Dual generalized Bernstein basis

Abstract

The generalized Bernstein basis in the space Π 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 ; ω | q ) ≔ 1 ( ω ; q ) n n i q x i ( ω x - 1 ; q ) i ( x ; q ) n - i ( i = 0 , 1 , … , n ) . We give explicitly the dual basis functions D k n ( x ; a , b , ω | q ) for the polynomials B i n ( x ; ω | q ) , in terms of big q-Jacobi polynomials P k ( 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 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].