n-j-m} \widehat{C}_{m,q}^{(k)}(\lambda ,0)x^{j} \\ &=\sum_{j=0}^{n} \Biggl(\sum _{\ell=j}^{n}\sum_{m=0}^{\ell-j}(-1)^{j} \binom{\ell }{j} S_{1}(n,\ell)S_{2}(\ell-j,m)q^{n-m-j} \widehat{C}_{m,q}^{(k)}(\lambda ,0) \Biggr)x^{j}, \end{aligned}$$
Next, the transfer formula will be invoked. To do this, one observes that for any power series \(g(t)=\sum_{m\geq0}b_{m}\frac{t^{m}}{m!}\), \(n\geq0\), \(a\neq0\), and \(p(x)=g(t)x^{n}\), \(g(at)x^{n}=a^{n}p(x/a)\). Recall that the Bernoulli polynomials
\(B_{n}^{(s)}(x)\)
of order
s (see [29, 30]) are defined by the generating function \((\frac{t}{e^{t}-1} )^{s} e^{xt}=\sum_{n\geq 0}B_{n}^{(s)}(x)\frac{t^{n}}{n!}\), or equivalently,
$$\begin{aligned} B_{n}^{(s)}(x)\sim \biggl( \biggl(\frac{e^{t}-1}{t} \biggr)^{s},t \biggr). \end{aligned}$$
(2.6)
Theorem 2.3
For all
\(n\geq1\),
$$\begin{aligned} &C_{n,q}^{(k)}(\lambda,x) =\sum_{j=0}^{n} \Biggl(\sum_{\ell=0}^{n-j}\sum _{m=0}^{n-j-\ell}\binom {n-1}{\ell}\binom{n-\ell}{j}S_{2}(n- \ell-j,m) \lambda^{n-\ell-j-m}q^{n-j-m}\frac{B_{\ell}^{(n)}}{(m+1)^{k}} \Biggr)x^{j}, \\ &\widehat{C}_{n,q}^{(k)}(\lambda,x) =\sum _{j=0}^{n} \Biggl(\sum_{\ell=0}^{n-j} \sum_{m=0}^{n-j-\ell }(-1)^{m-j} \binom{n-1}{\ell}\binom{n-\ell}{j}S_{2}(n-\ell-j,m)\\ &\hphantom{\widehat{C}_{n,q}^{(k)}(\lambda,x) =}{}\times\lambda^{n-\ell-j-m}q^{n-j-m}\frac{B_{\ell}^{(n)}}{(m+1)^{k}} \Biggr)x^{j}. \end{aligned}$$
Proof
By (2.4) and the fact that \(x^{n}\sim(1,t)\), one obtains
By (2.6), one gets
Thus, by (1.1) and (2.2), one has
which completes the proof of the first formula.
$$\frac{1}{\operatorname{Lif}_{k} (\frac{e^{q\lambda t}-1}{q\lambda} )}C_{n,q}^{(k)}(\lambda,x) =x \biggl( \frac{qt}{e^{qt}-1} \biggr)^{n}x^{-1}x^{n}= x \biggl( \frac{qt}{e^{qt}-1} \biggr)^{n}x^{n-1}. $$
$$\frac{1}{\operatorname{Lif}_{k} (\frac{e^{q\lambda t}-1}{q\lambda} )}C_{n,q}^{(k)}(\lambda,x) =x\sum _{\ell=0}^{n-1}B_{\ell}^{(n)} \frac{q^{\ell}}{\ell!}t^{\ell}x^{n-1} =\sum _{\ell=0}^{n-1}\binom{n-1}{\ell}B_{\ell}^{(n)}q^{\ell}x^{n-\ell}. $$
$$\begin{aligned} C_{n,q}^{(k)}(\lambda,x) ={}&\sum_{\ell=0}^{n-1} \binom{n-1}{\ell}B_{\ell}^{(n)}q^{\ell}\operatorname{Lif}_{k} \biggl(\frac{e^{q\lambda t}-1}{q\lambda} \biggr) x^{n-\ell} \\ ={}&\sum_{\ell=0}^{n-1}\sum _{m=0}^{n-\ell}\sum_{j=m}^{n-\ell} \binom {n-1}{\ell}\binom{n-\ell}{j}S_{2}(j,m)B_{\ell}^{(n)}q^{\ell}\frac{(q\lambda )^{j}}{(m+1)^{k}(q\lambda)^{m}}x^{n-\ell-j} \\ ={}&\sum_{\ell=0}^{n}\sum _{m=0}^{n-\ell}\sum_{j=0}^{n-\ell-m} \binom{n-1}{\ell}\binom{n-\ell }{j}S_{2}(n-\ell-j,m)B_{\ell}^{(n)}q^{\ell}\frac{(q\lambda)^{n-\ell -j}}{(m+1)^{k}(q\lambda)^{m}}x^{j} \\ ={}&\sum_{j=0}^{n} \Biggl(\sum _{\ell =0}^{n-j}\sum_{m=0}^{n-j-\ell} \binom{n-1}{\ell}\binom{n-\ell }{j}S_{2}(n-\ell-j,m) \\ &{}\times\lambda^{n-\ell-j-m}q^{n-j-m}\frac{B_{\ell}^{(n)}}{(m+1)^{k}} \Biggr)x^{j}, \end{aligned}$$
(2.7)
Theorem 2.4
For all
\(n\geq1\),
$$\begin{aligned} &C_{n,q}^{(k)}(\lambda,x) =\sum_{j=0}^{n} \Biggl(\sum_{\ell=0}^{n-j}\sum _{m=0}^{n-j-\ell}\binom {n-1}{\ell}\binom{n-\ell}{j}S_{2}(n- \ell-j,m) q^{n-m-j} B_{\ell}^{(n)}C_{m,q}^{(k)}( \lambda,0) \Biggr)x^{j}, \\ &\widehat{C}_{n,q}^{(k)}(\lambda,x) =\sum _{j=0}^{n} \Biggl(\sum_{\ell=0}^{n-j} \sum_{m=0}^{n-j-\ell}(-1)^{j} \binom {n-1}{\ell}\binom{n-\ell}{j}S_{2}(n-\ell-j,m) q^{n-m-j} B_{\ell}^{(n)}\widehat{C}_{m,q}^{(k)}( \lambda,0) \Biggr)x^{j}. \end{aligned}$$
Proof
