Aptekarev, Alexander I.; Dobrokhotov, Sergey Yu.; Tulyakov, Dmitrii N.; Tsvetkova, Anna V. Plancherel-Rotach type asymptotic formulae for multiple orthogonal Hermite polynomials and recurrence relations. (English. Russian original) Zbl 1510.33008 Izv. Math. 86, No. 1, 32-91 (2022); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 86, No. 1, 36-97 (2022). Summary: We study the asymptotic properties of multiple orthogonal Hermite polynomials which are determined by the orthogonality relations with respect to two Hermite weights (Gaussian distributions) with shifted maxima. The starting point of our asymptotic analysis is a four-term recurrence relation connecting the polynomials with adjacent numbers. We obtain asymptotic expansions as the number of the polynomial and its variable grow consistently (the so-called Plancherel-Rotach type asymptotic formulae). Two techniques are used. The first is based on constructing expansions of bases of homogeneous difference equations, and the second on reducing difference equations to pseudodifferential ones and using the theory of the Maslov canonical operator. The results of these approaches agree. Cited in 1 Document MSC: 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\) 39A06 Linear difference equations 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis Keywords:asymptotic formulae; special functions; recurrence relations; pseudodifferential operators; Maslov canonical operator PDFBibTeX XMLCite \textit{A. I. Aptekarev} et al., Izv. Math. 86, No. 1, 32--91 (2022; Zbl 1510.33008); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 86, No. 1, 36--97 (2022) Full Text: DOI References: [1] Plancherel, M.; Rotach, W., Sur les valeurs asymptotiques des polynômes d’Hermite \(H_n(x)=(-1)^ne^{\frac{x^2}2}\frac{d^n}{dx^n}\bigl(e^{-\frac{x^2}2}\bigr)\), Comment. Math. Helv., 1, 227-254 (1929) · JFM 55.0799.02 [2] Szegö, G., Amer. Math. Soc. Colloq. Publ., 23 (1959), Amer. Math. Soc.: Amer. Math. Soc., Providence, RI · Zbl 0089.27501 [3] Aptekarev, A. I.; Bleher, P. M.; Kuijlaars, A. B. J., Large \(n\) limit of Gaussian random matrices with external source. II, Comm. Math. Phys., 259, 2, 367-389 (2005) · Zbl 1129.82014 [4] Deift, P.; Zhou, X., A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. (2), 137, 2, 295-368 (1993) · Zbl 0771.35042 [5] Deift, P.; Kriecherbauer, T.; McLaughlin, K. T.-R.; Venakides, S.; Zhou, X., Strong asymptotics of orthogonal polynomials with respect to exponential weights, Comm. Pure Appl. Math., 52, 12, 1491-1552 (1999) · Zbl 1026.42024 [6] Deift, P.; Kriecherbauer, T.; McLaughlin, K. T.-R.; Venakides, S.; Zhou, X., Uniform asymptotics of polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, Comm. Pure Appl. Math., 52, 11, 1335-1425 (1999) · Zbl 0944.42013 [7] Bleher, P.; Its, A., Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model, Ann. of Math. (2), 150, 1, 185-266 (1999) · Zbl 0956.42014 [8] Dimofte, T.; Gukov, S., Quantum field theory and the volume conjecture, Interactions between hyperbolic geometry, quantum topology and number theory, 541, 41-67 (2011) · Zbl 1236.57001 [9] Garoufalidis, S.; Lê, Thang T. Q., The colored Jones function is \(q\)-holonomic, Geom. Topol., 9, 1253-1293 (2005) · Zbl 1078.57012 [10] Kashaev, R. M., The hyperbolic volume of knots from quantum dilogarithm, Lett. Math. Phys., 39, 3, 269-275 (1997) · Zbl 0876.57007 [11] Deift, P., Courant Lect. Notes Math., 3 (2000), Courant Inst. Math. Sci.: Courant Inst. Math. Sci., New York [12] Tulyakov, D. N., Plancherel-Rotach type asymptotics for solutions of linear recurrence relations with rational coefficients, Mat. Sb., 201, 9, 111-158 (2010) · Zbl 1242.39002 [13] Dobrokhotov, S. Yu.; Tsvetkova, A. V., Lagrangian manifolds related to the asymptotics of Hermite polynomials, Mat. Zametki, 