A new infinite series identities involving modified Bessel functions and Hermite polynomials

  • Mikhail Trifonov Sechenov Institute of Evolutionary Physiology and Biochemistry of the Russian Academy of Sciences, Saint-Petersburg 194223, Russia
Article ID: 2324
Keywords: modified Bessel functions of the first kind; Hermite polynomials; probability density functions; infinite series identities; integral representations

Abstract

Some new infinite series identities and their analogues—integral representations involving the modified Bessel functions of the first kind and the Hermite polynomials were obtained using a probabilistic approach.

References

[1]Holt DR, Crow EI. Tables and graphs of the stable probability density functions. Journal of research of the National Bureau of Standards - B. Mathematical Sciences. 1973; 77B(3–4): 143–198.

[2]Rice SO. Statistical properties of a sine wave plus random noise. The Bell System Technical Journal. 1948; 27(1): 109–157. doi: 10.1002/j.1538-7305.1948.tb01334.x

[3]Middleton D. An Introduction to Statistical Communication Theory. McGraw-Hill; 1960.

[4]Ho KP, Chan CK, Tong F, Chen LK. Exact analysis of homodyne crosstalk induced penalty in optical networks. Proceedings SPIE, Optical Fiber Communication. 1998; 3420: 72–77. doi: 10.1117/12.312832

[5]Trifonov M. Bessel representation for amplitude distribution of noisy sinusoidal signals. Statistical Papers. 2022; 63(3): 983–988. doi: 10.1007/s00362-021-01262-z

[6]Olver FWJ, Lozier DW, Boisvert RF, Clark CW. NIST Handbook of Mathematical Functions. Cambridge University Press; 2010.

[7]Szegö G. Orthogonal Polynomials, 4th ed. American Mathematical Society Colloquium Publications; 1975. Volume 23.

[8]Gradshteyn IS, Ryzhik IM. Table of Integrals, Series, and Products, 7th ed. Academic Press; 2007.

[9]Prudnikov AP, Brychkov YA, Marichev OI. Integrals and Series. In: Special functions. Gordon and Breach Science Publishers; 1986.

[10]Marcum JI. A statistical theory of target detection by pulsed radar. IRE Transactions on Information Theory. 1960; 6(2): 59–267. doi: 10.1109/TIT.1960.1057560

[11]Esposito R, Wilson L. Statistical properties of two sine waves in Gaussian noise. IEEE Transactions on Information Theory. 1973; 19(2): 176–183. doi: 10.1109/TIT.1973.1054978

[12]Watson GN. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge University Press; 1966.

Published
2025-03-10
How to Cite
Trifonov, M. (2025). A new infinite series identities involving modified Bessel functions and Hermite polynomials. Journal of AppliedMath, 3(1), 2324. https://doi.org/10.59400/jam2324
Section
Article