By using similar arguments to the proof of Theorem 2.2 together with (2.7) (or with the analog of (2.7) in the case of \(\widehat{C}_{n,q}^{(k)}(\lambda,x)\)), one obtains
and
which completes the proof. □
$$\begin{aligned} C_{n,q}^{(k)}(\lambda,x) &=\sum_{\ell=0}^{n} \sum_{m=0}^{n-\ell}\sum _{j=0}^{n-\ell-m}\binom {n-1}{\ell}\binom{n-\ell}{j}B_{\ell}^{(n)}S_{2}(n- \ell-j,m) q^{n-m-j} C_{m,q}^{(k)}(\lambda,0)x^{j} \\ &=\sum_{j=0}^{n} \Biggl(\sum _{\ell=0}^{n-j}\sum_{m=0}^{n-j-\ell} \binom {n-1}{\ell}\binom{n-\ell}{j}S_{2}(n-\ell-j,m) q^{n-m-j} B_{\ell}^{(n)}C_{m,q}^{(k)}(\lambda,0) \Biggr)x^{j} \end{aligned}$$
$$\begin{aligned} \widehat{C}_{n,q}^{(k)}(\lambda,x) &=\sum _{\ell=0}^{n}\sum_{m=0}^{n-\ell} \sum_{j=0}^{n-\ell-m}(-1)^{j} \binom {n-1}{\ell}\binom{n-\ell}{j}B_{\ell}^{(n)}S_{2}(n- \ell-j,m) q^{n-m-j}\widehat{C}_{m,q}^{(k)}( \lambda,0)x^{j} \\ &=\sum_{j=0}^{n} \Biggl(\sum _{\ell=0}^{n-j}\sum_{m=0}^{n-j-\ell}(-1)^{j} \binom {n-1}{\ell}\binom{n-\ell}{j}S_{2}(n-\ell-j,m) q^{n-m-j} B_{\ell}^{(n)}\widehat{C}_{m,q}^{(k)}( \lambda,0) \Biggr)x^{j}, \end{aligned}$$
Before proceeding recall here that the Bernoulli polynomials
\(b_{n}(x)\) (see [31]) of the second kind are defined by
When \(x=0\), \(b_{n}=b_{n}(0)\) are called Bernoulli numbers of the second kind. With a q parameter, one has
When \(x=0\), we write \(C_{n,q}=C_{n,q}(0)\). Also, it is well known (see [32]) that, for \(k\geq1\),
By induction on k, one has
for all \(k\geq2\), and
Thus, by changing variables, one obtains
for all \(k\geq2\), and
$$\frac{t}{\log(1+t)}(1+t)^{x}=\sum_{n\geq0}C_{n}^{(1)}(x) \frac{x^{n}}{n!}=\sum_{n\geq0}b_{n}(x) \frac{x^{n}}{n!}. $$
$$\frac{q((1+qt)^{\frac{1}{q}}-1)}{\log(1+qt)} (1+t)^{x} =\sum_{n\geq0}C_{n,q}^{(1)}(x) \frac{t^{n}}{n!}=\sum_{n\geq 0}C_{n,q}(x) \frac{t^{n}}{n!}. $$
$$\begin{aligned} &\operatorname{Lif}_{k} \biggl(\frac{\log(1+qt)}{q} \biggr) \\ &\quad=\frac{q}{\log(1+qt)} \underbrace{\int_{0}^{t} \frac{q}{(1+qt)\log (1+qt)}\cdots \int_{0}^{t} \frac{q}{(1+qt)\log(1+qt)}}_{(k-1)\ \mathrm{times}} \bigl((1+qt)^{\frac{1}{q}}-1\bigr)\,dt\cdots\,dt. \end{aligned}$$
$$\operatorname{Lif}_{k} \biggl(\frac{\log(1+qt)}{q} \biggr)=\sum _{j_{1},\ldots,j_{k}\geq 0}t^{j_{1}+\cdots+j_{k}} \frac{b_{j_{k}}q^{j_{k}}}{j_{k}!}\frac{C_{j_{1},q}(-q)}{j_{1}!(j_{1}+1)} \prod_{i=2}^{k-1}\frac{b_{j_{i}}(-1)q^{j_{i}}}{j_{i}!(j_{1}+\cdots+j_{i}+1)}, $$
$$\operatorname{Lif}_{1} \biggl(\frac{\log(1+qt)}{q} \biggr)=\sum _{j_{1}\geq0}C_{j_{1},q}\frac {t^{j_{1}}}{j_{1}!}. $$
$$\begin{aligned} &\operatorname{Lif}_{k} \biggl(\frac{(1+qt)^{\lambda}-1}{q\lambda} \biggr) \\ &\quad=\sum_{j_{1},\ldots,j_{k}\geq0} \biggl(\frac{e^{\frac{(1+qt)^{\lambda}-1}{\lambda}}-1}{q} \biggr)^{j_{1}+\cdots+j_{k}} \frac{b_{j_{k}}q^{j_{k}}}{j_{k}!} \frac{C_{j_{1},q}(-q)}{j_{1}!(j_{1}+1)} \prod _{i=2}^{k-1}\frac{b_{j_{i}}(-1)q^{j_{i}}}{j_{i}!(j_{1}+\cdots +j_{i}+1)}, \end{aligned}$$
(2.8)
$$\begin{aligned} \operatorname{Lif}_{1} \biggl(\frac{(1+qt)^{\lambda}-1}{q\lambda} \biggr)=\sum _{j_{1}\geq 0}C_{j_{1},q}\frac{(e^{\frac{(1+qt)^{\lambda}-1}{\lambda }}-1)^{j_{1}}}{j_{1}!q^{j_{1}}}. \end{aligned}$$
(2.9)
Theorem 2.5
Let
\(n\geq0\). Then
for all
\(k\geq2\), and
$$\begin{aligned} C_{n,q}^{(k)}(\lambda,x)={}&\sum_{j_{1}+\cdots+j_{k}\leq n} \sum_{\ell =j_{1}+\cdots+j_{k}}^{n}\sum _{m=0}^{\ell}(-1)^{\ell-m}\frac{\binom{\ell }{m}}{\ell!\lambda^{\ell}} \frac{(j_{1}+\cdots+j_{k})!}{q^{j_{1}+\cdots+j_{k}}}\frac {c_{j_{1},q}(-q)}{j_{1}!(j_{1}+1)}\frac{b_{j_{k}}q^{j_{k}}}{j_{k}!}\\ &{}\times S_{2}(\ell,j_{1}+ \cdots+j_{k}) (x+\lambda qm|q)_{n}\prod _{i=2}^{k-1}\frac{b_{j_{i}}(-1)q^{j_{i}}}{j_{i}!(j_{1}+\cdots+j_{i}+1)}, \end{aligned}$$
