Several generalizations and applications of incomplete poly-Bernoulli polynomials and incomplete poly-Cauchy polynomials (Analytic Number Theory and Related Areas)

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(1)20. 数理解析研究所講究録第2014巻 2017年 20-34. generalizations and applications of incomplete poly‐Bernoulli polynomials and incomplete poly‐Cauchy polynomials Several. Takao Komatsu. School of Mathematics and. Statistics,. Wuhan. University. Introduction. 1 The. poly‐Bernoulli polynomials. function. B_{n}^{( $\mu$)}(x) ([6]). are. defined. \displaystyle\frac{\mathrm{L}\mathrm{i}_{$\mu$}(1-e^{-t}){1-e^{-t}e^{-tx}=\sum_{n=0}^{\infty}B_{n}^{($\mu$)}(x)\frac{t^n}{n!}. where. polylogarithm. replaced by e^{tx} numbers Bernoulli. ([11]).. in. [1, 2].. function. Note that e^{-tx}. on. (1). the left‐hand side is. the poly‐Bernoulli B_{n}^{( $\mu$)}(0)=B_{n}^{( $\mu$)} the classical $\mu$=1, B_{n}^{(1)}(0)=B_{n}. When x=0,. When x=0 and. the generating. ,. \displaystle\mathrm{L}\mathrm{i}_ $\mu$}(z)=\sum_{7n=1}^{\infty}\frac{z^rn}{m^ $\mu$}. is the $\mu$‐th. by. are. are. numbers, defined by the generating function. \displaystyle\frac{t}1-e^{-t}=\sum_{n=0}^{\infty}B_{n^{\frac{t^{n}{n!} generating function of poly‐Cauchy polynomials is 2]) given by. The rem. \displayst le\frac{\mathrm{L}\mathrm{j}\mathrm{f}_{$\mu$}(\log(1+t)}{(1+t)^{x}=\sum_{n=0}^{\infty}c_{n}^{($\mu$)}(x)\frac{t^n}{n!} where. \displayst le\mathrm{L}\mathrm{i}\mathrm{f}_{$\mu$}(z)=\sum_{n=0}^{\infty}\frac{z^n}{n!( +1)^{$\mu$}. ,. c_{m}^{( $\mu$)}(x) ([12,. Theo‐. (2).

(2) 21. polylogarithm factorial or polyfactorial function. Note that (1+t)^{x} on the left‐hand side is replaced by (1+t)^{-x} in [10]. When x=0, c_{n}^{( $\mu$)}(0)=c_{n}^{( $\mu$)} are the poly‐Cauchy numbers c_{n}^{( $\mu$)} ([12, Theorem 2 given by is the. \displaystyle\mathrm{L}\mathrm{i}\mathrm{f}_{$\mu$}(\log(1+t)=\sum_{n=0}^{\infty}c_{n}^{($\mu$)}\frac{t^n}{n!} $\mu$=1, c_{n}^{(1)}(0)=c_{n} generating function. When x=0 and. defined. by. the. are. (3). .. the classical. Cauchy numbers,. \displaystyle\frac{t} \log(1+t)}=\sum_{n=0}^{\infty}c_{n}\frac{t^{n} {n!}. (4). In this paper, by using the restricted and associated Stirling numbers of the first kind, we define the restricted and associated poly‐Cauchy polyno‐. By using the restricted and associated Stirling numbers of the sec‐ ond kind, we define the restricted and associated poly‐Bernoulli polynomials. These polynomials are generalizations of original poly‐Cauchy polynomials and original poly‐Bernoulli polynomials, respectively. We also study their characteristic and combinatorial properties. mials.. 2. Restricted and associated. Stirling. numbers. of the second kind place of the classical Stirling numbers \left\{ begin{ar y}{l n\ k \end{ar y}\right\} we substitute the Stirling numbers and the associated Stirling numbers, denoted uy. In. \left\{ begin{ar y}{l n\ k \end{ar y}\right\}. and. restricted. \left\{ begin{ar y}{l n\ k \end{ar y}\right\}. Some combinatorial and modular properties of these numbers can be found in [21], and other properties can be found in the papers from the list of. references of. [21].. The generating fUmctions of these numbers. \displayst le\sum_{n=k}^{rnk}\left\{ begin{ar y}{l n\ k \end{ar y}\right\} frac{x^n}{n!}=\frac{1}k!}(E_{$\tau$n}(x)-1^{\text{鳶}. are. given Uy. (5).

