Barnes-type Narumi polynomials

In this paper, we study the Barnes-type Narumi polynomials with umbral calculus viewpoint. From our study, we derive various identities of the Barnes-type Narumi polynomials.MSC:05A19, 05A40, 11B68.


Introduction
As is well known, the Narumi polynomials of order α are defined by the generating function to be Let r ∈ Z > . We consider the polynomials N n (x|a  ,...,a r )a n dN n (x|a  ,...,a r ), respectively, called the Barnes-type Narumi polynomials of the first kind and those of the second kind and respectively given by In the previous paper [], N (α) n (x)wasdenotedbyN (-α) n and called the Narumi polynomial of order α.
The Bernoulli polynomials are defined by the generating function to be Let C be the complex number field and let F be the set of all formal power series in the variable t: Let P = C[x]a n dl e tP * be the vector space of all linear functionals on P. L|p(x) denotes the action of the linear functional L on p(x) which satisfies L + M|p(x) = L|p(x) + M|p(x) ,and cL|p(x) = c L|p(x) ,wherec is a complex constant. The linear functional f (t)|· on P is defined by f (t)|x n = a n (n ≥ ), where f (t) ∈ F . Thus, we note that t k |x n = n!δ n,k (n, k ≥ ), where δ n,k is the Kronecker symbol (see [-] For f L (t)= ∞ k= L|x k k! t k ,w eh a v e f L (t)|x n = L|x n .S o ,t h em a pL → f L (t)i sav e c t o r space isomorphism from P * onto F .H e n c e f o rt h ,F denotes both the algebra of formal power series in t and the vector space of all linear functionals on P, and so an element f (t)o fF will be thought of as both a formal power series and a linear functional. We call F the umbral algebra. The order o(f (t)) of a power series f (t) =  is the smallest integer for which the coefficient of t k does not vanish. If o(f (t)) = , then f (t) is called a delta series; if o(f (t)) = , then g(t) is called an invertible series. Let f (t), g(t) ∈ F with o(f (t)) =  and o(g(t)) = . Then there exists a unique sequence s n (x)(deg s n (x)=n)such that g(t)f (t) k |s n (x) = n!δ n,k for n, k ≥ . The sequence s n (x) is called the Sheffer sequence for (g(t), f (t)) which is denoted by s n (x) ∼ (g(t), f (t)).
For f (t), g(t) ∈ F and p(x) ∈ P,wehave and From (), we can derive the following equation (): Let s n (x) ∼ (g(t), f (t)). Then the following will be used: and Let us assume that s n (x) ∼ (g(t), f (t)) and r n (x) ∼ (h(t), l(t)). Then we have where From (), ()and(), we note that In this paper, we study the Barnes-type Narumi polynomials with umbral calculus viewpoint. From our study, we derive various identities of the Barnes-type Narumi polynomials.

Barnes-type Narumi polynomials
and we recall (). Thus, we havê N n (x|a  ,...,a r )= r j= e a j t - te a j t (x) n = e -r j= a j t r j= e a j t - t (x) n = e -r j= a j t N n (x|a  ,...,a r ) Therefore, by (), ()and(), we obtain the following theorem.
Now, we observe that where r -r j= a j te a j t e a j t - = r j= i =j (e a i t -){e a j t --a j te a j t } r j= (e a j t -) has at least the order . By ()and(), we get From (), we note that, for n ≥ , N n (y|a  ,...,a r ) Now, we observe that is a series with order greater than or equal to . By ()and(), we get Now we compute the following formula () in two different ways: On the one hand, On the other hand, x n-l From ()and(), we note that