Geodesic
Studies on the zeros of Bessel functions and methods for their computation: 3. Some new works on monotonicity, convexity, and other properties
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 12, pp. 1986-2030 Cet article a éte moissonné depuis la source Math-Net.Ru

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This paper continues the study of real zeros of Bessel functions begun in the previous parts of this work (see M. K. Kerimov, Comput. Math. Math. Phys. 54 (9), 1337–1388 (2014); 56 (7), 1175–1208 (2016)). Some new results regarding the monotonicity, convexity, concavity, and other properties of zeros are described. Additionally, the zeros of $q$-Bessel functions are investigated. 
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M. K. Kerimov. Studies on the zeros of Bessel functions and methods for their computation: 3. Some new works on monotonicity, convexity, and other properties. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 12, pp. 1986-2030. http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_12_a1/

[1] Abramowitz M., Stegun I. N. (eds.), Handbook of mathematical functions, Applied Mathematical Series, 55, 10th edit., National Bureau of Standards, Washington, 1964 ; Dover Publications, Inc., New York, 1972; V. A. Ditkin, L. N. Karmazina (red.), Spravochnik po spetsialnym funktsiyam, Nauka, M., 1979 | MR

[2] Askey R. A., “The q-gamma and q-beta-functions”, Applicable Anal., 8 (1978), 125–141 | DOI | MR | Zbl

[3] Blatt J. M., Butler S. T., “Superfluidity of an ideal Bose–Einstein gas”, Phys. Rev., 100 (1995), 476–480 | DOI

[4] Bocher M., “On certain methods of Sturm and their application to the roots of Bessel's functions”, Bull. Amer. Math. Soc., 3 (1897), 205–213 | DOI | MR | Zbl

[5] Elbert A., “Some inequalities concerning Bessel functions of firs kind”, Studia Sci. Math. Hungarica, 6 (1971), 277–285 | MR

[6] Elbert A., “Concavity of the zeros of Bessel functions”, Studia Sci. Math. Hungarica, 12 (1977), 81–88 | MR | Zbl

[7] Elbert A., Gatteschi L., Laforgia A., “On the concavity of zeros of Bessel functions”, Applicable Analysis, 16:4 (1983), 261–278 | DOI | MR | Zbl

[8] Elbert A., Laforgia A., “On the zeros of derivatives of Bessel functions”, Z. angew. Math. und Phys., 34:6 (1983), 774–786 | DOI | MR | Zbl

[9] Elbert A., Laforgia A., “On the square of the zeros of Bessel functions”, SIAM J. Math. Anal., 15 (1984), 206–212 | DOI | MR | Zbl

[10] Elbert A., Laforgia A., “An asymptotic relation for the zeros of Bessel functions”, J. Math. Anal. and Appl., 98 (1984), 502–511 | DOI | MR | Zbl

[11] Elbert A., Laforgia A., “Further results on the zeros of Bessel functions”, Analysis, 5 (1985), 71–86 | DOI | MR | Zbl

[12] Elbert A., Laforgia A., “On the convexity of the zeros of Bessel functions”, SIAM J. Math. Anal., 16:3 (1985), 614–619 | DOI | MR | Zbl

[13] Elbert A., Laforgia A., “Monotonicity properties of the zeros of Bessel functions”, SIAM J. Math. Anal., 17:6 (1986), 1483–1488 | DOI | MR | Zbl

[14] Elbert A., Laforgia A., “Some concequences of a lower found for the second derivative of the zeros of Bessel functions”, J. Math. Anal. and Appl., 125:1 (1987), 1–5 | DOI | MR | Zbl

[15] Elbert A., Laforgia A., “On the power of the zeros of Bessel functions”, Rendiconti Circolo Mat. Palermo, 36:3 (1987), 507–517 | DOI | MR | Zbl

[16] Elbert A., Laforgia A., “A convexity property of zeros of Bessel functions”, J. Appl. Math. and Phys. (ZAMP), 41:5 (1990), 734–737 | DOI | MR | Zbl