(3) 22. and. respectively,. where. \displaystyle\sum_{n=mk}^{\infty}\left\{ begin{ar ay}{l n\ k \end{ar ay}\right\} frac{x^{n} {n!}=\frac{1}{k!}(e^{x}-E_{rn-1}(x) ^{k}. (6). E_{rn}(t)=\displaystyle\sum_{k=0}^{rn}\frac{t^{k} {k!}. partial sum of the exponential function. These give the number of the k‐partitions of an n\rightarrow element set, such that each block contains at most or at least m elements, respectively. Notice that as particular cases, these numbers when m=2 have been considered by several authors (e.g., [5, 8, 22, 23 Since the generating function of \left\{ begin{ar y}{l n\ k \end{ar y}\right\} is given by is the mth. \displayst le\sum_{n=k}^{\infty}\left\{ begin{ar y}{l n\ k \end{ar y}\right\} frac{x^n}{ !}=\frac{(e^{x}-1)^{\tex{ん} {k!} (see. e.g.,. [9]), by E_{\infty}(x)=e^{x}. and. E_{0}(x)=1. ,. we. (7). have. \left\{ begin{ar y}{l n\ k \end{ar y}\right\}=\left\{ begin{ar y}{l n\ k \end{ar y}\right\}=\left\{ begin{ar y}{l n\ k \end{ar y}\right\} 3. Restricted and associated. Stirling numbers. of the first kind Denote. by. \left{\begin{ar y}{l n\ k \end{ar y}\ight}. the. (unsigned) Stirling. numbers of the first. coefficients of the rising factorial. x( +1)\displaystyle \ldots(x+n-1)=\sum_{i=0}^{n}\left\{\begin{ar ay}{l} n\ i \end{ar ay}\right\}x^{i} the. generating function of. \left{\begin{ar y}{l n\ k \end{ar y}\ight}. is. given by. \displayst le\sum_{n=k}^{\infty}\left\{ begin{ar y}{l n\ k \end{ar y}\right\} frac{x^n}{n!}=\frac{(-\log(1-x)^{k}{k!}.. kind, arising. as.

(4) 23. place of the classical (unsigned) Stirling numbers of the first kind [_{k}^{n} ], we (unsigned) restricted Stirling numbers of the first kind and the (unsigned) associated Stirling numbers of the first kind, denoted by. In. substitute the. \left\{ begin{ar y}{l n\ k \end{ar y}\right\} respectively.. The associated. and. Stirling. \left\{ begin{ar y}{l n\ k \end{ar y}\right\},. number of the first kind. \left\{ begin{ar y}{l n\ k \end{ar y}\right\}. equals. the number of permutations of a set N(|N|=n) with k orbits such that each block contains at least m elements ([5, p.256‐257], [23]). The restricted. Stirling. of. n. number of the first kind. \left\{ begin{ar ay}{l n\ k \end{ar ay}\right\}. equals. the number of permutations m elements.. with k orbits such that each block contains at most. generating functions of the restricted Stirling \left\{ begin{ar y}{l n\ k \end{ar y}\right\} and the associated Stirling numbers of the. For k\geq 1 and m\geq 1 , the numbers of the first kind. first kind mth. \left\{ begin{ar y}{l n\ k \end{ar y}\right\}. partial. sum. given by the following ( [4, of the logarithm function by. are. p.. F_{m}(t)=\displaystyle \sum_{k=1}^{m}(-1)^{k+1}\frac{t^{k} {k}. 467, (12.8)] ) Denote the .. (8). .. \displaystyle\sum_{n=0}^{\infty}\left\{ begin{ar ay}{l n\ k \end{ar ay}\right\} frac{x^{n}{n!}=\frac{1}{k!}(-F_{rn}(-x)^{k} \displaystyle \sum_{n=0}^{\infty}\left\{\begin{ar ay}{l} n\ k \end{ar ay}\right\}\frac{x^{n} {n!}=.\frac{1}{k!}(-\log(1-x)+F_{rn-1}(-x) ^{k}. (9). ,. It is trivial to. see. (10). that. \left\{ begin{ar y}{l n\ k \end{ar y}\right\}= left\{ begin{ar y}{l n\ k \end{ar y}\right\}= left\{ begin{ar y}{l n\ k \end{ar y}\right\} Incomplete poly‐Cauchy polynomials. 4 In x. [10], the concept of the poly‐Cauchy polynomials is introduced by replacing. by. -x. in the definition. For. integers. n. and $\mu$ with n\geq 0 and. $\mu$\geq 1 define ,.

