An asymptotic expansion for an integral variant of the Wright operators
DOI:
https://doi.org/10.64700/mmm.98Keywords:
Mittag-Leffler Functions, asymptotic expansion, hypergeometric series, pointwise estimationsAbstract
In this work, we investigate an integral variant of the Wright operators arising in approximation theory. Our main contribution is the derivation of a complete pointwise asymptotic expansion for these operators \(W_n^{\beta, \gamma}\). To establish this expansion, we first develop several structural properties of the operators, emphasizing their intrinsic connections with special functions, including the Mittag-Leffler and confluent hypergeometric functions. The analysis requires a careful study of the moments of the associated integral operators and relies on a collection of auxiliary results, such as a localization theorem. The obtained asymptotic formula provides refined convergence estimates and yields deeper insight into the limiting behavior of the operators as \(n \rightarrow \infty\).
References
[1] U. Abel and P. G. Patel: An asymptotic expansion for the Mittag-Leffler operators, Indian J. Pure Appl. Math., (2025).
[2] R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogozin: Mittag-Leffler functions, related topics and applications, Springer, Berlin (2020).
[3] F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark: NIST handbook of mathematical functions, Cambridge Univ. Press, New York (2010).
[4] J. Paneva-Konovska: Inequalities and asymptotic formulae for the three parametric Mittag-Leffler functions, Math. Balk., 26 (2012), 203–210.
[5] P. G. Patel: On positive linear operators linking gamma, Mittag-Leffler and Wright functions, Int. J. Appl. Comput. Math., 10 (2024), Article ID:152.
[6] P. G. Patel: Some properties of Wright operators, arXiv:2404.04651, (2024).
[7] P. G. Patel: Generalized gamma-Wright integral operators: Approximation properties, Proc. Int. Conf. Math. Adv. Appl., 2 (2025), 40–48.
[8] T. R. Prabhakar: A singular integral equation with a generalized Mittag Leffler function in the kernel, Yokohama Math. J., 19 (1) (1971), 7–15.
[9] P. C. Sikkema: On some linear positive operators, Indag. Math., 73 (1970), 327–337.
[10] E. M.Wright: On the coefficients of power series having exponential singularities, J. Lond. Math. Soc., 8 (1) (1933), 71–79.
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Copyright (c) 2026 Ulrich Abel, Prashantkumar G. Patel
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