Some Summation Formulae Over The Set Of Partitions

. Some algebraic identities with independent variables are established by means of the calculus on formal power series. Applications to special functions and classical polynomials are also demonstrated. 1. Introduction Let oe(n) denote the set of partitions of n (a nonnegative integer) usually denoted by 1 k1 2 k2 : : : n kn with P ik i = n; k i is, of course, the number of partitions of size i. If the number of parts for the partition set of n is restricted to k, i.e., P k i = k; then the corresponding subset of oe(n) is denoted by oe(n; k). For nonnegative integral vector ¯ k = (k 1 ; : : : ; kn ), the multinomial coefficient \Gamma x ¯ k \Delta as usual, is defined by (1.1) ` x ¯ k ' = (x) j ¯ kj Q (k i ) ! ; where the finite product Q runs over i from 1 to n, (x) k stands for the all factorial notation, and j ¯ kj represents the coordinate sum for the vector ¯ k = (k 1 ; : : : ; kn ). In his recent paper, Chu [3] obtained a useful algebraic identity in ..