104, 6, 835-850 (2018) · Zbl 1409.42019 [14] Aptekarev, A. I., Asymptotics of orthogonal polynomials in a neighborhood of the endpoints of the interval of orthogonality, Mat. Sb., 183, 5, 43-62 (1992) · Zbl 0772.33003 [15] Tulyakov, D. N., Local asymptotics of the ratio of orthogonal polynomials in the neighbourhood of an end-point of the support of the orthogonality measure, Mat. Sb., 192, 2, 139-160 (2001) · Zbl 1019.42016 [16] Tulyakov, D. N., Difference equations having bases with powerlike growth which are perturbed by a spectral parameter, Mat. Sb., 200, 5, 129-158 (2009) · Zbl 1184.39007 [17] Aptekarev, A. I.; Tulyakov, D. N., Asymptotics of Meixner polynomials and Christoffel-Darboux kernels, Trans. Moscow Math. Soc., 73, 1, 87-132 (2012) · Zbl 1271.33003 [18] Aptekarev, A. I.; Tulyakov, D. N., The leading term of the Plancherel-Rotach asymptotic formula for solutions of recurrence relations, Mat. Sb., 205, 12, 17-40 (2014) · Zbl 1317.39001 [19] Aptekarev, A. I.; Tulyakov, D. N., Preprints of Keldysh Inst. Appl. Math. (2018) [20] Heading, J., An introduction to phase-integral methods (1962), Methuen: Methuen, London · Zbl 0115.07102 [21] Babič, V. M.; Buldyrev, V. S., Springer Ser. Wave Phenomena, 4 (1972), Nauka: Nauka, Moscow [22] Slavyanov, S. Yu., Transl. Math. Monogr., 151 (1990), Leningrad Univ. Press: Leningrad Univ. Press, Leningrad [23] Maslov, V. P., Operational methods (1973), Nauka: Nauka, Moscow · Zbl 0288.47042 [24] Maslov, V., The characteristics of pseudo-differential operators and difference schemes, Actes du Congrès International des Mathematiciens, 755-769 (1971) · Zbl 0244.35072 [25] Danilov, V. G.; Maslov, V. P., The Pontryagin duality principle for computing a Cherenkov type effect in crystals and difference schemes. II, Proc. Steklov Inst. Math., 167, 96-107 (1985) · Zbl 0582.65045 [26] Maslov, V. P., Études mathematiques (1965), Moscow Univ. Press: Moscow Univ. Press, Moscow [27] Maslov, V. P.; Fedoriuk, M. V., Math. Phys. Appl. Math., 7 (1976), Nauka: Nauka, Moscow [28] Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G., Higher transcendental functions (1953), McGraw-Hill: McGraw-Hill, New York-Toronto-London · Zbl 0052.29502 [29] NIST Digital Library of Mathematical Functions [30] Dobrokhotov, S. Yu.; Tsvetkova, A. V., An approach to finding the asymptotics of polynomials given by recurrence relations, Russ. J. Math. Phys., 28, 2, 198-223 (2021) · Zbl 1467.42041 [31] Aptekarev, A. I.; Branquinho, A.; Assche, W. Van, Multiple orthogonal polynomials for classical weights, Trans. Amer. Math. Soc., 355, 10, 3887-3914 (2003) · Zbl 1033.33002 [32] Anikin, A. Yu.; Dobrokhotov, S. Yu.; Nazaikinskii, V. E.; Tsvetkova, A. V., Uniform asymptotic solution in the form of an Airy function for semiclassical bound states in one-dimensional and radially symmetric problems, Teor. Mat. Fiz., 201, 3, 382-414 (2019) · Zbl 1441.81091 [33] Dobrokhotov, S. Yu.; Nazaikinskii, V. E., Lagrangian manifolds and efficient short-wave asymptotics in a neighborhood of a caustic cusp, Mat. Zametki, 108, 3, 334-359 (2020) · Zbl 1483.53094 [34] Buslaev, V. S.; Fedotov, A. A., The complex WKB method for the Harper equation, Algebra i Analiz, 6, 3, 59-83 (1994) · Zbl 0839.34066 [35] Fedotov, A.; Shchetka, E., Complex WKB method for the difference Schrödinger equation with the potential being a trigonometric polynomial, Algebra i Analiz, 29, 2, 193-219 (2017) · Zbl 1385.39001 [36] Belov, V. V.; Dobrokhotov, S. Yu.; Tudorovskiy, T. Ya., Operator separation of variables for adiabatic problems in quantum and wave mechanics, J. Engrg. Math., 55, 1-4, 183-237 (2006) · Zbl 1110.81080 [37] Karasev, M. V.; Maslov, V. P., Transl. Math. Monogr., 119 (1991), Nauka: Nauka, Moscow This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.