In the paper, we introduce a quantum random walk polynomial (QRWP) that can be defined as a polynomial {Pn(x)}\documentclass[12pt]{minimal}
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\begin{document}$$\{P_{n}(x)\}$$\end{document}, which is orthogonal with respect to a quantum random walk measure (QRWM) on [-1,1]\documentclass[12pt]{minimal}
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\begin{document}$$[-1, 1]$$\end{document}, such that the parameters αn,ωn\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _{n},\omega _{n}$$\end{document} are in the recurrence relations Pn+1(x)=(x-αn)Pn(x)-ωnPn-1(x)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} P_{n+1}(x)= (x - \alpha _{n})P_{n}(x) - \omega _{n}P_{n-1}(x) \end{aligned}$$\end{document}and satisfy αn∈R,ωn>0\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _{n}\in \mathfrak {R},\omega _{n}> 0$$\end{document}. We firstly obtain some results of QRWP and QRWM, in which case the correspondence between measures and orthogonal polynomial sequences is one-to-one. It shows that any measure with respect to which a quantum random walk polynomial sequence is orthogonal is a quantum random walk measure. We next collect some properties of QRWM; moreover, we extend Karlin and McGregor’s representation formula for the transition probabilities of a quantum random walk (QRW) in the interacting Fock space, which is a parallel result with the CGMV method. Using these findings, we finally obtain some applications for QRWM, which are of interest in the study of quantum random walk, highlighting the role played by QRWP and QRWM.