(5) 24. the. poly‐Cauchy polynomials of the first. kind. \mathrm{c}_{n}^{( $\mu$)}(x)( $\mu$\geq 1). as. c_{n}^{( $\mu$)}(x)\displaystyle \cdot=\int_{0^{1} \ldots\int_{0}^{1}(t_{1}t_{2}\cdots t_{ $\mu$}-x)(t_{1}t_{2}\cdots t_{ $\mu$}-x-1) 一 $\mu$ .. c_{n}^{( $\mu$)}(x) \left{\begin{ar y}{l n\ k \end{ar y}\ight}. Then, kind. ([10,. can. be. expressed. .. (t_{1}t_{2}\cdots t_{ $\mu$}-x-n+1)dt_{1}dt_{2}\cdots dt_{ $\mu$}.. in terms of the. Stirling. numbers of the first. :. c_{n}^{($\mu$)}(x)=\displayst le\sum_{k=0}^{n}\left\{ begin{ar y}{l n\ k \end{ar y}\right\}(-1)^{n-k}\sum_{i=0}^{k}\left(\begin{ar y}{l k\ i \end{ar y}\right)\frac{(-x)^{i} (k-i+1)^{$\mu$}. Theorem 1. Hence, it is natural to define two types of incomplete poly‐Cauchy poly‐ nomials of the first kind c_{n,\leq m}^{( $\mu$)}(x) and c_{n,\geq m}^{( $\mu$)}(x) by. c_{n,\leqm}^{($\mu$)}(x)=\displayst le\sum_{k=0}^{n}\left\{ begin{ar y}{l n\ k \end{ar y}\right\}(-1)^{n-k}\sum_{i=0}^{k}\left(\begin{ar y}{l k\ i \end{ar y}\right)\frac{(-x)^{i} (k-i+1)^{$\mu$} c_{n,\geqm}^{($\mu$)}(x)=\displayst le\sum_{k=0}^{n}\left\{ begin{ar y}{l n\ k \end{ar y}\right\}(-1)^{n-k}\sum_{i=0}^{k}\left(\begin{ar y}{l k\ i \end{ar y}\right)\frac{(-x)^{i} (k-i+1)^{$\mu$}. (11). ,. (12). ,. \cdot. respectively. The generating function of poly‐Cauchy polynomials is given by (3). The generating functions of two types of incomplete poly‐Cauchy polynomials of the first kind can be also given by using the polylogarithm factorial functions in terms of the mth partial sum of the logarithm function F_{m}(t) .. Theorem 1. For. integers n\geq 1, m\geq 1 and $\mu$\geq 1. we. have. \displayst le\frac{\mathrm{L}\mathrm{j}\int_{$\mu$}(F_{$\tau$n}(t)}{e^xF_{m}(t)}=\sum_{n=0}^{\infty}c_{n,\leqn}^{($\mu$)_{7}(x)\frac{t^n}{n!} \displaystyle\frac{\mathrm{L}\mathrm{i}\mathrm{f}_{$\mu$}(\log(1+t)-F_{rn-1}(t)}{e^{x(\log(1+t)-F_{m}-1(t)}=\sum_{n=0}^{\infty}\mathrm{c}_{n,\geqrn}^{($\mu$)}(x)\frac{t^n}{n!}. ,. .. (13). (14).

(6) 25. Since. Remark.. (13). F_{\infty}(t)=\log(1.+t) and F_{0}(t)=0 if we take m\rightarrow\infty in (14), both of (13) and (14) are reduced to the generating ,. and m=1 in. function in. ([17,. Theorem 3. The generating function of two types of the incomplete poly‐Cauchy poly‐ can be written in the form of iterated integrals.. nomials of the first kind. Corollary. 1. For. $\mu$\geq 1. we. have. \displaystyle\frac{1}{e^{xF_{m}(t)}F_{7n}(t)}(e^{F_{m}(t)}-1)dt_{\check{$\mu$-1}dt\frac{\int_{0}^{t}\frac{1-(t)^{m}{(1+t)F_{rn}(t)}\cdots\int_{0}^{t}\frac{1-(t)^{m}{(1+t)F_{rn}(t)}{$\mu$-1}\cdots =\displayst le\sum_{n=0}^{\infty}c_{n,\leqm}^{($\mu$)}(x)\frac{t^n}{.n!}. (15). ,. 1. e^{x(\log(1+t)-F_{m}-1(t))(}\log(1+t)-F_{rn-1}(t)). \displaystyle\times\int0_{$\mu$-1}^{t}\frac{(-t)^{m-1} {(1+t)(\log(1+t)-F_{m-1}(t) }\ldots\int_{0}^{t}\frac{(-t)^{m-1} {(1+t)(\log(1+t)-F_{rn-1}(t) }\ovalbox{\t \smal REJECT} \times(e(1\circ \mathrm{g}(1+t)-F_{m-1}(t) _{-1)dt_{\check{ $\mu$-1} dt.=\sum_{n=0}^{\infty}c_{n,\geq m}^{( $\mu$)}(x)\frac{t^{n} {n!}. Remark. If. we. take. m\rightarrow\infty. in. (15). and m=1 in. reduced to the generating function in [10, Corollary (1) is reduced to [15, Corollary 1].. are. For. and $\mu$ with n\geq 0 and the second. integers. polynomials of. n. (16),. (16). .. both of. Corollary 1].. (15). and. (16). When x=0,. $\mu$\geq 1 define the poly‐Cauchy. kind\hat{c}_{n}^{\langle $\mu$)}(x)( $\mu$\geq 1). ,. as. \displaystyle \hat{c}_{n}^{\langle $\mu$)}(x)=\int_{0^{1} \ldots\int_{0}^{1}(-t_{1}t_{2}\cdots t_{ $\mu$} +x)(-t_{1}t_{2}\cdot\cdot:t_{ $\mu$}+x-1) 一 $\mu$ .. \hat{c}_{n}^{\langle $\mu$)}(x) \left{\begin{ar y}{l n\ k \end{ar y}\ight}. Then, kind. can. be. expressed. .. .. (-t_{1}t_{2}\cdots t_{ $\mu$}+x-n+1)dt_{1}dt_{2}\cdots dt_{ $\mu$}.. in terms of the. Stirling. numbers of the first. :. \displayst le\hat{c}_{n}^{\langle$\mu$)}(x=\sum_{k=0}^{n}\left\{ begin{ar y}{l n\ k \end{ar y}\right\}(-1)^{n}\sum_{i=0}^{k}\left(\begin{ar y}{l k\ i \end{ar y}\right)\frac{(.-x)^{i} (k-i+1)^{$\mu$}.

