A symbolic operator approach to several summation formulas for power series

This paper deals with the summation problem of power series of the form S a b ( f ; x ) = ∑ a ⩽ k ⩽ b f ( k ) x k , where 0 ⩽ a < b ⩽ ∞ , and { f ( k ) } is a given sequence of numbers with k ∈ [ a , b ) or f ( t ) is a differentiable function defined on [ a , b ) . We present a symbolic summation operator with its various expansions, and construct several summation formulas with estimable remainders for S a b ( f ; x ) , by the aid of some classical interpolation series due to Newton, Gauss and Everett, respectively.