$$\begin{aligned} C_{n,q}^{(1)}(\lambda,x) &=\sum_{j_{1}=0}^{n} \sum_{\ell=j_{1}}^{n}\sum _{m=0}^{\ell}(-1)^{\ell-m}\frac {\binom{\ell}{m}}{\ell!\lambda^{\ell}} \frac{C_{j_{1},q}}{q^{j_{1}}}S_{2}(\ell,j_{1}) (x+\lambda qm|q)_{n}. \end{aligned}$$
Proof
By (1.2), one has \(C_{n,q}^{(k)}(\lambda,y)= \langle \operatorname{Lif}_{k} (\frac{(1+qt)^{\lambda }-1}{q\lambda} )(1+qt)^{\frac{y}{q}}|x^{n} \rangle\). Thus by (2.8), one gets
which, by (2.2), implies
By using the fact that \(\langle(1+qt)^{\frac{y}{q}+\lambda m}|x^{n} \rangle=(y+\lambda qm|q)_{n}\), the proof is completed for the case \(k\geq2\).
$$\begin{aligned} C_{n,q}^{(k)}(\lambda,y)={}&\sum_{j_{1}+\cdots+j_{k}\leq n} \frac{c_{j_{1},q}(-q)}{q^{j_{1}+\cdots+j_{k}}j_{1}!(j_{1}+1)}\frac {b_{j_{k}}q^{j_{k}}}{j_{k}!} \\ &{}\times\prod_{i=2}^{k-1}\frac{b_{j_{i}}(-1)q^{j_{i}}}{j_{i}!(j_{1}+\cdots+j_{i}+1)} \bigl\langle \bigl(e^{\frac{(1+qt)^{\lambda}-1}{\lambda}}-1 \bigr)^{j_{1}+\cdots+j_{k}}(1+qt)^{\frac{y}{q}}|x^{n} \bigr\rangle , \end{aligned}$$
$$\begin{aligned} &C_{n,q}^{(k)}(\lambda,y) \\ &\quad=\sum_{j_{1}+\cdots+j_{k}\leq n} \frac{c_{j_{1},q}(-q)}{q^{j_{1}+\cdots+j_{k}}j_{1}!(j_{1}+1)}\frac {b_{j_{k}}q^{j_{k}}}{j_{k}!} \prod_{i=2}^{k-1}\frac {b_{j_{i}}(-1)q^{j_{i}}}{j_{i}!(j_{1}+\cdots+j_{i}+1)} \\ &\qquad{}\times(j_{1}+\cdots+j_{k})!\sum _{\ell=j_{1}+\cdots +j_{k}}^{n}\frac{S_{2}(\ell,j_{1}+\cdots+j_{k})}{\ell!\lambda^{\ell}} \bigl\langle \bigl((1+qt)^{\lambda}-1\bigr)^{\ell}(1+qt)^{\frac{y}{q}}|x^{n} \bigr\rangle \\ &\quad=\sum_{j_{1}+\cdots+j_{k}\leq n} \frac{c_{j_{1},q}(-q)}{q^{j_{1}+\cdots+j_{k}}j_{1}!(j_{1}+1)}\frac {b_{j_{k}}q^{j_{k}}}{j_{k}!} \prod_{i=2}^{k-1}\frac {b_{j_{i}}(-1)q^{j_{i}}}{j_{i}!(j_{1}+\cdots+j_{i}+1)} \\ &\qquad{}\times(j_{1}+\cdots+j_{k})!\sum _{\ell=j_{1}+\cdots +j_{k}}^{n}\sum_{m=0}^{\ell}\binom{\ell}{m}(-1)^{\ell-m}\frac{S_{2}(\ell,j_{1}+\cdots+j_{k})}{\ell!\lambda ^{\ell}} \bigl\langle (1+qt)^{\frac{y}{q}+\lambda m}|x^{n} \bigr\rangle . \end{aligned}$$
For \(k=1\), by (1.2), one obtains
which, by (2.2), implies
By using the fact that \(\langle(1+qt)^{\frac{y}{q}+\lambda m}|x^{n} \rangle=(y+\lambda qm|q)_{n}\), the proof is completed for the case \(k=1\). □
$$\begin{aligned} C_{n,q}^{(1)}(\lambda,y) &= \biggl\langle \operatorname{Lif}_{1} \biggl(\frac{(1+qt)^{\lambda}-1}{q\lambda} \biggr) (1+qt)^{\frac{y}{q}}\Big|x^{n} \biggr\rangle =\sum _{j_{1}=0}^{n}\frac {C_{j_{1},q}}{j_{1}!q^{j_{1}}} \bigl\langle \bigl(e^{\frac{(1+qt)^{\lambda}-1}{\lambda }}-1\bigr)^{j_{1}}(1+qt)^{\frac{y}{q}}|x^{n} \bigr\rangle , \end{aligned}$$
$$\begin{aligned} C_{n,q}^{(1)}(\lambda,x) &=\sum_{j_{1}=0}^{n} \sum_{\ell=j_{1}}^{m}\frac{C_{j_{1},q}S_{2}(\ell ,j_{1})}{q^{j_{1}}\lambda^{\ell}\ell!} \bigl\langle \bigl((1+qt)^{\lambda}-1\bigr)^{\ell }(1+qt)^{\frac{y}{q}}|x^{n} \bigr\rangle \\ &=\sum_{j_{1}=0}^{n}\sum _{\ell=j_{1}}^{m}\sum_{m=0}^{\ell}\binom{\ell }{m}(-1)^{\ell-m}\frac{C_{j_{1},q}S_{2}(\ell,j_{1})}{q^{j_{1}}\lambda^{\ell}\ell !} \bigl\langle (1+qt)^{\frac{y}{q}+\lambda m}|x^{n} \bigr\rangle . \end{aligned}$$
By similar arguments to the proof of Theorem 2.5 for the degenerate poly-Cauchy polynomials with a q parameter of the first kind, one has the following result.
