×

zbMATH — the first resource for mathematics

Spectral functions of zeros in the Bessel \(q\)-functions. (English. Russian original) Zbl 1041.33502
Theor. Math. Phys. 107, No. 3, 740-754 (1996); translation from Teor. Mat. Fiz. 107, No. 3, 397-414 (1996).
Summary: The author studies zeta functions \(\zeta_\nu(z;q)=\sum_{n=1}^\infty[j_{\nu n}(q)]^{-z}\) and partition functions \(Z_\nu(t;q)_=\sum_n\exp[-tj_{\nu n}^2(q)]\) related to the zeros \(j_{\nu n}(q)\) of the \(q\)-Bessel functions \(J_\nu(x;q)\) and \(J_\nu^{(2)}(x;q)\). He obtains explicit formulas for \(\zeta_\nu(2n;q)\) at \(n=\hat A\pm1,\hat A\pm 2,\dots\). He finds poles of \(\zeta_\nu(z;q)\) in the complex plane and corresponding residues and derives the asymptotics of the partition functions \(Z_\nu(t;q)\) as \(t\downarrow0\).
MSC:
33E20 Other functions defined by series and integrals
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] P. B. Gilkey,Invariance Theory, the Heat Equation, and the Atiya-Singer Index Theorem, Publish or Perish, Wilmington (1984).
[2] C. Itzykson, P. Moussa, and J. M. Luck,J. Phys. A,19, L111-L115 (1986). · Zbl 0605.35068 · doi:10.1088/0305-4470/19/3/004
[3] M. V. Berry and M. Robnik,J. Phys. A,19, 649–668 (1986). · doi:10.1088/0305-4470/19/5/019
[4] M. V. Berry,J. Phys. A,19, 2281–2296 (1986). · Zbl 0623.58044 · doi:10.1088/0305-4470/19/12/015
[5] S. Leseduarte and A. Romeo,J. Phys. A,27, 2483–2495 (1994); E. Elizalde, S. Leseduarte, and A. Romeo,J. Phys. A,26, 2409–2419 (1993). · Zbl 0838.58042 · doi:10.1088/0305-4470/27/7/025
[6] G. N. Watson,A Treatise on the Theory of Bessel Functions (2nd edn.), Cambridge University Press, Cambridge (1944). · Zbl 0063.08184
[7] N. Kishore,Proc. Am. Math. Soc.,14, 527–533 (1963). · doi:10.1090/S0002-9939-1963-0151649-2
[8] F. H. Jackson,Trans. Roy. Soc. Edinburgh,41, 1–28 (1903).
[9] F. H. Jackson,Trans. Roy. Soc. Edinburgh,41, 105–118 (1903).
[10] F. H. Jackson,Proc. London Math. Soc. (2.2, 192–220 (1903–1904);3, 1–20 (1904–1905). · JFM 35.0487.02 · doi:10.1112/plms/s2-2.1.192
[11] W. Hahn,Z. Angew. Math. Mech.,33, 270–272 (1953). · Zbl 0051.15502 · doi:10.1002/zamm.19530330811
[12] H. Exton,q-Hypergeometric Functions and Applications, Ellis Horwood, Chichester (1983). · Zbl 0514.33001
[13] T. H. Koornwinder and R. F. Swarttouw,Trans. Am. Math. Soc.,333, 445–461 (1992). · Zbl 0759.33007 · doi:10.2307/2154118
[14] H. T. Koelink and R. F. Swarttouw,J. Math. Anal. Appl.,186, 690–710 (1994). · Zbl 0811.33013 · doi:10.1006/jmaa.1994.1327
[15] W. Hahn,Math. Nachr.,2, 340–379 (1949). · Zbl 0033.05703 · doi:10.1002/mana.19490020604
[16] M. E. H. Ismail,J. Math. Anal. Appl.,86, 1–19 (1982). · Zbl 0483.33004 · doi:10.1016/0022-247X(82)90248-7
[17] M. E. H. Ismail and M. E. Muldon,J. Math. Anal. Appl.,135, 187–207 (1982). · Zbl 0651.33007 · doi:10.1016/0022-247X(88)90148-5
[18] B. Ja. Levin,Distribution of Zeros of Entire Functions, Am. Math. Soc., Providence (1964). · Zbl 0152.06703
[19] A. A. Kvitsinsky, ”Spectral zeta functions forq-Bessel equations,” to appear inJ. Phys. A. · Zbl 0856.58047
[20] D. Dickinson,Proc. Am. Math. Soc.,5, 946–956 (1954). · doi:10.1090/S0002-9939-1954-0086897-8
[21] A. A. Kvitsinsky, ”Zeta functions, heat kernel expansions and asymptotics forq-Bessel functions,” to appear inJ. Math. Anal. Appl. · Zbl 0842.33010
[22] G. Gasper and M. Rahman,Basic Hypergeometric Series, Cambridge University Press, Cambridge (1990). · Zbl 0695.33001
[23] A. B. Olde Daalhuis,J. Math. Anal. Appl.,186, 896–913 (1994). · Zbl 0809.33008 · doi:10.1006/jmaa.1994.1339
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.