In this paper, we present a unified family of polynomials including not only the Apostol–Bernoulli, Euler and Genocchi polynomials, but also a general family of polynomials suggested by Özden et al. ...H. Ozden, Y. Simsek, H.M. Srivastava, A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials. Comput. Math. Appl. 60 (10) (2010) 2779–2787. We obtain the explicit representation of this unified family, in terms of a Gaussian hypergeometric function. Some symmetry identities and multiplication formula are also given.
Recently an extension of beta function containing an extra parameter, which proved to be useful earlier, was used to extend the hypergeometric functions (Aslam Chaudhry, 2004
6). In this paper, ...linear and bilinear generating relations for extended hypergeometric functions are obtained by defining the extension of the fractional derivative operator. Some properties of the extended fractional derivative operator are also presented.
In this work, giving a modification of the well-known Szasz–Mirakjan operators, we prove that the error estimation of our operators is better than that of the classical Szasz–Mirakjan operators. ...Furthermore, we obtain a Voronovskaya type theorem for these modified operators.
In the present paper, we introduce a modification of the Meyer-König and Zeller
$\left( \text{MKZ} \right)$
operators which preserve the test functions
${{f}_{0}}\left( x \right)=1$
and
...${{f}_{2}}\left( x \right)={{x}^{2}}$
, and we show that this modification provides a better estimation than the classical
$\left( \text{MKZ} \right)$
operators on the interval
$\left \frac{1}{2},1 \right)$
with respect to the modulus of continuity and the Lipschitz class functionals. Furthermore, we present the
$r$
-th order generalization of our operators and study their approximation properties.
Almost four decades ago, H.M. Srivastava considered a general family of univariate polynomials, the Srivastava polynomials, and initiated a systematic investigation for this family
10. In 2001, B. ...González, J. Matera and H.M. Srivastava extended the Srivastava polynomials by inserting one more parameter
4. In this study we obtain a family of linear generating functions for these extended polynomials. Some illustrative results including Jacobi, Laguerre and Bessel polynomials are also presented. Furthermore, mixed multilateral and multilinear generating functions are derived for these polynomials.
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