[17] Elbert A., Laforgia A., Lorch L., “Additional monotonicity properties of the zeros of Bessel functions”, Analysis, 11:4 (1991), 293–299 | DOI | MR | Zbl

[18] Elbert A., Siafarikas P. D., “On the square of the first zero of Bessel function $J_{\nu}(z)$”, Canad. Math. Bull., 20:1 (1998), 1–12 | MR

[19] Giordano C., Laforgia A., “Further properties of the zeros of Bessel functions”, Le Matematiche (Catania), 42 (1987), 19–28 | MR | Zbl

[20] Giordano C., Rodono L. G., “Further monotonicity and convexity properties of the zeros of cylinder functions”, J. of Comput. and Appl. Math., 42:2 (1992), 245–251 | DOI | MR | Zbl

[21] Hahn W., “Beitrage zur Theorie der Heineschen Reihen”, Math. Nachrichten, 2 (1949), 340–379 | DOI | MR | Zbl

[22] Hurwitz A., “Ueber die Nullstellen Bessel schen Functionen”, Math. Anal., 33 (1880), 246–266 | DOI | MR

[23] Ifantis E. K., Siafarikas P. D., “A differential equations for the zeros of Bessel functions”, Applicable Anal., 20 (1985), 269–281 | DOI | MR | Zbl

[24] Ifantis E. K., Siafarikas P. D., “Differential inequalities for the positive zeros of Bessel functions”, J. of Comput. and Appl. Math., 30:2 (1990), 139–143 | DOI | MR | Zbl

[25] Ifantis E. K., Siafarikas P. D., “A differential inequality for the positive zeros of Bessel functions”, J. of Comput. and Appl. Math., 44:1 (1992), 115–120 | DOI | MR | Zbl

[26] Ismail M. E. H., “The zeros of basic Bessel functions, the functions $J_{\nu+\alpha x}(x)$ and associated orthogonal polynomials”, J. Math. Anal. and Appl., 86 (1982), 1–19 | DOI | MR | Zbl

[27] Ismail M. E. H., “On zeros of Bessel functions”, Applicable Analysis, 22 (1986), 167–168 | DOI | MR | Zbl

[28] Ismail M. E. H., Muldoon M. E., “On the variation with respect to a parameter of zeros of Bessel and $q$-Bessel functions”, J. Math. Anal. and Appl., 135 (1988), 187–207 | DOI | MR | Zbl

[29] Ismail M. E. H., Muldoon M. E., “Bounds for the small real and purely imaginary zeros of Bessel and related functions”, Methods of Appl. Anal., 2:1 (1995), 1–21 | MR | Zbl

[30] Jackson F. H., “The basic gamma function and elliptic function”, Proc. Royal Soc. London. Serie A, 76 (1905), 127–144 | DOI

[31] Kerimov M. K., “The Rayleigh Function:Theory and Methods for its Calculation”, Comput. Math. and Math. Phys., 39:12 (1999), 1962–2006 | MR | Zbl

[32] Kerimov M. K., “Overview of some results concerning the theory and applications of the Rayleigh special functions”, Comput. Math. and Math. Phys., 48:9 (2008), 1454–1507 | DOI | MR

[33] Kerimov M. K., “Studies on the zeros of Bessel Functions and Methods for their Computation”, Comput. Math. and Math. Phys., 53:9 (2014), 1387–1441 | MR

[34] Kokologiannaki C. G., Muldoon M. E., Siafarikas P. D., “A unimodal property of purely imaginary zeros of Bessel and related functions”, Canad. Math. Bull., 37:3 (1994), 365–373 | DOI | MR | Zbl

[35] Laforgia A., Muldoon M. E., “Inequalities and approximations for zeros of Bessel functions of small order”, SIAM J. Math. Anal., 14:2 (1983), 383–388 | DOI | MR | Zbl

[36] Laforgia A., Muldoon M. E., “Monotonicity and concavity properties of zeros of Bessel functions”, J. Math. Anal. and Appl., 98:2 (1984), 470–477 | DOI | MR | Zbl

