Dual generalized Bernstein basis

The generalized Bernstein basis in the space Πn 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], Bin(x;ω|q)≔1(ω;q)nniqxi(ωx-1;q)i(x;q)n-i(i=0,1,…,n).We give explicitly the dual basis functions Dkn(x;a,b,ω|q) for the polynomials Bin(x;ω|q), in terms of big q-Jacobi polynomials Pk(x;a,b,ω/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 Djn (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 Dkn, 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].