(7) 26. ([10,. Theorem 4. Hence, nomials. of. it is natural to define two. the second. kind\hat{c}_{n,\leq m}^{\langle $\mu$)}(x). types of incomplete poly‐Cauchy poly‐. and. \hat{c}_{n,\geq m}^{\langle $\mu$)}(x) by. \displayst le\hat{c}_{n,\leqm}^{\langle$\mu$)}(x)=\sum_{k=0}^{n}\left\{ begin{ar y}{l n\ k \end{ar y}\right\}(-1)^{n}\sum_{i=0}^{k}\left(\begin{ar y}{l k\ i \end{ar y}\right)\frac{(-x)^{i} (k-i+1)^{$\mu$} \displayst le\hat{c}_{n,\geq$\tau$n}^{($\mu$)}(x)=\sum_{k=0}^{n}\left\{ begin{ar y}{l n\ k \end{ar y}\right\}(-1)^{n}\sum_{i=0}^{k}\left(\begin{ar y}{l k\ i \end{ar y}\right)\frac{(-x)^{i} (k-i+1)^{$\mu$} respectively. The generating function. \mathrm{o}\mathrm{f}\hat{c}_{n}^{\langle $\mu$)}(x). is. (17). ,. (18). ,. given by. (1+x)^{z}\displaystyle\mathrm{L}\mathrm{i}\mathrm{f}_{$\mu$}(-\ln(1+t) =\sum_{n=0}^{\infty}\hat{c}_{n}^{\text{く$\mu$}) (x)\frac{t^{n} {n!} ([10,. Theorem. 5].. Note that. z. is. replaced by. -z. (19). ).. The generating functions of two types of incomplete poly‐Cauchy poly‐ nomials of the second kind can be also given by using the polylogarithm. F_{\infty}(t)=\log(1+t) and F_{0}(t)=0 if we take m\rightarrow\infty in (20) and m=1 in (21), both of (20) and (21) are reduced to the generating function in [10, Theorem 5]. On the other hand, if x=0 then Theorem 2 is reduced to [15, Theorem 2]. factorial functions. \mathrm{L}\mathrm{i}\mathrm{f}_{ $\mu$}(z). in terms of. F_{7n}(t). .. Since. ,. ,. Theorem 2. For integers n\geq 1, m\geq 1 and. $\mu$\geq 1. we. have. e^{xF_{m}(t)}\displaystyle\mathrm{L}\mathrm{i}\mathrm{f}_{$\mu$}(-F_{rn}(t)=\sum_{n=0}^{\infty}\hat{c}_{n,\leqn}^{($\mu$)_{7}(x)\frac{t^n}{n!} e^{x(\log(1+t)-F_{m-1}(t) }\displaystyle \mathrm{L}\mathrm{i}\mathrm{f}_{ $\mu$}(-\log(1+qt)+F_{m-1}(t) =\sum_{n=0}^{\infty}\hat{c}_{n,\geq \mathrm{z}n}^{\{ $\mu$)}(x)\frac{t^{n} {n!} The. generating function of. mials of the second kind. can. two. ,. .. (20) (21). types of the incomplete Cauchy polyno‐. be also written in the form of iterated. integrals..

