Overview of some new results concerning the theory and applications of the Rayleigh special function
- M. K. Kerimov
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The author’s previous work provided a detailed overview of the results concerning the theory and applications of the Rayleigh special function starting from its appearance in science until recent years. Its numerous applications in various areas of mathematics, physics, and other fields were described, and an extensive bibliography was presented. This work overviews the studies not covered in the previous one and addresses new results published in many monographs and journals. Additionally, results concerning the estimation of zeros of some special polynomials and functions closely related to the Rayleigh function are described. The overview embraces the issues addressed in the scientific literature up to the last years.
- M. K. Kerimov, “The Rayleigh Function: Theory and Methods of Its Calculation,” Zh. Vychisl. Mat. Mat. Fiz. 39 1962–2006 (1999) [Comput. Math. Math. Phys. 39, 1883–1925 (1999)].
- J. W. Rayleigh (Strutt), “Note on the Numerical Calculus of the Roots of Fluctuating Functions,” Proc. London Math. Soc. 5, 112–194 (1874).
- G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge Univ. Press, Cambridge, 1944; Inostrannaya Literatura, Moscow, 1949).
- M. K. Kerimov, “Some Remarks on Works Concerning the Summation of Series with Inverse Powers of Zeros of First-Kind Bessel Functions,” Zh. Vychisl. Mat. Mat. Fiz. 47, 21–23 (2007) [Comput. Math. Math. Phys. 47, 180–182 (2007)].
- P. L. Kapitsa, “Heat Conduction and Diffusion in a Liquid with Periodic Flow: I. Determination of the Wave Transport Coefficient in a Pipe, Slit, and Cannel,” Zh. Eksp. Teor. Fiz. 21, 964–978 (1951).
- P. L. Kapitsa, “Computation of Negative Even Power Sums of the Zeros of Bessel Functions,” Dokl. Akad. Nauk SSSR, No. 2, 561–564 (1951).
- N. N. Meiman, “On Recurrence Formulas for Power Sums of the Zeros of Bessel Functions,” Dokl. Akad. Nauk SSSR 108, 190–193 (1956).
- C. C. Grossjean, “On Orthogonality Property of the Lommel Polynomials and Twofold Infinity of Relations between Rayleigh’s σ-Sums,” J. Comput. Appl. Math. 10, 355–382 (1984). CrossRef
- G. Szegö, Orthogonal Polynomials (Am. Math. Soc., Providence. R.I., 1975; Fizmatlit, Moscow, 1962).
- V. S. Vladimirov, Generalized Functions in Mathematical Physics (Mir, Moscow, 1979; Nauka, Moscow, 1979).
- H. M. Schwartz, “A Class of Continued Fractions,” Duke Math. J. 6, 48–65 (1940). CrossRef
- O. Perron, Die Lehre von den Kettenbruchen. Zweite Auflage (Teubner-Verlag, Leipzig, 1929).
- J. Shohat, Theorie Generale des Polynomes Orthogonaux de Tchebichef (Gauthier-Villars, Paris, 1934).
- D. Dickinson, “On Lommel and Bessel Polynomials,” Proc. Am. Math. Soc. 5, 946–956 (1954). CrossRef
- W. Hahn, “Uber Orthogonal Polynome mit drei Parameter,” Deutsche Math. 5, 273–278 (1940).
- I. M. Sheffer, “Note on Functionally-Orthogonal Polynomials,” Töhoku Math. J. 33, 3–11 (1930).
- E. S. Titchmarsh, The Theory of Functions (Oxford Univ. Press, London, 1939; Fizmatlit, Moscow, 1980).
- H. L. Krall and O. Frink, “A New Class of Orthogonal Polynomials: The Bessel Polynomials,” Trans. Am. Math. Soc. 65(1), 100–115 (1949). CrossRef
- J. Favard, “Sur les polynomes de Tchebicheff,” C.R. Acad. Sci. Paris 200, 2052–2053 (1935).
- D. J. Dickinson, H. O. Pollak, and G. H. Wannier, “On a Class of Polynomials Orthogonal over a Denumerable Set,” Pacific J. Math. 6, 239–247 (1956).
- J. L. Goldberg, “Polynomials Orthogonal over a Denumerable Set,” Pacific J. Math. 15, 1171–1186 (1965).
- A. Hurwitz, “Über die Nullstellen der Besselschen Funktionen,” Math. Ann. 33, 246–266 (1889). CrossRef
- M. E. Muldoon and A. Raza, “Convolution Formulas for Functions of Rayleigh Type,” J. Phys. A: Math. Gen. 31, 9327–9330 (1998). CrossRef
- A. McD. Mercer, “The Zeros of az 2 J″v + bzJ′v + cJ v(z) as Function of Order,” Int. J. Math. Math. Sci. 15, 319–322 (1992). CrossRef
- A. Zygmund, “Two Notes on the Summability of Infinite Series,” Colloq. Math. 1, 225–229 (1948).
- L. Lorch, “The Limits of Indetermination for Riemann Summation in Terms of Bessel Functions,” Colloq. Math. 15, 313–318 (1966).
- A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Bateman Manuscript Project (McGraw-Hill, New York, 1953; Nauka, Moscow, 1966), Vol. 2.
- S. Minakshisundaram, “A New Summation Process,” Math. Student 11, 21–27 (1943).
- N. Kishore, “Congruence Properties of the Rayleigh Functions and Polynomials,” Duke Math. J. 35, 557–562 (1968). CrossRef
- N. Liron, “Some Infinite Sums,” SIAM J. Math. Anal. 2, 105–112 (1971). CrossRef
- K. Knopp, Theory and Application of Infinite Series (Blackie, London, 1954).
- L. Carlitz, “Recurrences for the Rayleigh Functions,” Duke Math. J. 34, 581–590 (1967). CrossRef
- N. Kishore, “The Rayleigh Function,” Proc. Am. Math. Soc. 14, 527–533 (1963). CrossRef
- N. Kishore, “A Class of Formulas for the Rayleigh Function,” Duke Math. J. 34, 573–579 (1967). CrossRef
- N. Liron, “A Recurrence Concerning Rayleigh Functions,” SIAM J. Math. Anal. 2, 496–499 (1971). CrossRef
- N. Liron, “Sums of Roots for a Class of Transcendental Equations and Bessel Functions of Order One-Half,” Math. Comput. 25, 769–781 (1971). CrossRef
- R. P. Boas, Entire Functions (Academic, New York, 1954).
- D. Gupta and M. E. Muldoon, “Riccati Equations and Convolution Formulae for Functions of Rayleigh Type,” J. Phys. Ser. A: Math. Gen. 33, 1360–1368 (2000).
- M. E. H. Ismail and M. E. Mildoon, “Bounds for the Small and Purely Imaginary Zeros of Bessel and Related Functions,” Methods Appl. Anal. 2(1), 1–21 (1995).
- Overview of some new results concerning the theory and applications of the Rayleigh special function
Computational Mathematics and Mathematical Physics
Volume 48, Issue 9 , pp 1454-1507
- Cover Date
- Print ISSN
- Online ISSN
- SP MAIK Nauka/Interperiodica
- Additional Links
- Rayleigh special function
- overview of new results related to the Rayleigh function
- Lommel polynomials
- estimates for zeros of Bessel functions
- orthogonal polynomials
- M. K. Kerimov (1)
- Author Affiliations
- 1. Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119333, Russia