Theorem 2.6
Let
\(n\geq0\). Then
for all
\(k\geq2\), and
$$\begin{aligned} \widehat{C}_{n,q}^{(k)}(\lambda,x)={}&\sum _{j_{1}+\cdots+j_{k}\leq n}\sum_{\ell=j_{1}+\cdots+j_{k}}^{n}\sum _{m=0}^{\ell}(-1)^{\ell-m} \frac{\binom{\ell }{m}}{\ell!(-\lambda)^{\ell}} \frac{(j_{1}+\cdots+j_{k})!}{q^{j_{1}+\cdots+j_{k}}}\frac {c_{j_{1},q}(-q)}{j_{1}!(j_{1}+1)}\frac{b_{j_{k}}q^{j_{k}}}{j_{k}!}\\ &\qquad{}\times S_{2}(\ell,j_{1}+ \cdots+j_{k}) (\lambda qm-x|q)_{n}\prod _{i=2}^{k-1}\frac{b_{j_{i}}(-1)q^{j_{i}}}{j_{i}!(j_{1}+\cdots+j_{i}+1)}, \end{aligned}$$
$$\begin{aligned} \widehat{C}_{n,q}^{(1)}(\lambda,x) &=\sum _{j_{1}=0}^{n}\sum_{\ell=j_{1}}^{n} \sum_{m=0}^{\ell}(-1)^{\ell-m} \frac {\binom{\ell}{m}}{\ell!(-\lambda)^{\ell}} \frac{C_{j_{1},q}}{q^{j_{1}}}S_{2}(\ell,j_{1}) ( \lambda qm-x|q)_{n}. \end{aligned}$$
3 Recurrences
Note that the sequences of polynomials \(C_{n,q}^{(k)}(\lambda,x)\) and \(\widehat{C}_{n,q}^{(k)}(\lambda,x)\) are Sheffer sequences. Thus they satisfy the Sheffer identity
Next, one shows several recurrences for the sequence of poly-Cauchy polynomials with a q parameter of the first kind and of the second kind.
$$\begin{aligned} &C_{n,q}^{(k)}(\lambda,x+y)=\sum_{j=0}^{n} \binom {n}{j}C_{j,q}^{(k)}(\lambda,x) (y|q)_{n-j}, \\ &\widehat{C}_{n,q}^{(k)}(\lambda,x+y)=\sum _{j=0}^{n}\binom{n}{j}\widehat {C}_{j,q}^{(k)}(\lambda,x) (-y|q)_{n-j}. \end{aligned}$$
Theorem 3.1
For all
\(n\geq1\),
$$\begin{aligned} C_{n,q}^{(k)}(\lambda,x+q)=C_{n,q}^{(k)}( \lambda ,x)+nqC_{n-1,q}^{(k)}(\lambda,x),\quad \widehat{C}_{n,q}^{(k)}(\lambda,x-q)=\widehat{C}_{n,q}^{(k)}( \lambda ,x)+nq\widehat{C}_{n-1,q}^{(k)}(\lambda,x). \end{aligned}$$
Proof
Note that \(f(t)S_{n}(x)=nS_{n-1}(x)\) for any \(S_{n}(x)\sim(g(t),f(t))\) (see [14, 15]). Hence, by (1.4), one has
which implies
as required. □
$$\begin{aligned} \frac{e^{qt}-1}{q}C_{n,q}^{(k)}(\lambda,x)=nC_{n-1,q}^{(k)}( \lambda ,x),\quad \frac{e^{-qt}-1}{q}\widehat{C}_{n,q}^{(k)}( \lambda,x)=n\widehat {C}_{n-1,q}^{(k)}(\lambda,x), \end{aligned}$$
$$\begin{aligned} C_{n,q}^{(k)}(\lambda,x+q)=C_{n,q}^{(k)}( \lambda ,x)+nqC_{n-1,q}^{(k)}(\lambda,x),\quad \widehat{C}_{n,q}^{(k)}(\lambda,x-q)=\widehat{C}_{n,q}^{(k)}( \lambda ,x)+nq\widehat{C}_{n-1,q}^{(k)}(\lambda,x), \end{aligned}$$
Theorem 3.2
For
\(n\geq0\),
where
\(d_{\ell,q}^{(k)}(\lambda)=C_{\ell,q}^{(k)}(\lambda,0)-C_{\ell ,q}^{(k-1)}(\lambda,0)\)
and
\(\widehat{d}_{\ell,q}^{(k)}(\lambda)=\widehat{C}_{\ell,q}^{(k)}(\lambda ,0)-\widehat{C}_{\ell,q}^{(k-1)}(\lambda,0)\).