(8) 27. Corollary. 2. For. $\mu$\geq 1. have. we. \displaystyle\frac{e^{xF_{m}(t)}{F_{m}(t)}(1-e^{-F_{m}(t)}dt_{\tilde{$\mu$-1}dt\frac{\int_{0}^{t}\frac{1-(t)^{m}{(1+t)F_{rn}(t)}\cdots\int_{0}^{t}\frac{1-(t)^{m}{(1+t)F_{rn}(t)}{$\mu$-1}\ldots =\displayst le\sum_{n=0}^{\infty}\hat{c}_{n,\leqm}^{\langle$\mu$)}(x\frac{t^n}{n!}. (22). ,. e^{x(\log(1+t)-F_{m-1}(t))}. \overline{\log(1+t)-F_{ $\tau$ n-1}(t)}. \displaystyle\mathrm{x}\int0_{$\mu$-1}^{t}\frac{(-t)^{7n-1}{(1+t)(\log(1+t)-F_{m-1}(t) }\ldots\int_{0}^{t}\frac{(-t)^{m-1}{(1+t)(\log(1+t)-F_{rn-1}(t) }\ovalbox{\t\smal REJECT} \displaystyle\times(1-e^{-\log(1+t)+F_{m-1}(t)} dt_{\check{$\mu$-1} dt=\sum_{n=0}^{\infty}\hat{c}_{n,\geqm}^{($\mu$)_{(r1\frac{t^{n} {n!} \cdots. Remark. If we take are. m\rightarrow\infty. in. (22). and m=1 in. (23),. both of. [10, Corollary 2]. [15, Corollary 2].. reduced to the generating function in. Corollary. 5. 2. are. Some. properties polynomials. It is known that. B_{k}^{(-n)}. reduced to. poly‐Uernoulli. of incomplete. numbers. for n, k\geq 0 [ 11 , Theorem. 2]. satisfy. (23). (22). and. (23). If x=0 then ,. poly‐Cauchy. duality theorem B_{n}^{(-k)}= symmetric formula. the. because of the. \displaystyle\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}B_{n}^{(-k)}\frac{x^{n}y^{k} {n!k}=\frac{e^{x+y} {e^{x}+e^{y}-e^{x+y}. .. (24). However, incomplete poly‐Cauchy polynomials do not satisfy the duality the‐ orem for any integer m\geq 1 by the following results. ,. Theorem 3. For nonnegative integers n, k , and. a. positive integer. m,. we.

(9) 28. have. \displaystyle \sum_{n=0}^{\infty}\sum_{k=0}^{\infty}c_{n,\leq m}^{(-k)}(z)\frac{x^{n} {n!}\frac{y^{k} {k!}=\exp( e^{y}-z)F_{m}(x)+y) \displayst le\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}c_{n,\geqm}^{(-k)}(z\frac{x^n}{n!}\frac{y^k}{k!} =\exp((e^{y}-z)(\log(1+x)-F_{7n-1}(x))+y). Remark. If the first to. [15,.. m\rightarrow\infty. identity. in. Theorem. 4],. in. (25). or. m=1 in. [10, Proposition 1].. where. z. is. (26),. both identities. (25). ,. (26). .. are. reduced to. If z=0 , then Theorem 3 is reduced. replaced by. -z.. Similarly, incomplete poly‐Cauchy polynomials of the second kind have the following properties. If m\rightarrow\infty in (27) or m=1 in (28), both identities If z=0 then are reduced to the second identity in [10, Proposition 1]. Theorem 3 is reduced to [15, Theorem 5], where z is replaced by -z. ,. Theorem 4. For. nonnegative integers. n, k , and. a. positive integer. m_{f}. we. have. \displaystyle \sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\hat{c}_{n,\leq m}^{(-k)}(z)\frac{x^{n}y^{k} {n!k!}=\exp( z-e^{y})F_{m}(x)+y) \displayst le\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\hat{c}_{n,\geq7n}^{(-k)}(z)\frac{x^n}y^{k}{n!k}. =\exp((z-e^{y})(\log(1+x)-F_{m-1}(x))+y). ,. .. (27). (28). By using Theorem 3, we have explicit expressions of incomplete poly‐ Cauchy polynomials of the first kind with negative indices. Theorem 5. For nonnegative integers n, k , and. a. positive integer. m,. we.

