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We introduce polynomials Bni(x;ω|q), depending on two parameters qand ω, which generalize classical Bernstein polynomials, discrete Bernstein polynomials defined by Sablonnière, as well as q-Bernstein polynomials introduced by Phillips. Basic properties of the new polynomials are given. Also, formulas relating Bni(x;ω|q), big q-Jacobi and q-Hahn (or dual q-Hahn) polynomials are presented.

Let {Pk} be a sequence of the semi-classical orthogonal polynomials. Given a function fsatisfying a linear second-order differential equation with polynomial coefficients, we describe an algorithm to construct a recurrence relation satisfied by the coefficients ak[f] in f=∑kak[f]Pk. A systematic use of basic properties (including some nonstandard ones) of the polynomials {Pk} results in obtaining a recurrence of possibly low order. Recurrences for connection or linearization coefficients related to the first associated generalized Gegenbauer, Bessel-type and Laguerre-type polynomials are given explicitly.

We propose an algorithm to construct recurrence relations for the coefficients of the Fourier series expansions with respect to the q-classical orthogonal polynomials pk(x;q). Examples dealing with inversion problems, connection between any two sequences of q-classical polynomials, linearization of ϑm(x) pn(x;q), where ϑm(x) is xmor (x;q)m, and the expansion of the Hahn-Exton q-Bessel function in the little q-Jacobi polynomials are discussed in detail.

We show that the generalized hypergeometric function

$\displaystyle {P_n}:{ = _{p + 3}}{F_{p + 2}}\left( {\left. {\begin{array}{*{20}... ...} \\ {{b_{p + 2}}} \\ \end{array} } \right\vert 1} \right)\quad (n \geqslant 0)$ satisfies a nonhomogeneous recurrence relation of order $ p + \sigma $ $ \sigma = 0$ $ _{p + 3}{F_{p + 2}}(1)$ $ \sigma = 1$

$\displaystyle {U_n}: = \frac{{{{({c_{q + 1}})}_n}}}{{{{({d_q})}_n}{{(n + \lambd... ...,2n + \lambda + 1} \\ \end{array} } \right\vert 1} \right)\quad (n \geqslant 0)$ a homogeneous recurrence relation of order is given.

Summation of series of the form $ \sum\nolimits _{k = 1}^\infty {k^{\nu - 1}}r(k)$ $ 0 \leq \nu \leq 1$r is a rational function. By an application of the Euler-Maclaurin summation formula, the problem is reduced to the evaluation of Gauss' hypergeometric function. Examples are given.

A method is presented for convergence acceleration of the generalized hypergeometric series $ _3{F_2}$, using analytic properties of their terms. Iterated transformation of the series is performed analytically, which results in obtaining new fast converging expansions for some special functions and mathematical constants.

It is known that a near minimax polynomial approximation tof ∈C [−1, 1] is provided by a finite carrier projectionMn fromC[−1, 1] onto the subspace of all polynomials of degree ≤n, such thatMnf is a weighted least squares approximation off on the set consisting of the extreme points of the Chebyshev polynomialT2n + 1. In this paper, upper bounds for the error ∥f−Mnf∥∞ are given in terms of divided differences.