chebЧебышевский сборникChebyshevskii Sbornik2226-8383Tula State Lev Tolstoy Pedagogical University10.22405/2226-8383-2021-22-1-225-233cheb-944Research ArticleСтатьиArticleО двух формулах для функций Макдональда и их теоретико-групповом смыслеOn two formulas for Macdonald functions and their group-theoretical senseНижниковАлександр ИвановичNizhnikovAlexander Ivanovichainizhnikov@mail.ruШилинИлья АнатольевичShilinIlya Anatolyevichilyashilin@li.ruМосковский педагогический государственный университетРоссияMoscow State Pedagogical UniversityRussian FederationНациональный исследовательский унтверситет МЭИ; Московский педагогический государственный университетРоссияNational Research University MPEI; Moscow State Pedagogical UniversityRussian Federation202106042021221225233Copyright © Нижников А.И., Шилин И.А., 20212021Нижников А.И., Шилин И.А.Nizhnikov A.I., Shilin I.A.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.chebsbornik.ru/jour/article/view/944

С помощью интегральных билинейных функционалов, определенных на паре пространств представления трехмерной лоренцовой группы или квадрате такого пространства, получены две формулы для функций Макдональда — частном случае цилиндрических функции, широко используемом в математике и приложениях

Two formulas for Macdonald functions (which are a widely known in mathematics and applications particular case of cylinder functions) are obtained by using some integral bilinear functionals defined on a pair of representation spaces or a square of these spaces