(10) 29. have. c_{n,\leqrn}^{(-k)}(z)=\displayst le\sum_{j=0}^{k}\sum_{i=0}^{n}\sum_{\mathrm{t}=0}^{i}j!\left\{ begin{ar y}{l k&+1\ j+1& \end{ar y}\right\} displayst le\tex{ぐ_{}j^{+j})\left(\begin{ar y}{l i\ l \end{ar y}\right)\left\{ begin{ar y}{l n&\ i+&j \end{ar y}\right\}(-1)^{n-i +t_{Z}t,. (29). c_{n,\geqm}^{(-k)}(z=\displayst le\sum_{j=0}^{k}\sum_{i=0}^{n}\sum_{l=0}^{ij!\left\{ begin{ar y}{l k+1\ +1j \end{ar y}\right\} left(\begin{ar y}{l i+&j\ j& \end{ar y}\right)\displayst le\left(\begin{ar y}{l i\ l \end{ar y}\right)\left\{ begin{ar y}{l n&\ i+&j \end{ar y}\right\}(-1)^{n-i +l_{Z}l.. (30). Remark. If 5. are. -z ,. m\rightarrow\infty. (29),. or. if m=1 in. and. [15,. (30),. identity [10, only classical Stirling numbers of both. If z=0 , then in. in. reduced to the first. Theorem. Corollary. we. have the. both identities in Theorem. Theorem. in. following.. These. 8],. kinds. are. where are. z. is. replaced by. used.. different expressions. seen. 6].. 3. For. nonnegative integers. n, k , and. a. positive integer. m,. we. have. c_{n,\leqm}^{(-k)}=\displayst le\sum_{j=0}^{k}.\sum_{i=0}^{n}j!\left\{ begin{ar y}{l k+1\ j+\mathrm{l} \end{ar y}\right\} left(\begin{ar y}{l i+&j\ j& \end{ar y}\right)\displayst le\left\{ begin{ar y}{l n&\ i+&j \end{ar y}\right\}(-1)^{n-ij} c_{n,\geqm}^{(-k)}=\displayst le\sum_{j=0}^{k}\sum_{i=0}^{n}j!\left\{ begin{ar y}{l k+1\ j+1 \end{ar y}\right\} left(\begin{ar y}{l i+&j\ j& \end{ar y}\right)\displayst le\left\{ begin{ar y}{l n&\ i+&j \end{ar y}\right\}(-1)^{n-ij}. (31). ,. (32). .. Similarly, using Theorem 4, we have explicit expressions of incomplete poly‐Cauchy polynomials of the second kind with negative indices. If m\rightarrow\infty in. (33),. second. or. if m=1 in. identity classical Stirling. in. [10,. (34),. any. Theorem. identity. 8],. where. numbers of both kinds. Theorem 6. For. nonnegative integers. are. in Theorem 6 is reduced to the z. is. replaced by. -z ,. and. only. used.. n, k , and. a. positive integer. m,. we. have. \displayst le\hat{c}_n,\leqm}^{(-k)}(z=\sum_{j=0}^{k}\sum_{i=0}^{n}\sum_{l=0}^{ij!\left\{ begin{ar y}{l k+1\ j+1 \end{ar y}\right\} left(\begin{ar y}{l i+&j\ j& \end{ar y}\right)\displayst le\left(\begin{ar y}{l i\ l \end{ar y}\right)\left\{ begin{ar y}{l n&\ i+&j \end{ar y}\right\}(-1)^{n-l_{Z}l \displaystle\hat{c}_n,\geq,n}^{\langle-k)}(\mathrm{z})=\sum_{j=0}^{k\sum_{i=0}^{n\sum_{l=0}^{ij!\left\{ begin{ar y}{l k+\mathrm{l}\ j+\mathrm{l} \end{ar y}\right\} left(\begin{ar y}{l i+&j\ j& \end{ar y}\right)\displaystle\ ft(\begin{ar y}{l i\ l \end{ar y}\right)\left\{ begin{ar y}{l n&\ i+&j \end{ar y}\right\}(-1)^{n-l_{Z}l. ,. .. (33) (34).

(11) 30. If z=0 , then then seen. in. [15,. Corollary. Theorem 4. For. we. have the. following.. These. are. different expressions. 7].. nonnegative integers. n, k , and. a. positive integer. m,. we. have. \displaystle\hat{\mathrm{c}_{n,\leqrn}^{\langle-k)}=(-1)^{n}\sum_{j=0}^{k\sum_{i=0}^{nj!\left\{ begin{ar y}{l k&+1\ j\cdot+1& \end{ar y}\right\} displaystle\ ft(\begin{ar y}{l i+&j\ j& \end{ar y}\right)\left\{ begin{ar y}{l n&\ i+&j \end{ar y}\right\} \displayst le\hat{c}_n,\geqm}^{\-k)}=(-1)^{n}\sum_{j=0}^{k}\sum_{i=0}^{n}j!\left\{ begin{ar y}{l k+1\ j+1 \end{ar y}\right\} left(\begin{ar y}{l i+&j\ j& \end{ar y}\right)\displayst le\left\{ begin{ar y}{l n&\ i+&j \end{ar y}\right\}. (36). Incomplete poly‐Bernoulli polynomials. 6 In. (35). [6], poly‐Bernoulli polynomials B_{n}^{( $\mu$)}(x). are. defined in. (1).. In. [1, 2],. x. and. interchanged in the definition of B_{n}^{( $\mu$)}(x) In [3], an extended concept named poly‐Bernoulli polynomials with a q parameter is introduced. In [16], still different poly‐Bernoulli polynomials B_{n}^{( $\mu$)}(x) are defined. Poly‐Bernoulli polynomials have an explicit expression in terms of the Stirling numbers of -x are. .. the second kind:. B_{n}^{($\mu$)}(x)=\displayst le\sum_{k=0}^{n}\frac{(-1)^{n-k} !{(k+1)^{$\mu$}\sum_{l=0}^{n}\left(\begin{ar y}{l n\ l \end{ar y}\right)\left\{ begin{ar y}{l l\ k \end{ar y}\right\}x^{n-l} When x=0 ,. we. get the expression of poly‐Bernoulli numbers. (37). B_{n}^{( $\mu$)}(0)=B_{n}^{( $\mu$)}. :. B_{n}^{($\mu$)}=\displayst le\sum_{k=0}^{n}\frac{(-1)^{n-k} !{(k+1)^{$\mu$}\left\{ begin{ar y}{l n\ k \end{ar y}\right\} ([11,. Theorem 1. By using two types Bernoulli polynomials. B_{n,\geq $\tau$ n}^{( $\mu$)}.(x) by. incomplete Stirling numbers, define restricted poly‐ B_{n,\leq m}^{( $\mu$)}(x) and associated poly‐Bernoulli polynomials. of. B_{n,\leq rn}^{( $\mu$)}(x)=\displaystyle \sum_{k=0}^{n}\frac{(-1)^{n-k} !}{(k+1)^{ $\mu$} \sum_{t=0}^{n}\left(\begin{ar ay}{l} n\ l \end{ar ay}\right) (n\geq 0). (38).

