Abstract
Integro-differential methods, currently exploited in calculus, provide an inexhaustible source of tools to be applied to a wide class of problems, involving the theory of special functions and other subjects. The use of integral transforms of the Borel type and the associated formalism is shown to be a very effective mean, constituting a solid bridge between umbral and operational methods. We merge these different points of view to obtain new and efficient analytical techniques for the derivation of integrals of special functions and the summation of associated generating functions as well.
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Notes
The RMT may be formulated as follows: if f(x) admits the expansion \(f(x) = \sum _{n=0}^{\infty } \frac{\phi (n)(-x)^{n}}{n!}\) with \(\phi (0) \ne 0\) in a neighborhood of \(x = 0\), then \(\int _{0}^{\infty } x^{\nu -1} f(x) {{\mathrm{d}}}x = \Gamma (\nu ) \phi (-\nu )\).
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Acknowledgements
The authors express their sincere appreciation to Dr. D. Babusci for interesting and enlightening discussions on the topics treated in this paper. It is also a pleasure to recognize the interest and the encouragement of Prof. V. Strehl. K. G., A. H. and K. A. P. were supported by the PAN-CNRS program for the French-Polish collaboration. Moreover, K. G. thanks for the support from MNiSW, Warsaw (Poland), under “Iuventus Plus 2015–2016”, program no IP2014 013073.
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Dattoli, G., Palma, E.d., Sabia, E. et al. Operational Versus Umbral Methods and the Borel Transform. Int. J. Appl. Comput. Math 3, 3489–3510 (2017). https://doi.org/10.1007/s40819-017-0315-7
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DOI: https://doi.org/10.1007/s40819-017-0315-7