Geodesic
Two approaches to the asymptotics of the zeros of a class of hypergeometric-type polynomials
Sbornik. Mathematics, Tome 83 (1995) no. 2, pp. 483-494 
A. Martínez; A. Sarso; R. Yan'es. Two approaches to the asymptotics of the zeros of a class of hypergeometric-type polynomials. Sbornik. Mathematics, Tome 83 (1995) no. 2, pp. 483-494. http://geodesic.mathdoc.fr/item/SM_1995_83_2_a11/
@article{SM_1995_83_2_a11,
     author = {A. Mart{\'\i}nez and A. Sarso and R. Yan'es},
     title = {Two approaches to the~asymptotics of the~zeros of a~class of hypergeometric-type polynomials},
     journal = {Sbornik. Mathematics},
     pages = {483--494},
     year = {1995},
     volume = {83},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1995_83_2_a11/}
}
TY  - JOUR
AU  - A. Martínez
AU  - A. Sarso
AU  - R. Yan'es
TI  - Two approaches to the asymptotics of the zeros of a class of hypergeometric-type polynomials
JO  - Sbornik. Mathematics
PY  - 1995
SP  - 483
EP  - 494
VL  - 83
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_1995_83_2_a11/
LA  - en
ID  - SM_1995_83_2_a11
ER  - 
%0 Journal Article
%A A. Martínez
%A A. Sarso
%A R. Yan'es
%T Two approaches to the asymptotics of the zeros of a class of hypergeometric-type polynomials
%J Sbornik. Mathematics
%D 1995
%P 483-494
%V 83
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1995_83_2_a11/
%G en
%F SM_1995_83_2_a11

Voir la notice de l'article provenant de la source Math-Net.Ru

A class of hypergeometric-type differential equations is considered. It is shown that its polynomial solutions $y_n$ exhibit an orthogonality with respect to a ‘varying measure’ (a sequence of measures) on $\mathbb R$. From this relation the asymptotic distribution of zeros is obtained by means of a potential theory approach. Moreover, the WKB or semiclassical approximation is used to construct an asymptotically exact sequence of absolutely continuous measures that approximate the zero distribution of $y_n$.

[1] Aldaya V., Bisquert J., Navarro-Salas J., “The quantum relativistic harmonic oscillator: generalized Hermite polynomials”, Phys. Lett. A, 156 (1991), 381–385 | DOI | MR

[2] Aldaya V., Bisquert J., Guerrero J., Navarro-Salas J., Theory of the quantum relativistiv harmonic oscillator, Preprint, Univ. of Granada, 1993

[3] Arriola E. R., Dehesa J. S., “The distribution of zeros of spherical Bessel functions”, Nuovo Cimento B, 103 (1991), 611–616 | DOI | MR

[4] Arriola E. R., Dehesa J. S., Zarzo A., “Spectral properties of the biconfluent Heun differential equation”, J. Comp. Appl. Math., 37 (1991), 161–169 | DOI | MR | Zbl

[5] Bitsadze A. B., Uravneniya matematicheskoi fiziki, Nauka, Moskva, 1976 | MR

[6] Coddington E. A., Levinson N., Theory of Ordinary Differential Equations, MacGraw-Hill, New York, 1955 | MR

[7] Froman N., Froman P. O., JWKB Approximation: Contributions to the Theory, North Holland, Amsterdam, 1965 | MR

[8] Gonchar A. A., Lopes G., “O teoreme Markova dlya mnogotochechnykh approksimatsii Pade”, Matem. sb., 105 (4) (1978), 512–524 | MR | Zbl

[9] Gonchar A. A., Rakhmanov E. A., “Ravnovesnaya mera i raspredelenie nulei ekstremalnykh mnogochlenov”, Matem. sb., 125(167):1(9) (1984), 117–127 | MR | Zbl

[10] Krall H. L., Scheffer I. M., “Differential equations of infinite order for orthogonal polynomials”, Ann. Mat. Pura Appl. (4), LXXIV (1966), 135–172 | MR

[11] Kwon K. H., Littlejohn L. L., Sobolev orthogonal polynomials and second-order differential equations, Report 5-93-66

[12] Landkov N. S., Osnovy sovremennoi teorii potentsiala, Nauka, Moskva, 1966 | MR

[13] Littlejohn L. L., “On the classification of differential equations having orthogonal polynomial solutions”, Ann. Mat. Pura Appl. (4), CXXXVIII (1984), 35–53 | DOI | MR

[14] Lopes G., “O skhodimosti mnogotochechnykh approksimatsii Pade k funktsiyam stiltesovskogo tipa”, DAN SSSR, 239:4 (1978) | MR

[15] Nikiforov A. F., Uvarov V. B., Special Functions of Mathematical Physics, Birkhauser Verlag, Basel, 1988 | MR | Zbl

[16] Rakhmanov E. A., “Ob asimptoticheskikh svoistvakh mnogochlenov, ortogonalnykh na veschestvennoi osi”, Matem. sb., 119(2) (1982), 163–203 | MR | Zbl

[17] Srivastava H. M., “Some orthogonal polynomials representing the energy spectral functions for a family of isotropic turbulence fields”, ZAMM, 64 (1984), 255–257 | DOI | MR | Zbl

[18] Zarzo A., Martinez A., “The quantum relativistic harmonic oscillator: spectrum of zeros of its wave functions”, J. Math. Phys., 34 (6) (1993) | DOI | MR | Zbl

[19] Zarzo A., Dehesa J. S., “Spectral properties of solutions of hypergeometric-type differential equations”, J. Comp. Appl. Math., 1993 (to appear) | MR

[20] Zarzo A., Dehesa J. S., Torres J., “On a new set of polynomials representing the wave functions of the quantum relativistic harmonic oscillator”, J. Phys. A.: Math. Gen., 1993 (to appear)