Dual generalized Bernstein basis

Abstract

The generalized Bernstein basis in the space ${\mathrm{\Pi }}_n$ of polynomials of degree at most n, being an extension of the q-Bernstein basis introduced by Philips [Bernstein polynomials based on the q-integers, Ann. Numer. Math. 4 (1997) 511–518], is given by the formula [S. Lewanowicz, P. Woźny, Generalized Bernstein polynomials, BIT Numer. Math. 44 (2004) 63–78],

$$B_i^n(x;\omega |q)≔\frac{1}{(\omega ;q{)}_n}{\left[\begin{array}{c}\hfill n\hfill \\ \hfill i\hfill \end{array}\right]}_q{x}^i(\omega x^{-1};q{)}_i(x;q{)}_{n-i}(i=0,1,\dots ,n)\text{.}$$

We give explicitly the dual basis functions $D_k^n(x;a,b,\omega |q)$ for the polynomials $B_i^n(x;\omega |q)$, in terms of big q-Jacobi polynomials $P_k(x;a,b,\omega /q;q)$, a and b being parameters; the connection coefficients are evaluations of the q-Hahn polynomials. An inverse formula—relating big q-Jacobi, dual generalized Bernstein, and dual q-Hahn polynomials—is also given. Further, an alternative formula is given, representing the dual polynomial $D_j^n$ $(0⩽j⩽n)$ as a linear combination of $\min (j,n-j)+1$ big q-Jacobi polynomials with shifted parameters and argument. Finally, we give a recurrence relation satisfied by $D_k^n$, as well as an identity which may be seen as an analogue of the extended Marsden's identity [R.N. Goldman, Dual polynomial bases, J. Approx. Theory 79 (1994) 311–346].