On biorthogonal polynomials suggested by the Jacobi polynomials (Q1067058)

\textit{H. C. Madhekar} and the reviewer [Pac. J. Math. 100, 417-424 (1982; Zbl 0456.33009)] constructed the pair of biorthogonal polynomials \(J_ n^{(\alpha,\beta)}(x;k)\) and \(K_ n^{(\alpha,\beta)}(x;k)\) with respect to the Jacobi weight function \((1- x)^{\alpha}(1+x)^{\beta}\quad (Re \alpha >-1,\quad Re \beta >-1),\) though earlier others had discussed the first set in different contexts. \textit{L. Toscano} [Mathematiche 15, 41-53 (1960; Zbl 0154.065)] had also studied \(J_ n^{(\alpha,\beta)}(x;k)\) by defining them as a Rodrigues type formula involving the difference operator \(\Delta_ k\). In this note the author gives a recurrence relation and a shift operator formula for the first set \(J_ n^{(\alpha,\beta)}(x,k)\) using \(\Delta_ k\).