(12) 31. and. B_{n,\geqrn}^{($\mu$)}(x)=\displaystyle\sum^{n}\frac{(-1)^{n-k} !{(k+1)^{$\mu$}\sum^{n}k=0l=0\left(\begin{ar ay}{l n\ l \end{ar ay}\right)\left\{ begin{ar ay}{l l\ k \end{ar ay}\right\}x^{n-t}(n\geq0). (39). ,. incomplete poly‐Bernoulli polynomials. they are reduced to restricted poly‐Bernoulli numbers and associated poly‐Bernoulli numbers, respectively ([18]).. respectively.. We call these numbers. as. If x=0 , then One. can. deduce that the numbers. erating functions by using. the. B_{n,<m}^{( $\mu$)}(x). and. polylogarithms.. B_{n,\geq m}^{( $\mu$)}(x) have the gen‐. Theorem 7. We have. \displayst le\frac{\mathrm{L}\mathrm{i}_{$\mu$}(1-E_{m}(-t)}{1-E_{\ovalbox{\t smal REJ CT}n(-t)}e^{-tx}=\sum_{n=0}^{\infty}B_{n,\leqm}^{($\mu$)}(x)\frac{t^n}{n!}. (40). and. \displaystyle\frac{\mathrm{L}\mathrm{i}_{$\mu$}(E_{m-1}(-t)e^{-t}) {E_{m-1}(-t)e^{-i} e^{-tx}=\sum_{n=0}^{\infty}B_{n,\geqm}^{($\mu$)}(x)\frac{t^{n} {n!} Remark. Theorem. If. If x=0 ,. 1].. m\rightarrow\infty. function in. [3,. in. (40). they or. Theorem. are. reduced to the. m=1 in. (41), they. (41). .. generating functions. are. reduced to the. in. [18,. generating. 2.1] by putting q=1.. For $\mu$\geq 1 the generating functions can be written in the form of iterated integrals. We set E_{-1}(-t)=0 for convemence. ,.

(13) 32. Theorem 8.. \displaystyle\frac{e^{-tx}{1-E_{m}(-t)}\cdot(-\log(E_{7n}(-t) dt_{\check{$\mu$-1}dt\frac{\int_{0}^{t}\frac{E_{m-1}(-t)}{1-E_{$\tau$n}(-t)}\cdots\int_{0}^{t}\frac{E_{7n-1}(-t)}{1-E_{m}(-t)}{$\mu$-1}\ldots =\displayst le\sum_{n=0}^{\infty}B_{n,\leqn}^{($\mu$)_{7}(x)\frac{t^n}{n!}. (42). ,. \displayst le\frac{e^{-tx}{E_{m-1}(-t)e^{-t}.\int0_{$\mu$-1}^{t}\frac{e^{-t}E_{7n-2}(-t)}{E_{m-1}(-t)e^{-t}\ldots\int_{0}^{t}\frac{e^{-t}E_{m-2}(-t)}{E_{m-1}(-t)e^{-t}\ovalbox{\t smal REJ CT} \times(-\log(1+e^{-t}-E_{rn-1}(-t) )dt_{\check{ $\mu$-1} dt. =\displayst le\sum_{n=0}^{\infty}B_{n,\geqm}^{($\mu$)}(x)\frac{t^n}{n!}. (43). .. Remark. If x=0 , then Theorem 8 is reduced to if. m\rightarrow\infty. reduced to. [18,. Theorem. (43), (42), by E_{\infty}(-t)=e^{-t} the iterated form (2) in ([11]) by setting t=0.. in. ,. and if m=1 in. 2].. In. addition,. both of them. symmetric property does not hold for generalized poly‐Bernoulli bers with negative indices because of the following. A. are. num‐. Theorem 9. We have. n=0\displaystyle \sum^{\infty}\sum_{k=0}^{\infty}B_{n,\leq.m}^{(-k)}(z)\frac{x^{n} {n!}\frac{y^{k} {k!}=\frac{e^{y-xz} {1-e^{y}(1-E_{m}(-x) } \displaystyle \sum_{n=0}^{\infty}\sum_{k=0}^{\infty}B_{n,\geq rn}^{(-k)}(z)\frac{x^{n}y^{k} {n!k }=\frac{e^{y-xz} {1-e^{y}(E_{m-1}(-x)-e^{-x}). (44). ,. Remark. If. (24). m\rightarrow\infty. in. (44). or. m=1 in. (45),. both identities. .. are. (45).. reduced to. with z=0.. References. [1]. A.. Bayad and \mathrm{Y} : Hamahata, Polylogarthms mials, Kyushu J. Math. 65 (2011), 15‐24.. and. poly‐Bemoulli polyno‐.

