# Indices of Turing machine on different arity inputs

Let $$\Sigma$$ be an alphabet, and denote by $$\mathrm{ind}(\varphi_M^{(k)})$$ the index set (w.r.t some numbering) of the $$k$$-ary partial computable function $$\varphi_M^{(k)} : (\Sigma^*)^k \rightarrow \Sigma^*$$, which represents the $$k$$-ary proper function of the Turing machine $$M$$. Is it the case that $$\mathrm{ind}(\varphi_M^{(k)}) \subseteq \mathrm{ind}(\varphi_M^{(k + 1)})$$ for every $$k \geq 1$$? I think this is true simply because every $$k$$-ary input to $$M$$ is realizable as a $$(k+1)$$-ary input to $$M$$ but with one input word blank. However, to formally show the inclusion is evading me. I thought applying the $$S^m_n$$ theorem would be helpful, but I think that merely establishes the inclusion for some Turing machine (not necessarily $$M$$). Any help would be appreciated.

You can define a machine implementing the computable function $$(\Sigma^*)^k \rightarrow (\Sigma^*)^{k+1}$$ for adding the blank to the set of inputs, then compose the machine implementing this function with your machine taking $$k+1$$ inputs to produce the required machine with $$k$$ inputs.
The $$S^m_n$$ theorem can be used to construct the index for the composed machine.