Actual directions of scientific researches of the XXI century: theory and practiceActual directions of scientific researches of the XXI century: theory and practiceАктуальные направления научных исследований XXI века: теория и практика2308-8877288310.12737/4776Секция: Теория функций и функциональный анализSection: Theory of functions and functional analysisСекция: Теория функций и функциональный анализRising factorial powers and a nonelementary function of the dawson’s integral typeВозрастающие факториальные степени и неэлементарная функция типа интеграла ДоусонаГойТ. П.GoyT. П.tarasgoy@yahoo.com101020141010201424343346https://zh-szf.ru/en/nauka/article/2883/view
Исследована новая неэлементарная функция действительного переменного типа интеграла Доусона, построенная в виде степенного ряда с помощью возрастающих факториальных степеней. Установлена ее связь с функцией ошибок (функцией вероятностей). Показано, что новая функция есть решением уравнения Риккати.
A new nonelementary real-valued function of the Dawson’s integral type is studied. It is constructed as a power series with the help of rising factorial powers. Its connection with the error function is determined. It is proved that the new function is a solution of the Riccati equation.
Introduction. Duality of rising and falling factorial powers is a common feature in the combinatorial analysis. In other words, if a problem leads to some combinatorial identity constructed with the help of falling factorial powers, then often there is a dual combinatorial problem, which leads to a dual combinatorial identity involving rising factorial powers. One can find some examples of these dual combinatorial identities in [1-3].
Graham R. Concrete Mathematics: a Foundation for Computer Science / R. Graham, D. Knuth, O. Patashnik. - Addison-Wesley Professional, 1994. - 657 pp.Graham R. Concrete Mathematics: a Foundation for Computer Science / R. Graham, D. Knuth, O. Patashnik. - Addison-Wesley Professional, 1994. - 657 pp.Jordan C. Calculus of Finite Differences / C. Jordan. - New York : Chelsea Publishing, 1939. - 652 pp.Jordan C. Calculus of Finite Differences / C. Jordan. - New York : Chelsea Publishing, 1939. - 652 pp.Zatorsky R. A. Calculus of Triangular Matrices and Its Applications / R. A. Zatorsky. - Ivano-Frankivsk : Simyk, 2010. - 508 pp. (in Ukrainian)Zatorsky R. A. Calculus of Triangular Matrices and Its Applications / R. A. Zatorsky. - Ivano-Frankivsk : Simyk, 2010. - 508 pp. (in Ukrainian)Goy T. P. New functions generated by rising factorials and their properties / T. P. Goy, R. A. Zatorsky // Bukovyna Math. J. - 2013. - 1, № 1-2. - 28-33 p. (in Ukrainian).Goy T. P. New functions generated by rising factorials and their properties / T. P. Goy, R. A. Zatorsky. Bukovyna Math. J. - 2013. - 1, № 1-2. - 28-33 p. (in Ukrainian).Oldham K. B. An Atlas of Functions: with Equator, the Atlas Function Calculator / K. B. Oldham, J. Myland, J. Spanier. - Springer, 2008. - 762 pp.Oldham K. B. An Atlas of Functions: with Equator, the Atlas Function Calculator / K. B. Oldham, J. Myland, J. Spanier. - Springer, 2008. - 762 pp.