Truncated Homogeneous Symmetric Functions

Extending the elementary and complete homogeneous symmetric functions, we introduce the truncated homogeneous symmetric function $h_{\lambda }^{\text{dd}}$ in $(\text{ref}THSF)$ for any integer partition $\lambda$, and show that the transition matrix from $h_{\lambda }^{\text{dd}}$ to the power sum symmetric functions $p_{\lambda }$ is given by $$M(h^{\text{dd}},p)=M^{\mathrm{\prime }}(p,m)z^{-1}{D}^{\text{dd}},$$ where $D^{\text{dd}}$ and $z$ are nonsingular diagonal matrices. Consequently, $\{h_{\lambda }^{\text{dd}}\}$ forms a basis of the ring $\mathrm{\Lambda }$ of symmetric functions. In addition, we show that the generating function $H^{\text{dd}}(t)={\text{ssum}}_{n\ge 0}{h}_n^{\text{dd}}(x)t^n$ satisfies $$\omega (H^{\text{dd}}(t))={(H^{\text{dd}}(-t))}^{-1},$$ where $\omega$ is the involution of $\mathrm{\Lambda }$ sending each elementary symmetric function $e_{\lambda }$ to the complete homogeneous symmetric function $h_{\lambda }$.