I want to figure out the trace of gamma matrices relating with $\gamma^{(d+1)}$ for even $d$ dimensional case.
First define $\gamma^{(d+1)}$ as \begin{align} \gamma^{(d+1)} = \gamma^1 \gamma^2 \cdots \gamma^d \end{align} What i want to obtain is \begin{align} tr[ \gamma^{(d+1)} \gamma^{\nu_1 \cdots \nu_n}] = \textrm{something} \end{align} by restricting $0 \leq n \leq d$, i think, there is some formula related with this.
What i know is \begin{align} tr[\gamma^{(d+1)}]=0 \end{align} I think there is some general formula related with \begin{align} \gamma^{(d+1)} \gamma^{\nu_1 \cdots \nu_n} =\cdots \end{align}
If you know the formula, please let me know. Actually i know for $d=4$, $tr[\gamma^5 \gamma^{\mu_1} \gamma^{\mu_2}\gamma^{\mu_3}\gamma^{\mu_4}] = -4i\epsilon^{\mu_1 \mu_2 \mu_3\mu_4}$ and of course the $6$ product($n>d=4$) of gamma matrices can be achieved recursively, but i am not sure about its generalization.