Abstract
The first aim of this paper is to construct new generating functions for the generalized {\lambda}-Stirling type numbers of the second kind, generalized array type polynomials and generalized Eulerian type polynomials and numbers, attached to Dirichlet character. We derive various functional equations and differential equations using these generating functions. The second aim is provide a novel approach to deriving identities including multiplication formulas and recurrence relations for these numbers and polynomials using these functional equations and differential equations. Furthermore, by applying p-adic Volkenborn integral and Laplace transform, we derive some new identities for the generalized {\lambda}-Stirling type numbers of the second kind, the generalized array type polynomials and the generalized Eulerian type polynomials. We also give many applications related to the class of these polynomials and numbers.
Highlights
The first aim of this paper is to construct new generating functions for the generalized λ-Stirling type numbers of the second kind, generalized array type polynomials and generalized Eulerian type polynomials and numbers
In the literature, one can find extensive investigations related to the generating functions for the Bernoulli, Euler and Genocchi numbers and polynomials and their generalizations, the λ-Stirling numbers of the second kind, the array polynomials and the Eulerian polynomials, related to nonnegative real parameters, have not been studied yet
We investigate and introduce fundamental properties and many new identities for the generalized λ-Stirling type numbers of the second kind, the generalized array type polynomials and the generalized Eulerian type polynomials and numbers
Summary
On both sides of the above equation, we arrive at the desired result. Generating functions for the polynomial Svn(x; a, b, c; λ) can be defined as follows. The generalized array type polynomials Svn(x; a, b; λ) are defined by means of the following generating function: gv(x, t; a, b; λ) =. By using ( ) and binomial theorem, we arrive at the desired result. Which yields the generating function for the array polynomials Svn(x) studied by Chang and Ha [ ]; see (cf [ , ]). Comparing the coefficients of tn on both sides of the above equation, we arrive at the desired result. ∂j ∂ xj gv(x, t; tj(ln b)j gv(x, t; λ) From this equation, we arrive at higher order derivative of the array type polynomials by the following theorem.
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