$$\begin{aligned}& \begin{aligned}[b] C_{n+1;q}^{(k)}(\lambda,x)={}&xC_{n,q}^{(k)}( \lambda,x-q)-\sum_{m=0}^{n}\sum _{\ell=0}^{m+1}\sum_{j=0}^{m+1-\ell} \frac{\binom {m+1}{j}}{m+1}\lambda^{j}q^{n-\ell+1} S_{1}(n,m)\\ &{}\times S_{2}(m+1-j, \ell)d_{\ell,q}^{(k)}(\lambda)B_{j} \biggl( \frac {x+(\lambda-1)q}{q\lambda} \biggr), \end{aligned}\\& \begin{aligned}[b] \widehat{C}_{n+1;q}^{(k)}(\lambda,x)={}&{-}x\widehat{C}_{n,q}^{(k)}( \lambda ,x+q)-\sum_{m=0}^{n}\sum _{\ell=0}^{m+1}\sum_{j=0}^{m+1-\ell} \frac{\binom {m+1}{j}}{m+1}\lambda^{j}q^{n-\ell+1} S_{1}(n,m)\\ &{}\times S_{2}(m+1-j, \ell)\widehat{d}_{\ell,q}^{(k)}(\lambda)B_{j} \biggl( \frac {-x+(\lambda-1)q}{q\lambda} \biggr), \end{aligned} \end{aligned}$$
Proof
Recall that
and \(S_{n+1}(x)=(x-\frac{g'(t)}{g(t)})\frac{1}{f'(t)}S_{n}(x)\) for any \(S_{n}(x)\sim(g(t),f(t))\) (see [14, 15]). Thus, in the case of (1.4), one obtains
where \(g(t)=\frac{1}{\operatorname{Lif}_{k} (\frac{e^{q\lambda t}-1}{q\lambda} )}\). Note that \(\frac{g'(t)}{g(t)}=(\log(g(t)))'=- (\log \operatorname{Lif}_{k} (\frac {e^{q\lambda t}-1}{q\lambda} ) )'\), which leads to
Thus,
where \(A_{k}-A_{k-1}=\frac{\lambda qte^{(\lambda-1) qt}}{e^{\lambda qt}-1}\operatorname{Lif}_{k} (\frac{e^{q\lambda t}-1}{q\lambda} )-\frac {\lambda qte^{(\lambda-1)qt}}{e^{\lambda qt}-1}\operatorname{Lif}_{k-1} (\frac {e^{q\lambda t}-1}{q\lambda} )\) has order at least one (the order of a non-zero power series \(f(t)\) is the smallest integer k for which the coefficient of \(t^{k}\) in \(f(t)\) does not vanish). So, by the fact that \(\frac{1}{\operatorname{Lif}_{k} (\frac{e^{q\lambda t}-1}{q\lambda} )}C_{n,q}^{(k)}(\lambda,x) =(x|q)_{n}=\sum_{m=0}^{n}S_{1}(n,m)q^{n-m}x^{n}\) (see (2.3)), one has
On the other hand, by (2.2), one gets
Hence,
where \(d_{\ell,q}^{(k)}(\lambda)=C_{\ell,q}^{(k)}(\lambda,0)-C_{\ell ,q}^{(k-1)}(\lambda,0)\), which completes the proof of the first recurrence.
$$\begin{aligned} \bigl(\operatorname{Lif}_{k}(x)\bigr)'= \frac{\operatorname{Lif}_{k-1}(x)-\operatorname{Lif}_{k}(x)}{x}, \end{aligned}$$
(3.1)
$$C_{n+1;q}^{(k)}(\lambda,x)=xC_{n,q}^{(k)}( \lambda,x-q)-e^{-qt}\frac {g'(t)}{g(t)}C_{n,q}^{(k)}( \lambda,x), $$
$$\frac{g'(t)}{g(t)}=\frac{-1}{\operatorname{Lif}_{k} (\frac{e^{q\lambda t}-1}{q\lambda } )} \biggl(\operatorname{Lif}_{k-1} \biggl(\frac{e^{q\lambda t}-1}{q\lambda} \biggr)-\operatorname{Lif}_{k} \biggl( \frac{e^{q\lambda t}-1}{q\lambda} \biggr) \biggr)\frac{\lambda qe^{\lambda qt}}{e^{\lambda qt}-1}. $$
$$\begin{aligned} e^{-qt}\frac{g'(t)}{g(t)}C_{n;q}^{(k)}(\lambda,x)= \frac{1}{t} (A_{k}-A_{k-1} )\frac{1}{\operatorname{Lif}_{k} (\frac{e^{q\lambda t}-1}{q\lambda } )}C_{n,q}^{(k)}( \lambda,x), \end{aligned}$$
$$\begin{aligned} e^{-qt}\frac{g'(t)}{g(t)}C_{n;q}^{(k)}(\lambda,x) &= \sum_{m=0}^{n}S_{1}(n,m)q^{n-m} \frac{1}{t} (A_{k}-A_{k-1} )x^{m} \\ &=\sum_{m=0}^{n}\frac{S_{1}(n,m)}{m+1}q^{n-m} (A_{k}-A_{k-1} )x^{m+1}. \end{aligned}$$
$$\begin{aligned} A_{k}x^{m+1} &=\frac{\lambda qte^{(\lambda-1) qt}}{e^{\lambda qt}-1}\operatorname{Lif}_{k} \biggl(\frac{(1+qs)^{\lambda}-1}{q\lambda} \biggr)\bigg|_{s=\frac {e^{qt}-1}{q}}x^{m+1} \\ &=\frac{\lambda qte^{(\lambda-1) qt}}{e^{\lambda qt}-1}\sum_{\ell =0}^{m+1}C_{\ell,q}^{(k)}( \lambda,0)\frac{(e^{qt}-1)^{\ell}}{\ell!q^{\ell}}x^{m+1} \\ &=\frac{\lambda qte^{(\lambda-1) qt}}{e^{\lambda qt}-1}\sum_{\ell =0}^{m+1}\sum _{j=\ell}^{m+1}C_{\ell,q}^{(k)}( \lambda,0)S_{2}(j,\ell )q^{-\ell}\frac{(qt)^{j}}{j!}x^{m+1} \\ &=\sum_{\ell=0}^{m+1}\sum _{j=\ell}^{m+1}C_{\ell,q}^{(k)}(\lambda ,0)S_{2}(j,\ell)q^{j-\ell}\binom{m+1}{j} \frac{\lambda qte^{(\lambda-1) qt}}{e^{\lambda qt}-1}x^{m+1-j} \\ &=\sum_{\ell=0}^{m+1}\sum _{j=\ell}^{m+1}C_{\ell,q}^{(k)}(\lambda ,0)S_{2}(j,\ell)q^{j-\ell}\binom{m+1}{j} (\lambda q)^{m+1-j}B_{m+1-j} \biggl(\frac{x+(\lambda-1)q}{q\lambda} \biggr) \\ &=\sum_{\ell=0}^{m+1}\sum _{j=0}^{m+1-\ell}\binom{m+1}{j}\lambda ^{j}q^{m-\ell+1}C_{\ell,q}^{(k)}( \lambda,0)S_{2}(m+1-j,\ell) B_{j} \biggl(\frac{x+(\lambda-1)q}{q\lambda} \biggr). \end{aligned}$$
$$\begin{aligned} C_{n+1;q}^{(k)}(\lambda,x)={}&xC_{n,q}^{(k)}( \lambda,x-q)-\sum_{m=0}^{n}\sum _{\ell=0}^{m+1}\sum_{j=0}^{m+1-\ell} \frac{\binom {m+1}{j}}{m+1}\lambda^{j}q^{n-\ell+1} S_{1}(n,m)\\ &{}\times S_{2}(m+1-j, \ell)d_{\ell,q}^{(k)}(\lambda)B_{j} \biggl( \frac {x+(\lambda-1)q}{q\lambda} \biggr), \end{aligned}$$
By applying the above proof to the case of poly-Cauchy polynomials with a q parameter of the second kind together with using (1.4) for \(\widehat{C}_{n,q}^{(k)}(\lambda,x)\) instead of \(C_{n,q}^{(k)}(\lambda,x)\), one can obtain the second recurrence. □
In the next result one finds the expressions for \(\frac {d}{dx}C_{n,q}^{(k)}(\lambda,x)\) and \(\frac{d}{dx}\widehat {C}_{n,q}^{(k)}(\lambda,x)\).