(14) 33. [2].. A.. Bayad and Y. Hamahata, Arakawa‐Kaneko L ‐functions and gener‐ poly‐Bernoulli polynomials, J. Number Theory 131 (201\cdot 1) 1020‐. alized. ,. 1036.. [3]. M. Cenkci and T.. with. a. q. Komatsu, Poly‐Bemoulli numbers and polynomials parameter, J. Number Theory 152 (2015),. 38‐54.. [4]. C. A.. [5]. L.. [6]. M.‐A.. [7]. R. L. Graham, D. E. Knuth and O. Patashnik, Concrete 2nd edn. Addison‐Wesley, Reading, MA, 1994.. [8\mathrm{J}. F. T.. [9]. Ch.. Charalambides, Enumerative Combinatorics (Discrete Mathemat‐ Applications), Chapman and HaJl/CRC, 2002.. ics and Its. Comtet, Advanced Combinatorics, Reidel, Dordrecht,. Coppo and B. Candelpergher, The tions, Ramanujan J. 22 (2010), 153‐162.. 1974.. Arakawa‐Kaneko zeta. func‐. Mathematics,. Howard, Associated Stirling numbers, Fibonacci Quart.. (1980),. 18. 303‐315.. Jordan, Calculus of finite differences, Chelsea Publ. Co.,. New. York,. 1950.. [10]. K. Kamano and T.. Komatsu, Poly‐Cauchy polynomials, Mosc.. Number. (2013),. [11]. M.. [12]. T.. Theory. 3. J. Comb.. 183‐209.. Kaneko, Poly‐Bernoulli numbers,. (1997),. J Th. Nombres Bordeaux 9. 221‐228.. Komatsu, Poly‐Cauchy numbers, Kyushu J. Math.. 67. (2013),. 143‐. 153.. [13] [14]. T.. Komatsu, Poly‐Cauchy numbers with. 31. (2013),. T.. Komatsu, Hypergeometnc Cauchy numbers,. 9. [15]. T.. (2013),. a. q. parameter, Ramanujan J.. 353‐371. Int. J. Number. Theory. 545‐560.. Komatsu, Incomplete poly‐Cauchy numbers, Monatsh. Math.,. pear.. to ap‐.

(15) 34. [16]. Luca, Some relationships between poly‐Cauchy num‐ numbers, Ann. Math. Inform. 41 (2013), 99‐105.. T. Komatsu and F.. bers and poly‐Bernoulh. [17]. T.. Komatsu,. Liptai, A generalization of poly‐ properties, Abstr. Appl. Anal. 2013 (20\cdot 13). V. Laohakosol and K.. numbers and their. ,. Article ID 179841, 8 pages.. [18]. T.. \acute{}. Liptai and I. Mezó Incomplete poly‐Bernoulli numbers associated with incomplete Stirling numbers, Publ. Math. Debrecen, to Komatsu,. K.. ,. appear.. [19]. T.. [20]. T. Komatsu and L.. [21]. I. Mezó ,. [22]. \acute{}. Komatsu, I. Mezó and L. Szalay, Incomplete Cauchy numbers, Period. Math. Hungar., to appear. Math. J. 54. (2014),. Szalay, Shifted poly‐Cauchy numbers,. Lithuanian. 166‐181.. \acute{}. J.. Periodicity of the last digits of Integer Seq. 17 (2014), Article 14.1.1.. J.. Riordan, Combinatorial identities, John Wiley & Sons,. some. combinatorial sequences,. New. York,. 1968.. [23]. F.‐Z.. Zhao, Some properties of associated Stirling numbers, J. Integer Seq. 11 (2008), Article 08.1.7, 9 pages.. School of Mathematics and Statistics Wuhan. University. Wuhan 430072. CHINA address: komatsu@whu.edu.cn \mathrm{E}\left‐mail ar ow.

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