Fully degenerate poly-Bernoulli numbers and polynomials

In this paper, we introduce the new fully degenerate poly-Bernoulli numbers and polynomials and investigate some properties of these polynomials and numbers. From our properties, we derive some identities for the fully degenerate poly-Bernoulli numbers and polynomials.


Introduction
It is well known that the Bernoulli polynomials are defined by the generating function t e t 1 e xt D 1 X nD0 B n .x/ t n nŠ ; .see / : and B 0 D 1; B n .1/ B n D ı 1;n ; .n 2 N/ ; .see [1,19] where ı n;k is the Kronecker's symbol.
From (10), we note that By (1) and (11), we see that .n 0/ : The classical polylogarithm function Li k .x/ is defined by .k 2 Z/ ; .see [10,11]/ : It is known that the poly-Bernoulli polynomials are defined by the generating function ; .see [9,10,12]/ : When k D 1, we have By (14), we easily get B .1/ n .x/ D B n .x C 1/ ; .n 0/ : .k/ n .0/ are called the poly-Bernoulli numbers. In this paper, we introduce the new fully degenerate poly-Bernoulli numbers and polynomials and inverstigate some properties of these polynomials and numbers. From our investigation, we derive some identities for the fully degenerate poly-Bernoulli numbers and polynomials.

Fully degenerate poly-Bernoulli polynomials
For k 2 Z, we define the fully degenerate poly-Bernoulli polynomials which are given by the generating function When x D 0,ˇ. k/ n; Dˇ. k/ n; .0/ are called the fully degenerate poly-Bernoulli numbers. From (13) and (15), we have Thus, we get lim !0ˇ.
and lim where .x/ n D x .x 1/ .x n C 1/ D P n lD0 S 1 .n; l/ x l .
From (15), we can derive the following equation: Thus, by (21), we get where S 2 .n; l/ and S 1 .n; l/ are the Stirling numbers of the second kind and of the first kind, respectively. Therefore, by (22), we obtain the following theorem.
From (12), we can easily derive the following equation: Thus, by (23), the generating function of the fully degenerate poly-Bernoulli numbers is also written in terms of the following iterated integral: k/ n; t n nŠ : For k D 2, we have Therefore, by (25), we obtain the following theorem. . / n l C 1 : From (15), we have Therefore, by (26), we obtain the following theorem. From (23), we have On the other hand, k/ n; t n 1 .n 1/Š : By (27) and (28), we get  Therefore, by (30), we obtain the following theorem. Now, we observe that k/ n; t n nŠ : By (31), we get . 1/ j Cm j Š n m S 2 .m; j / S 1 .n; m/ .j C 1/ k 1 A x n nŠ y k kŠ : Therefore, by (32), we obtain the following theorem.

Further remarks
Let C be complex number field and let F be the set of all formal power series in the variable t over C with Let P be the algebra of polynomials in a single variable x over C and let P be the vector space of all linear functionals on P. The action of linear functional L 2 P on a polynomial p .x/ is denoted by h Lj p .x/i ; and linearly extended as˝c kŠ , we define a linear functional on P by setting f .t/j x n˛D a n for all n 0: Thus, by (39), we get D t kˇxn E D nŠı n;k ; .n; k 0/ ; .see [4,16,20]/ : For f L .t/ D P 1 kD0˝L j x k˛t k kŠ , by (40), we get h f L .t /j x n i D h Lj x n i : In addition, the mapping L 7 ! f L .t/ is a vector space isomorphism from P onto F. Henceforth, F denotes both the algebra of the formal power series in t and the vector space of all linear functionals on P and so an element f .t / of F can be regarded as both a formal power series and a linear functional. We refer to F umbral algebra. The umbral calculus is the study of umbral algebra (see [5,15,20]). The order o .f .t// of the non-zero power series f .t/ is the smallest integer k for which the coefficient of t k does not vanish.