Theorem 3.3
For all
\(n\geq0\),
$$\frac{d}{dx}C_{n,q}^{(k)}(\lambda,x)=n!\sum _{\ell=0}^{n-1}\frac {(-q)^{n-1-\ell}}{(n-\ell)\ell!}C_{\ell,q}^{(k)}( \lambda,x),\quad \frac {d}{dx}\widehat{C}_{n,q}^{(k)}( \lambda,x)=-n!\sum_{\ell=0}^{n-1} \frac {(-q)^{n-1-\ell}}{(n-\ell)\ell!}\widehat{C}_{\ell,q}^{(k)}(\lambda,x). $$
Proof
It is well known that \(\frac{d}{dx}S_{n}(x)=\sum_{\ell=0}^{n-1}\binom {n}{\ell} \langle\bar{f}(t)|x^{n-\ell} \rangle S_{\ell}(x)\), where \(S_{n}(x)\sim(g(t),f(t))\) and \(\bar{f}(t)\) is the compositional inverse of \(f(t)\) (see [14, 15]). In the present cases, see (1.4), one has either \(\bar{f}(t)=\frac{1}{q}\log(1+qt)\) or \(\bar{f}(t)=-\frac{1}{q}\log(1+qt)\). Note that \(\langle\frac{1}{q}\log(1+qt)|x^{n-\ell} \rangle =(-q)^{n-1-\ell}(n-1-\ell)!\). Thus, \(\frac{d}{dx}C_{n,q}^{(k)}(\lambda,x)=n!\sum_{\ell=0}^{n-1}\frac {(-q)^{n-1-\ell}}{(n-\ell)\ell!}C_{\ell,q}^{(k)}(\lambda,x)\) and \(\frac{d}{dx}\widehat{C}_{n,q}^{(k)}(\lambda,x)=-n!\sum_{\ell =0}^{n-1}\frac{(-q)^{n-1-\ell}}{(n-\ell)\ell!}\widehat{C}_{\ell ,q}^{(k)}(\lambda,x)\), as required. □
In the next theorem one uses the Korobov numbers. Recall that the Korobov numbers
\(K_{n}(\lambda)\) of the first kind are given by \(\sum_{n\geq0}K_{n}(\lambda)\frac{t^{n}}{n!}=\frac{\lambda t}{(1+t)^{\lambda}-1}\) (see [6, 7]).
Theorem 3.4
For all
\(n\geq1\),
$$\begin{aligned}& \begin{aligned}[b] C_{n,q}^{(k)}(\lambda,x)={}&xC_{n-1,q}^{(k)}( \lambda,x-q)\\ &{}+\frac{1}{n}\sum_{m=0}^{n}q^{n-m} \binom{n}{m}K_{n-m}(\lambda) \bigl( C_{m,q}^{(k-1)} \bigl(\lambda,x+(\lambda-1)q\bigr)-C_{m,q}^{(k)}\bigl( \lambda,x+(\lambda -1)q\bigr) \bigr), \end{aligned}\\& \begin{aligned}[b] \widehat{C}_{n,q}^{(k)}(\lambda,x)={}&{-}x\widehat{C}_{n-1,q}^{(k)}( \lambda ,x+q)\\ &{}+\frac{1}{n}\sum_{m=0}^{n}q^{n-m} \binom{n}{m}K_{n-m}(\lambda) \bigl( \widehat{C}_{m,q}^{(k-1)} \bigl(\lambda,x-(\lambda-1)q\bigr)-C_{m,q}^{(k)}\bigl(\lambda ,x-(\lambda-1)q\bigr) \bigr). \end{aligned} \end{aligned}$$
Proof
Here only the proof of the first recurrence will be provided. Let \(L_{k}=\operatorname{Lif}_{k} (\frac{(1+qt)^{\lambda}-1}{q\lambda} )\). By (1.2), we have \(C_{n,q}^{(k)}(\lambda,y)= \langle L_{k}(1+qt)^{\frac {y}{q}}|x^{n} \rangle=A+B\), where \(A= \langle L_{k}\frac{d}{dt}(1+qt)^{\frac{y}{q}}|x^{n-1} \rangle \) and \(B= \langle\frac{d}{dt}L_{k}(1+qt)^{\frac{y}{q}}|x^{n-1} \rangle\). The term A is given by \(A=y \langle L_{k}(1+qt)^{\frac{y-q}{q}}|x^{n-1} \rangle =yC_{n-1,q}^{(k)}(\lambda,y-q)\). By (3.1), the term B is given by
Note that the order of \(L_{k-1}-L_{k}\) is at least one. Thus,
Thus, by expressing the Korobov numbers of the first kind, one obtains