функции Макдональдатрехмерная лоренцева группаMacdonald functionsthree-dimensional Lorentz groupReferencesВиленкин Н. Я. Специальные функции и теория представлений групп. Наука, Москва, 1965.Vilenkin N. Ja. 1965, Special functions and group representations theory, Nauka, Moscow. (In Russian.)Виленкин Н. Я., Шлейникова М. А. Интегральные соотношения для функций Уиттекера и представления трехмерной группы Лоренца // Мат. сб. 1970. Т. 81. С. 185-191.Gelfand I. M., Graev M. I., Vilenkin N.Ya. 1962, Integral geometry and related aspects in group representation theory, Fizmatlit, Moscow. (In Russian.)Гельфанд И. М., Граев М. И., Виленкин Н. Я. Интегральная геометрия и связанные с ней вопросы теории представлений. Физматлит, Москва, 1962.Gradstein I. S., Ryzhik I.M˙ . 1963, Tables of integrals, sums, series and products, Fizmatlit, Moscow. (In Russian.)Градштейн И. С., Рыжик И. М. Таблицы интегралов, сумм, рядов и произведений. Физматлит, Москва, 1963.Prudnikov A. P., Brychkov Yu. A., Marichev O. I. 1986, Integrals and series: Advanced chapters, Nauka, Moscow. (In Russian.)Прудников А. П., Брычков Ю. А., Маричев О. И. Интегралы и ряды. Дополнительные главы. Наука, Москва, 1986.Prudnikov A. P., Brychkov Yu. A., Marichev O. I. 198s, Integrals and series: Special functions, Nauka, Moscow. (In Russian.)Прудников А. П., Брычков Ю. А., Маричев О. И. Интегралы и ряды. Специальные функции. Наука, Москва, 1983.Prudnikov A. P., Brychkov Yu. A., Marichev O. I. 1981, Integrals and series: Elementary functions, Nauka, Moscow. (In Russian.)Прудников А. П., Брычков Ю. А., Маричев О. И. Интегралы и ряды. Элементарные функции. Наука, Москва, 1981.Shilin I. A. 2012, “Double S0(2, 1)-invariant integrald and formulas for Whittaker functions“, Izv. VUZ., no. 5, pp. 56-66.Шилин И. А. Двойные S0(2, 1)-инвариантные интегралы и формулы для функций Уиттекера // Известия вузов. Математика. 2012. № 5. C. 56-66.Aurich R., Lustig S. & Steiner F. 2004, “Hyperbolic universes with a horned topology and the cosmic microwave background anisotropy“, Classical and Quantum Gravity, vol. 21, pp.Aurich R., Lustig S., Steiner F. Hyperbolic universes with a horned topology and the cosmic microwave background anisotropy // Classical and Quantum Gravity. 2004. Vol. 21. P. 4901-4925.-4925.Bolte J., Steil G., Steiner F. Arithmetical chaos and violation of universality in energy level statistics // Physical Review Letters. 1992. Vol. 69. P. 2188-2191.Bolte J., Steil G. & Steiner F. 1992, “Arithmetical chaos and violation of universality in energy level statistics“, Physical Review Letters, vol. 69, pp. 2188-2191.Booker A. R. A test for identifying Fourier coefficients of automorphic forms and applications to Kloosterman sums // Experimental Mathematics. 2000. Vol. 9. P. 571-581.Booker A. R. 2000, “A test for identifying Fourier coefficients of automorphic forms and applications to Kloosterman sums“, Experimental Mathematics, vol. 9, pp. 571-581.Bryson A. E. Control of spacecraft and aircraft. Princeton University Press, Princeton, 1994.Bryson A. E. 1994, Control of spacecraft and aircraft, Princeton University Press, Princeton.Heijhal D. A. The Selberg trace formula for PSL(2,R). – Lecture Notes in Mathematics, Volume 1001. Springer, Berlin, 1983.Heijhal D. A. 1983, The Selberg trace formula for PSL(2,R). Lecture Notes in Mathematics, Volume 1001, Springer, Berlin.Heijhal D. A., Rackner B. N. On the topography of Maass vaveforms for PSL(2, Z) // Experimental Mathematics. 1992. Vol. 1. P. 275-305.Heijhal D. A. & Rackner B. N. 1992, “On the topography of Maass vaveforms for PSL(2,Z)“, Experimental Math., vol. 1, pp. 275-305.Gil A., Seguna J., Temme N. M. Computation of the modified Bessel functions of the third kind of imaginary orders: uniform Airy-type asymptotic expansions // Journal Computational and Applied Mathematics. 2003. Vol. 153. P. 225-234.Gil A., Seguna J. & Temme N. M. 2003, “Computation of the modified Bessel functions of the third kind of imaginary orders: uniform Airy-type asymptotic expansions“, J. Comp. Appl.Iwaniec H. Introduction to the spectral theory of automorphic forms. Revista Matematica Iberoamericana, Madrid, 1995.Math., vol. 153, pp. 225-234.Shilin I. A., Junesang Choi. Certain connections between the spherical and hyperbolic bases on the cone and formulas for related special functions // Integral Transforms and Special Functions. 2014. Vol. 25. No. 5. P. 374 – 383.Iwaniec H. 1995, Introduction to the spectral theory of automorphic forms, Revista Matematica Iberoamericana, Madrid.Shilin I. A., Junesang Choi. Some connections between the spherical and parabolic bases on the cone expressed in terms of the Macdonald functions // Abstract and Applied Analysis. 2014. Article ID 741079.Shilin I. A., Junesang Choi. Some connections between the spherical and parabolic bases on the cone expressed in terms of the Macdonald functions // Abstract and Applied Analysis. 2014. Article ID 741079.I. A. Shilin, J. Choi, J. W. Lee. Some integrals involving Coulomb functions associated with the three-dimensional proper Lorentz group // AIMS Mathematics. 2020. Vol. 5. No. 6. P. 5664–5682.I. A. Shilin, J. Choi, J. W. Lee. Some integrals involving Coulomb functions associated with the three-dimensional proper Lorentz group // AIMS Mathematics. 2020. Vol. 5. No. 6. P. 5664–5682.Shilin I. A., Nizhnikov A. I. Some formulas for Legendre functions induced by the Poisson transform // Acta Polytechnica. 2011. Vol. 51. No. 1, P. 70-73.Shilin I. A., Nizhnikov A. I. Some formulas for Legendre functions induced by the Poisson transform // Acta Polytechnica. 2011. Vol. 51. No. 1, P. 70-73.Shilin I. A., Nizhnikov A. I. Some formulas for Legendre functions related to the Poisson transform and Lorentz group representation // Journal of Physics. Conference series. 2012.Shilin I. A., Nizhnikov A. I. Some formulas for Legendre functions related to the Poisson transform and Lorentz group representation // Journal of Physics. Conference series. 2012.Vol. 356. 012020.Vol. 356. 012020.Yakubivich S. V. Index transforms. World Scientific, Singapore, 1996.Yakubivich S. V. Index transforms. World Scientific, Singapore, 1996.Zhao P. Quantum variance of Maass-Hecke cusp forms // Communications in Mathematical Physics. 2010. Vol. 297. P. 475-514.Zhao P. Quantum variance of Maass-Hecke cusp forms // Communications in Mathematical Physics. 2010. Vol. 297. P. 475-514.

The authors declare that there are no conflicts of interest present.