Lie algebras and special functions related to the isotropic cone

I. Shilin ; J. Choi

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference «Classical and Modern Geometry» dedicated to the 100th anniversary of the birth of Professor Levon Sergeyevich Atanasyan (July 15, 1921—July 5, 1998). Moscow, November 1–4, 2021. Part 3, Tome 222 (2023), pp. 141-152.

Résumé

In this paper, we discuss the relationship between some maximal subalgebras of the Lie algebra of the proper three-dimensional Lorentz group $G$ and some special functions: Bessel and Bessel–Clifford functions, wave Coulomb functions, the Appel hypergeometric function $F_1$, etc. The kernels of integral operators in the space of representations are expressed in terms of the function introduced by the authors. For this function, we derive continual addition theorems, which, in turn, lead to integral formulas for special functions. We briefly discuss similar results related to groups similar to $G$.

Détail

Keywords: Lie algebra, Bessel–Clifford function, Coulomb wave function, Appel hypergeometric function, integral operator.

Bibliographie

[1] Vilenkin N. Ya., “Gipergeometricheskie funktsii ot neskolkikh peremennykh i vyrozhdennye predstavleniya gruppy $SL(n,\mathbb{R})$”, Izv. vuzov. Mat., 1970, no. 4, 50–55

[2] Vilenkin N. Ya., Shleinikova M. A., “Integralnye sootnosheniya dlya funktsii Uittekera i predstavleniya trekhmernoi gruppy Lorentsa”, Mat. sb., 81 (1970), 185–191

[3] Gradshtein I. S., Ryzhik I. M., Tablitsy integralov, ryadov i proizvedenii, Nauka, M., 1963

[4] Klyuchantsev M. I., “Singulyarnye differentsialnye operatory s $r-1$ parametrami i funktsii Besselya vektornogo indeksa”, Sib. mat. zh., 24:3 (1983), 47–62

[5] Krattser A., Frants V., Transtsendentnye funktsii, M., 1963

[6] Miller U., Simmetriya i razdelenie peremennykh, Mir, M., 1981

[7] Naimark M. A., Teoriya predstavlenii grupp, Nauka, M., 1976

[8] Nikolov A. V., “Struktura i parametrizatsiya grupp $O(p,q)$ i $U(p,q)$”, Bulg. J. Phys. II., 1976, no. 6, 537–545

[9] Prudnikov A. P., Brychkov Yu. A., Marichev O. I., Integraly i ryady. Dopolnitelnye glavy, Nauka, M., 1986

[10] Choi Dzh., Nizhnikov A. I., Shilin I. A., “Ob odnoi summe integralnykh preobrazovanii Gankelya-Klifforda funktsii Uittekera”, Chebyshev. sb., 20:3 (2019), 349–360

[11] Khasanov A., Ergashev T. G., “Formuly analiticheskogo prodolzheniya dlya gipergeometricheskikh funktsii Laurichella ot trekh peremennykh”, Bull. Inst. Math., 2019, no. 5, 50–58

[12] Shilin I. A., “Dvoinye $SO(2,1)$-invariantnye integraly i formuly dlya funktsii Uittekera”, Izv. vuzov. Mat., 2011, no. 5, 56–66

[13] Shilin I. A., Choi Dzh., “Metod kontinualnykh teorem slozheniya i integralnye sootnosheniya mezhdu funktsiyami Kulona i funktsiei Appelya $F_1$”, Zh. vychisl. mat. mat. fiz., 62:9 (2022), 1522–1531

[14] Shilin I. A., Choi Dzh., “Nekotorye formuly dlya obychnykh funktsii i giperfunktsii Besselya—Klifforda, svyazannye s sobstvennoi gruppoi Lorentsa”, Fundam. prikl. mat., 22:5 (2019), 195–208

[15] Shilin I. A., Choi Dzh., “O perekhodakh mezhdu bazisami prostranstva predstavleniya gruppy $SO(2,2)$”, Zh. vychisl. mat. mat. fiz., 61:8 (2020), 1235–1244

[16] Shilov G. E., Matematicheskii analiz. Vtoroi spetsialnyi kurs, Nauka, M., 1965

[17] Elliot Dzh., Dober P., Simmetriya v fizike. T. 1, Mir, M., 1983

[18] Bezrodnykh S. I., “Analytic continuation of the Lauricella function $F_D^{(N)}$ with arbitrary number of variables”, Int. Transforms Spec. Funct., 29:1 (2018), 21\cyrdash42

