The contiguous function relations for the basic hypergeometric series (Q915917)

The contiguous function relations for the basic hypergeometric function \({}_ r\phi_ s(z)\) comprise (i) \(r+s-1\) relations linking \({}_ r\phi_ s(z)\) and two contiguous functions, and (ii) \(r+1\) relations linking \({}_ r\phi_ s(z)\) and \(s+1\) contiguous functions. The variable z appears explicitly only in the latter ones, and in fact linearly. The relations are analogues of Rainville's contiguous function relations for \({}_ rF_ s(z)\), to which they reduce to the limit \(q\to 1\). The results are established by suitable series manipulations. An example is, in the customary notation, \[ (a_ 1-b_ kq^{-1})\phi =(1-b_ kq^{-1})a_ 1\phi (b_ k-)-(1-a_ 1)b_ kq^{-1}\phi (a_ 1+),\quad k\in \{1,...,s\}. \] As an application, q-Laguerre polynomials are considered.