On Generalized p , q -Euler Matrix and Associated Sequence Spaces

In this study, we introduce new BK -spaces b s r , t p , q and b ∞ r , t p , q derived by the domain of p , q -analogue B r , t p , q of the binomial matrix in the spaces ℓ s and ℓ ∞ , respectively. We study certain topological properties and inclusion relations of these spaces. We obtain a basis for the space b s r , t p , q and obtain Köthe-Toeplitz duals of the spaces b s r , t p , q and b ∞ r , t p , q . We characterize certain classes of matrix mappings from the spaces b s r , t p , q and b ∞ r , t p , q to space μ ∈ ℓ ∞ , c , c 0 , ℓ 1 , b s , c s , c s 0 . Finally, we investigate certain geometric properties of the space b s r , t p , q .

New Generalized Apostol-Frobenius-Euler polynomials and their Matrix Approach

In this paper, we introduce a new extension of the generalized Apostol-Frobenius-Euler polynomials ℋn[m−1,α](x; c,a; λ; u). We give some algebraic and differential properties, as well as, relationships between this polynomials class with other polynomials and numbers. We also, introduce the generalized Apostol-Frobenius-Euler polynomials matrix ????[m−1,α](x; c,a; λ; u) and the new generalized Apostol-Frobenius-Euler matrix ????[m−1,α](c,a; λ; u), we deduce a product formula for ????[m−1,α](x; c,a; λ; u) and provide some factorizations of the Apostol-Frobenius-Euler polynomial matrix ????[m−1,α](x; c,a; λ; u), which involving the generalized Pascal matrix.

On the degree of approximation of functions belonging to a Lipschitz class by Hausdorff means of its Fourier series

In a recent paper Lal and Yadav [1] obtained a theorem on the degree of approximation for a function belonging to a Lipschitz class using a triangular matrix transform of the Fourier series representation of the function. The matrix involved was the product of $ (C, 1) $, the Cesaro matrix of order one, with $ (E, 1) $, the Euler matrix of order one. In this paper we extend this result to a much wider class of Hausdorff matrices.