Truncated Homogeneous Symmetric Functions

Abstract

Extending the elementary and complete homogeneous symmetric functions, we introduce the truncated homogeneous symmetric function $h_{\lambda}^{\dd}$ in $(\ref{THSF})$ for any integer partition $\lambda$, and show that the transition matrix from $h_{\lambda}^{\dd}$ to the power sum symmetric functions $p_\lambda$ is given by \[M(h^{\dd},p)=M'(p,m)z^{-1}D^{\dd},\] where $D^{\dd}$ and $z$ are nonsingular diagonal matrices. Consequently, $\{h_{\lambda}^{\dd}\}$ forms a basis of the ring $\Lambda$ of symmetric functions. In addition, we show that the generating function $H^{\dd}(t)=\ssum_{n\ge 0}h_n^{\dd}(x)t^n$ satisfies \[\omega(H^{\dd}(t))=\left(H^{\dd}(-t)\right)^{-1},\] where $\omega$ is the involution of $\Lambda$ sending each elementary symmetric function $e_\lambda$ to the complete homogeneous symmetric function $h_\lambda$.