A New Approach to Multivariate q-Euler Polynomials Using the Umbral Calculus

We derive numerous identities for multivariate q-Euler polynomials by using the umbral calculus. 1 1 Preliminaries Throughout this paper, we use the following notation, where C denotes the set of complex numbers, F denotes the set of all formal power series in the variable t over C with F = { f(t) = ∑∞ k=0 ak t k! | ak ∈ C } , P = C[x] and P denotes the vector space of all linear functional on P , 〈L | p(x)〉 denotes the action of the linear functional L on the polynomial p(x), and it is well-known that the vector space operation on P is defined by 〈L+M | p(x)〉 = 〈L | p(x)〉+ 〈M | p(x)〉 , 〈cL | p(x)〉 = c 〈L | p(x)〉 , where c is some constant in C (for details, see [5, 6, 8, 11]). The formal power series are known by the rule f(t) = ∞ ∑ k=0 ak t k! ∈ F which defines a linear functional on P as 〈f(t) | x〉 = an for all n ≥ 0 (for details, see [5, 6, 8, 11]]). Additionally,