Hence,
as required. □
$$\begin{aligned} B&= \biggl\langle \frac{\lambda qt}{(1+qt)^{\lambda}-1}(1+qt)^{\frac {y+(\lambda-1)q}{q}} \frac{L_{k-1}-L_{k}}{t}\Big|x^{n-1} \biggr\rangle \\ &=\frac{1}{n} \biggl\langle \frac{\lambda qt}{(1+qt)^{\lambda}-1}(1+qt)^{\frac{y+(\lambda-1)q}{q}} (L_{k-1}-L_{k})\Big|x^{n} \biggr\rangle . \end{aligned}$$
$$\begin{aligned} B&=\frac{1}{n} \biggl\langle \frac{\lambda qt}{(1+qt)^{\lambda}-1}(1+qt)^{\frac{y+(\lambda-1)q}{q}}L_{k-1}\Big|x^{n} \biggr\rangle -\frac{1}{n} \biggl\langle \frac{\lambda qt}{(1+qt)^{\lambda}-1}(1+qt)^{\frac{y+(\lambda-1)q}{q}}L_{k}\Big|x^{n} \biggr\rangle \\ &=\frac{1}{n} \Biggl\langle \frac{\lambda qt}{(1+qt)^{\lambda}-1}\sum _{m=0}^{n} \bigl(C_{m,q}^{(k-1)} \bigl(\lambda,y+(\lambda-1)q\bigr) -C_{m,q}^{(k)}\bigl( \lambda,y+(\lambda-1)q\bigr) \bigr)\frac{t^{m}}{m!}\Big|x^{n} \Biggr\rangle \\ &=\frac{1}{n}\sum_{m=0}^{n} \binom{n}{m} \bigl( C_{m,q}^{(k-1)}\bigl(\lambda,y+( \lambda-1)q\bigr) -C_{m,q}^{(k)}\bigl(\lambda,y+(\lambda-1)q \bigr) \bigr) \biggl\langle \frac{\lambda qt}{(1+qt)^{\lambda}-1}\Big|x^{n-m} \biggr\rangle . \end{aligned}$$
$$\begin{aligned} B&=\frac{1}{n}\sum_{m=0}^{n} \binom{n}{m} \bigl( C_{m,q}^{(k-1)}\bigl(\lambda,y+( \lambda-1)q\bigr)-C_{m,q}^{(k)}\bigl(\lambda,y+(\lambda -1)q \bigr) \bigr) K_{n-m}(\lambda)q^{n-m}. \end{aligned}$$
$$\begin{aligned} C_{n,q}^{(k)}(\lambda,y)={}&yC_{n-1,q}^{(k)}( \lambda,y-q)\\ &{}+\frac{1}{n}\sum_{m=0}^{n}q^{n-m} \binom{n}{m}K_{n-m}(\lambda) \bigl( C_{m,q}^{(k-1)} \bigl(\lambda,y+(\lambda-1)q\bigr)-C_{m,q}^{(k)}\bigl( \lambda,y+(\lambda -1)q\bigr) \bigr), \end{aligned}$$
4 Connections with families of polynomials
Now, a few examples are presented on the connections with known families of polynomials. To do that, one uses the following fact from [14, 15]: 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
$$\begin{aligned} c_{n,k}=\frac{1}{k!} \biggl\langle \frac{h(\bar{f}(t))}{g(\bar{f}(t))}\bigl(\ell \bigl(\bar{f}(t)\bigr)\bigr)^{k}\Big|x^{n} \biggr\rangle . \end{aligned}$$
(4.1)
Let us start with the connection to Bernoulli polynomials
\(B_{n}^{(s)}(x)\)
of order
s. In the next result, one expresses the degenerate poly-Cauchy polynomials with a q parameter in terms of Bernoulli polynomials of order
s.
As analogs of (1.2) and (1.3), one defines the numbers \(\mathbb{C}_{n,q}^{(s)}\) and \(\widehat{\mathbb{C}}_{n,q}^{(s)}\) as \((\frac{q((1+qt)^{\frac{1}{q}}-1)}{\log(1+qt)} )^{s}=\sum_{m\geq 0}\mathbb{C}_{n,q}^{(s)}\frac{t^{n}}{n!}\) and \((\frac{q(1-(1+qt)^{-\frac{1}{q}})}{\log(1+qt)} )^{s} =\sum_{m\geq0}\widehat{\mathbb{C}}_{n,q}^{(s)}\frac{t^{n}}{n!}\).
