Abstract
Recently, Kim, Kwon, and Seo (J. Nonlinear Sci. Appl. 9:2380-2392, 2016) studied the degenerate q-Changhee polynomials and numbers. In this paper, we consider the Appell-type degenerate q-Changhee polynomials and give some new and explicit identities related to these polynomials
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1 Introduction
Let p be a fixed odd prime number. In this paper, we denote the ring of p-adic integers and the field of p-adic numbers by \(\mathbb{Z}_{p}\) and \(\mathbb{Q}_{p}\), respectively. The p-adic norm \(|\cdot|_{p}\) is normalized as \(|p|_{p} = \frac{1}{p}\). Let q be an indeterminate with \(|1-q|_{p} < p^{-\frac{1}{p-1}}\). We recall that \(\operatorname{UD}(\mathbb{Z}_{p})\) is the set of uniformly differentiable functions on \(\mathbb{Z}_{p}\). For each \(f \in \operatorname{UD}(\mathbb{Z}_{p})\), the p-adic q-Volkenborn integral on \(\mathbb{Z}_{p}\) is defined by Kim to be
where \([x]_{q} = \frac{1-q^{x}}{1-q}\) (see [1–4]). From (1.1), we have
(see [5–7]). Kwon-Kim-Seo [8] derived some identities of the degenerate Changhee polynomials which are given by the generating function
(see [1, 3, 4, 7–14]). We note that if \(x=0\), then \(\operatorname{Ch}_{n,\lambda }=\operatorname{Ch}_{n,\lambda}(0)\) are called the degenerate Changhee numbers. From (1.3), we note that
We recall that the gamma and beta functions are defined by the following definite integrals: for \(\alpha>0 \), \(\beta>0\),
and
(see [5, 15, 16]). From (1.4) and (1.5), we show that
The Bell polynomials are defined by the generating function
(see [6]).
Recently, Kim, Kwon, and Seo [1] defined the degenerate q-Changhee polynomials, a q-extension of (1.3), by
We note that if \(x=0\), then \(\operatorname{Ch}_{n,\lambda,q}=\operatorname{Ch}_{n,\lambda,q}(0)\) are called the degenerate q-Changhee numbers.
In this paper, we consider the Appell-type degenerate q-Changhee polynomials and give some explicit and new formulas for these polynomials.
2 The Appell-type degenerate q-Changhee polynomials
In this section, we define the Appell-type degenerate q-Changhee polynomials which are given by
If \(x=0\), then \(\widetilde{\operatorname{Ch}}_{n,\lambda,q} = \widetilde {\operatorname{Ch}}_{n,\lambda,q}(0)\) are called the Appell-type degenerate q-Changhee numbers. From (2.1), we note that
By (2.2), we obtain
From (2.3), we show that
We observe that
On the other hand, we derive
Thus, by (2.5) and (2.6), we give the first result.
Theorem 1
For \(n \in\mathbb{N}\cup\{0\} \), we have
In particular, \(x=0\);
We also observe that
Continuing this process consecutively yields
where \((n)_{m-1}=n(n-1) \cdots(n-m+2)\) and \(< n+1>_{m}=(n+1)(n+2)\cdots(n+m)\).
Thus, by (2.5) and (2.8), we give the second result.
Theorem 2
For \(n \in\mathbb{N}\) with \(n \geq3\), we have
For \(n \in\mathbb{N}\), we have
Therefore, by (2.5) and (2.9), we obtain the third result.
Theorem 3
For \(n \in\mathbb{N}\), we have
where \(B(n,m+1)\) is a beta function.
Now, we observe that, for \(n \in\mathbb{N}\cup\{0\}\), \(m \in\mathbb{N}\),
On the other hand,
Thus, by (2.10) and (2.11), we give the fourth result.
Theorem 4
For \(n \in\mathbb{N} \cup\{0\}\), \(m \in\mathbb{N}\), we have
By replacing t to \(\frac{1}{\lambda} ( e^{\lambda t}-1)\) in (2.1), we get
(see [6]). On the other hand,
where \(S_{2}(n,m)\) is for the Stirling numbers of the second kind, given by
By (2.12) and (2.13), we give the fifth result.