[37] Lewis J. T., Muldoon M. E., “Monotonicity and concavity properties of zeros of Bessel functions”, SIAM J. Math. Anal., 8:1 (1977), 171–178 | DOI | MR | Zbl

[38] Lewis J. T., Pule J. V., “The free Boson gas in rotating bucket”, Commun. Math. Phys., 45 (1975), 115–131 | DOI | MR

[39] Lorch L., “Elementary comparison techniques for certain classes of Sturm–Lionville equations”, Proc. International Conference on Differential Equations. Symposia Univ. Uppsaliensis Annum Quingentesimum Celebranis 7 (Uppsala, 1977), Acta Univ. Uppsaliensis, Uppsala, 1977, 125–133 | MR | Zbl

[40] Lorch L., “Turanians and Wronskians for the zeros of Bessel functions”, SIAM J. Math. Anal., 11 (1980), 223–227 | DOI | MR | Zbl

[41] Lorch L., Muldoon M. E., Szego P., “Some monotonicity properties of Bessel functions”, SIAM J. Math. Anal., 4 (1973), 385–392 | DOI | MR | Zbl

[42] Lorch L., Szego P., “Higher monotonicity properties of certain Sturm–Lionville functions”, Acta Math., 109 (1963), 53–73 | DOI | MR

[43] Lorch L., Szego P., “Higher monotonicity properties of certain Sturm–Liouoille functions. II”, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. and Phys., 11 (1963), 455–457 | MR | Zbl

[44] Lorch L., Szego P., “On the zeros of derivatives of Bessel functions”, SIAM J. Math. Anal., 19:6 (1988), 1450–1454 | DOI | MR | Zbl

[45] Olver F. W. J. (ed.), Bessel Functions, v. III, Zeros and associated values, Cambridge University Press, Cambridge, 1960 ; K. A. Karpov (red.), Tablitsy nulei funktsii Besselya, Perev. s angl. E. F. Chistovoi, Biblioteka matem. tablits, 44, Izd-vo VTs AN SSSR, M., 1967 | MR | Zbl

[46] Olver F. W. J., Asymptotics and Special Functions, Academic Press, New York, 1974 ; Olver F., Asimptotika i spetsialnye funktsii, Perev. s angl. Yu. A. Brychkova, ed. A. P. Prudnikov, Nauka, M., 1990 | MR | MR

[47] Piessens R., “A series expansion for the first positive zero of Bessel function”, Math. of Comput., 42 (1984), 195–197 | DOI | MR | Zbl

[48] Putterman S. J., Kas M., Uhlenbeck G. E., “Possible origin of the quantized vortices in He. II”, Phys. Rev. Letters, 29 (1972), 546–549 | DOI

[49] Ronveaux R., Monsiaux A., “A prior bound on the first zeros of some special functions”, Ann. Soc. Sci. Bruxelles. Serie II, 88 (1974), 169–175 | MR | Zbl

[50] Shohat J. A., Tamarkin J. D., The problem of Moments, Math. Surveys, 1, rev. edit., American Math. Soc., Providence, RI, 1950 | MR | Zbl

[51] Spigler R., “Alcuni risultati sugli zeri delle funzioni cilindriche e delle loro derivate”, Rendiconti Semin. Mat. Universita e Politechnica. Torino, 38:1 (1980), 67–85 | Zbl

[52] Tricomi F. G., “Sulle funzioni di Bessel di ordine e argomento pressoche uguali”, Atti Accad. Sci. Torino. Cl. Sci. Fis. Mat. Natur., 83 (1949), 3–20 | MR | Zbl

[53] Sturm J. C. F., “Sur les equations differentielles linearies du second order”, J. de Math., 1 (1836), 106–186

[54] Watson G. N., A treatise of the theory of Bessel functions, 2nd Edit, Cambridge University Press, Cambridge, 1944 ; G. N. Vatson, Teoriya besselevykh funktsii, Perev. s angl. V. S. Bermana, v. I, II, Izd-vo inostr. lit., M., 1949 | MR | Zbl