[19] Bezrodnykh S. I., “Analytic continuation of the Lauricella's functions $F_A^{(N)}$, $F_B^{(N)}$ and $F_D^{(N)}$”, Int. Transforms Spec. Funct., 31:11 (2020), 921–940

[20] Choi J., Shilin I. A., “A generalization of certain associated Bessel functions in connection with a group of shifts”, Commun. Math., 30:1 (2022), 103–118

[21] Clifford W. K., “On Bessel functions”, Mathematical Papers, Oxford Univ. Press, London, 1882, 346–349

[22] Datolli G., Maino G., Chiccoli C., Lorenzutta S., Torre A., “A inified point of view on the theory of the generalized Bessel functions”, Comput. Math. Appl., 30:7 (1995), 111–125

[23] Datolli G., Torre A., Lorenzutta S., Maino G., Chiccoli C., “Generalized Bessel functions within the group representation formalism”, Nuovo Cim. B., 111 (1996), 143–164

[24] Deleruer P., “Sur le calcul symbolique a $n$ variables et les fonctions hyperbesseliennes”, Ann. Soc. Sci. Bruxelles., 67:3 (1953), 229–274

[25] Dzieciol A., Yngve S., Froman P.O., “Coulomb wave functions with complex values of the variable and the parameters”, J. Math. Phys., 40 (1999), 6145–6166

[26] Gaspard D., “Connection formulas between Coulomb wave functions”, J. Math. Phys., 59 (2018), 112104

[27] Hayek N., “Sobre la transformación de Hankel”, Act. VIII Reunión An. Mat. Epanol., 1967, 47–60

[28] Kalnins E. G., Manocha H. L., Miller W., “The Lie theory of two-variable hypergeometric functions”, Stud. Appl. Math., 62:2 (1980), 143–173

[29] Kiryakova V., “From the hyper-Bessel operators of Dimovski to the generalized fractional calculus”, Fract. Calc. Appl. Anal., 17 (2014), 977–1000

[30] Kiryakova V., “Transmutation method for solving hyper-Bessel differential equations based on the Poisson–Dimovski transformation”, Fract. Calc. Appl. Anal., 11 (2008), 299–316

[31] Lipnevich V., Luchko Yu., “The Write function: its properties, applications, and numerical evaluation”, AIP Conf. Proc., 1301 (2010), 614–622

[32] Mainardi F., Fractional Calculus and Special Functions, Univ. of Bologna, Bologna, 2010

[33] Miller W., “Lie theory and the Lauricella functions $F_D$”, J. Math. Phys., 13 (1972), 1393–1399

[34] Miller W., “Lie theory and the Appell functions $F_1$”, SIAM J. Math. Anal., 4:4 (1973), 638–655

[35] Mendez Perez J. M. R., Socas Robayna M. M., “A pair of generalized Hankel–Clifford transformations and their applications”, J. Math. Anal. Appl., 154:2 (1991), 543–557

[36] Mushtaq S., Raza M., Din M. U. Certain geometric properties of Lommel and hyper-Bessel functions, Mathematics., 7 (2019), 240

[37] Paneva-Konovska J., “A family of hyper-Bessel functions and convergent series in them”, Fract. Calc. Appl. Anal., 17:4 (2014), 1001–1015

[38] Shilin I. A., Choi J., “Certain connections between the spherical and hyperbolic bases on the cone and formulas for related special functions”, Int. Transforms Spec. Funct., 25:5 (2014), 374–383

[39] Shilin I. A., Choi J., “Certain relations between Bessel and Whittaker functions related to some diagonal and block-diagonal $3\times3$-matrices”, J. Nonlin. Sci. Appl., 10 (2017), 560–574

[40] Shilin I. A., Choi J., “Certain relations between hyper Bessel–Clifford, Macdonald and Meijer functions and hyper Bessel–Clifford integral transforms”, submitted

[41] Shilin I. A., Choi J., “Maximal subalgebras in $\mathfrak{so}(2,1)$, additions theorems and Bessel–Clifford functions”, J. Anal., 2022

[42] Shilin I. A., Choi J., Lee J. W., “Some integrals involving Coulomb functions associated with the three-dimensional proper Lorentz group”, AIMS Math., 5:6 (2020), 5664–5682

[43] Tricomi F. G., Funzioni Ipergeometriche Confluenti, Cremonese, Rome, 1954

[44] Vilenkin N. Ya., Klimyk A. U., Representations of Lie Groups and Special Functions. Vol. 3. Classical and Quantum Groups and Special Functions, Kluwer, Dordrecht, 1992

[45] Yakubovich S. B., Index Transforms, World Scientific, Singapore, 1996