Theorem 4.1
For all
\(n\geq0\),
$$\begin{aligned} &C_{n,q}^{(k)}(\lambda,x)=\sum_{m=0}^{n} \Biggl(\sum_{\ell=m}^{n}\sum _{j=0}^{n-\ell}\binom{n}{\ell}\binom{n-\ell}{j}q^{\ell-m} S_{1}(\ell ,m)C_{j,q}^{(k)}(\lambda,0) \mathbb{C}_{n-\ell-j,q}^{(s)} \Biggr)B_{m}^{(s)}(x), \\ &\widehat{C}_{n,q}^{(k)}(\lambda,x)=\sum _{m=0}^{n} \Biggl( (-1 )^{m}\sum _{\ell=m}^{n}\sum_{j=0}^{n-\ell} \binom{n}{\ell}\binom{n-\ell }{j}q^{\ell-m} S_{1}(\ell,m) \widehat{C}_{j,q}^{(k)}(\lambda,0)\widehat { \mathbb{C}}_{n-\ell-j,q}^{(s)} \Biggr)B_{m}^{(s)}(x). \end{aligned}$$
Proof
Due to the similarity between the degenerate poly-Cauchy polynomials with a q parameter of the first kind and of the second kind, only the proof details of the first identity will be provided, where the proof details of the second one are omitted. Let \(C_{n,q}^{(k)}(\lambda,x)=\sum_{m=0}^{n}c_{n,m}B_{m}^{(s)}(x)\). Then by (1.4), (4.1) and (2.6), one obtains
which, by (2.1) and (1.2), implies
which implies \(c_{n,m} =\sum_{\ell=m}^{n}\sum_{j=0}^{n-\ell}\binom{n}{\ell}\binom{n-\ell }{j}q^{\ell-m} S_{1}(\ell,m)C_{j,q}^{(k)}(\lambda,0)\mathbb{C}_{n-\ell -j,q}^{(s)}\), as required. □
$$\begin{aligned} c_{n,m}&=\frac{1}{m!} \biggl\langle \biggl(\frac{q((1+qt)^{\frac {1}{q}}-1)}{\log(1+qt)} \biggr)^{s} \operatorname{Lif}_{k} \biggl(\frac{(1+qt)^{\lambda}-1}{q\lambda} \biggr)\Big|\biggl(\frac{1}{q}\log (1+qt)\biggr)^{m}x^{n} \biggr\rangle , \end{aligned}$$
$$\begin{aligned} c_{n,m} &=\frac{1}{q^{m}}\sum_{\ell=m}^{n} \binom{n}{\ell}q^{\ell}S_{1}(\ell,m) \biggl\langle \biggl( \frac{q((1+qt)^{\frac{1}{q}}-1)}{\log(1+qt)} \biggr)^{s}\Big|\operatorname{Lif}_{k} \biggl( \frac{(1+qt)^{\lambda}-1}{q\lambda} \biggr)x^{n-\ell} \biggr\rangle \\ &=\frac{1}{q^{m}}\sum_{\ell=m}^{n} \binom{n}{\ell}q^{\ell}S_{1}(\ell,m) \Biggl\langle \biggl( \frac{q((1+qt)^{\frac{1}{q}}-1)}{\log(1+qt)} \biggr)^{s}\Big|\sum_{j=0}^{n-\ell}C_{j,q}^{(k)}( \lambda,0)\frac{t^{j}}{j!}x^{n-\ell} \Biggr\rangle \\ &=\frac{1}{q^{m}}\sum_{\ell=m}^{n}\sum _{j=0}^{n-\ell}\binom{n}{\ell} \binom {n-\ell}{j}q^{\ell}S_{1}(\ell,m)C_{j,q}^{(k)}( \lambda,0) \biggl\langle \biggl(\frac{q((1+qt)^{\frac{1}{q}}-1)}{\log(1+qt)} \biggr)^{s}\Big|x^{n-\ell -j} \biggr\rangle , \end{aligned}$$
Using similar techniques to the proof of the previous theorem, one can express the degenerate poly-Cauchy polynomials in terms of other families, for instance, Frobenius-Euler polynomials (the proof is left to the interested reader). Note that the Frobenius-Euler polynomials
\(H_{n}^{(s)}(x|\mu)\) of order s are defined by the generating function \((\frac{1-\mu}{e^{t}-\mu} )^{s} e^{xt}=\sum_{n\geq 0}H_{n}^{(s)}(x|\mu)\frac{t^{n}}{n!}\) (\(\mu\neq1\)), or equivalently, \(H_{n}^{(s)}(x|\mu)\sim ( (\frac{e^{t}-\mu}{1-\mu} )^{s},t )\) (see [29, 30, 33, 34]).
Theorem 4.2
For all
\(n\geq0\),
$$\begin{aligned}& \begin{aligned}[b] C_{n,q}^{(k)}(\lambda,x)={}&\sum_{m=0}^{n} \Biggl(\frac{\mu^{s}}{(1-\mu)^{s}}\sum_{\ell=m}^{n} \sum_{j=0}^{n-\ell}\sum _{i=0}^{s} \frac{\binom{n}{\ell}\binom{n-\ell}{j}\binom{s}{i}q^{\ell-m}}{(-\mu )^{i}}S_{1}( \ell,m) (i|q)_{n-\ell-j}C_{j,q}^{(k)}(\lambda,0) \Biggr)\\ &{}\times H_{m}^{(s)}(x|\mu), \end{aligned}\\& \begin{aligned}[b] \widehat{C}_{n,q}^{(k)}(\lambda,x)={}& \sum _{m=0}^{n} \Biggl(\frac{\mu^{s}}{(\mu -1)^{s}}\sum _{\ell=m}^{n}\sum_{j=0}^{n-\ell} \sum_{i=0}^{s} \frac{(-1)^{m}\binom{n}{\ell}\binom{n-\ell}{j}\binom{s}{i}q^{\ell -m}}{(-\mu)^{i}}\\ &{}\times S_{1}( \ell,m) (-i|q)_{n-\ell-j}\widehat {C}_{j,q}^{(k)}(\lambda,0) \Biggr) H_{m}^{(s)}(x|\mu). \end{aligned} \end{aligned}$$
As another example, one can express our degenerate poly-Cauchy polynomials in terms of the rising factorials \((x|q)^{(m)}=x(x+q)\cdots (x+(m-1)q)\), as follows. Using the fact that \((x|q)^{(n)}\sim(1,\frac {1-e^{-qt}}{q})\) with (1.2), (1.3), and (4.1), one obtains the following result.
Theorem 4.3
For all
\(n\geq0\),
$$\begin{aligned} &C_{n,q}^{(k)}(\lambda,x)=\sum_{m=0}^{n} \binom {n}{m}C_{n-m,q}^{(k)}(\lambda,-qm) (x|q)^{(m)},\\ &\widehat {C}_{n,q}^{(k)}(\lambda,x)=\sum _{m=0}^{n}(-1)^{m} \binom{n}{m}\widehat {C}_{n-m,q}^{(k)}(\lambda,0) (x|q)^{(m)}. \end{aligned}$$
Acknowledgements
The second author was appointed as a chair professor at Tianjin polytechnic University by Tianjin city in China from August 2015 to August 2019. We would like to express our gratitude to Professor Toufik Mansour for his comments and improvements. Our thanks also go to the referees for their comments and suggestions, which improved the present paper greatly. The work reported in this paper was conducted during the sabbatical year of Kwangwoon University in 2014.
Open Access 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.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.