One-parameter family of Neville-Aitken algorithm on \(q\)-triangle (Q5952800)

Let \(n\) be a positive integer and \(q>0\), \(q\neq 1\). Then \[ S= \left\{\left( {1-q^i\over 1-q},{1-q^j \over 1-q}\right): i,j\geq 0,i+j\leq n \right\} \] is referred to as a `\(q\)-triangle'. The paper is concerned with interpolation of a function of two variables at the points of \(S\). In an earlier paper (On the polynomial interpolation at points of a geometric progression, SEAMS-GMU Proc. Math. Anal. \& Stats Conference, Yogyakarta 42-51 (1995)) the authors gave a forward difference formula for interpolation on \(S\) by a polynomial in two variables of degree at most \(n\). This paper gives a Lagrange type formula which provides a polynomial of degree at most \(2n\) which interpolates at the points of \(S\), and it is shown that it is also given by a generalised Neville-Aitken algorithm.