Theorem 5
For \(n \in\mathbb{N} \cup\{0\}\), we have
3 Remarks
In this section, we derive an explicit identity related to the Appell-type degenerate q-Changhee polynomials as follows. By (1.2), we get
On the other hand,
where \(S_{1}(m,k)\) is the Stirling numbers of the first kind which is given by
By (3.1) and (3.2), we give the final result.
Theorem 6
For \(n \geq0\), we have
4 Conclusions
We consider special numbers and polynomials such as Appell polynomials over the years: Bernoulli, Euler, Genocchi polynomials, and also Changhee polynomials and numbers have many applications in all most all branches of the mathematics and mathematical physics.
In Theorems 1, 2, 3, and 4, by using p-adic q-Volkenborn integral and generating functions, we derived many new and novel identities and relations related to the Appell-type degenerate q-Changhee polynomials and also q-Changhee numbers. In Theorem 5, we also gave some relations between q-Changhee type polynomials and the Stirling numbers of the first kind and Changhee numbers.
References
Kim, T, Kwon, H-I, Seo, JJ: Degenerate q-Changhee polynomials. J. Nonlinear Sci. Appl. 9, 2380-2392 (2016)
El-Desouky, BS, Mustafa, A: New results on higher-order Daehee and Bernoulli numbers and polynomials. Adv. Differ. Equ. 2016, 32 (2016)
Kwon, J, Park, J-W: A note on \((h,q)\)-Boole polynomials. Adv. Differ. Equ. 2015, 198 (2015)
Park, J-W: On the q-analogue of λ-Daehee polynomials. J. Comput. Anal. Appl. 19(6), 966-974 (2015)
Kim, DS, Kim, T: Some identities involving Genocchi polynomials and numbers. Ars Comb. 121, 403-412 (2015)
Kim, DS, Kim, T: On degenerate Bell numbers and polynomials. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 2016, 1-12 (2016). doi:10.1007/s13398-016-0304-4
Kwon, H-I, Kim, T, Seo, JJ: A note on Daehee numbers arising from differential equations. Glob. J. Pure Appl. Math. 12(3), 2349-2354 (2016)
Kwon, H-I, Kim, T, Seo, JJ: A note on degenerate Changhee numbers and polynomials. Proc. Jangjeon Math. Soc. 18(3), 295-305 (2015)
Jang, L-C, Ryoo, CS, Seo, JJ, Kwon, H-I: Some properties of the twisted Changhee polynomials and their zeros. Appl. Math. Comput. 274, 169-177 (2016)
Kim, T, Kim, DS: A note on nonlinear Changhee differential equations. Russ. J. Math. Phys. 23(1), 88-92 (2016)
Kim, T: Some properties on the integral of the product of several Euler polynomials. Quaest. Math. 38(4), 553-562 (2015)
Kim, T, Kim, DS, Seo, J-J, Kwon, H-I: Differential equations associated with λ-Changhee polynomials. J. Nonlinear Sci. Appl. 9, 3098-3111 (2016)
Lim, D, Qi, F: On the Appell type λ-Changhee polynomials. J. Nonlinear Sci. Appl. 9(4), 1872-1876 (2016)
Rim, S-H, Park, JJ, Pyo, S-S, Kwon, J: The n-th twisted Changhee polynomials and numbers. Bull. Korean Math. Soc. 52(3), 741-749 (2015)
Kim, DS, Kim, T, Seo, JJ: Higher-order Daehee polynomials of the first kind with umbral calculus. Adv. Stud. Contemp. Math. (Kyungshang) 24(1), 5-18 (2014)
Kim, T: A study on the q-Euler numbers and the fermionic q-integral of the product of several type q-Bernstein polynomials on \(\mathbb{Z}_{p} \). Adv. Stud. Contemp. Math. (Kyungshang) 23(1), 5-11 (2013)
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Qi, F., Jang, LC. & Kwon, HI. Some new and explicit identities related with the Appell-type degenerate q-Changhee polynomials. Adv Differ Equ 2016, 180 (2016). https://doi.org/10.1186/s13662-016-0912-5
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DOI: https://doi.org/10.1186/s13662-016-0912-5