Luca Guido MOLİNARİ

A note on trigonometric approximations of Bessel functions of the first kind, and trigonometric power sums

I reconsider the approximation of Bessel functions with finite sums of trigonometric functions, in the light of recent evaluations of Neumann-Bessel series with trigonometric coefficients. A proper choice of angle allows for an efficient choice of the trigonometric sum. Based on these series, I also obtain straightforward non-standard evaluations of new parametric sums with powers of cosine and sine functions.

Keywords:

Bessel functions, Neumann series, Trigonometric approximations Trigonometric power sum,

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[1] F. Olver, D. Lozier, R. Boisvert, C. Clark (Eds.), Handbook of Mathematical Functions, Cambridge University Press, 2010.
[2] G. Matviyenko, On the evaluation of Bessel functions,Appl. Comput. Harmon. Anal., 1 (1993), 116–135. https://doi.org/10.1006/acha.1993.1009
[3] C. Schwartz, Numerical calculation of Bessel functions, Int. J. Mod. Phys. C, 23(12) (2012), 1250084. https://doi.org/10.1142/S0129183112500842
[4] J. Bremer, An algorithm for the rapid numerical evaluation of Bessel functions of real orders and arguments, (2017), arXiv https://arxiv.org/abs/1705.07820
[5] E. A. Karatsuba, Calculation of Bessel functions via the summation of series, Numer. Analys. Appl., 12 (2019), 372–387. https://doi.org/10.1134/S1995423919040050
[6] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Dover Publ. New York, 1965.
[7] F. B. Gross, New approximations to J0 and J1 Bessel functions, IEEE transactions on antennas and propagation, 43(8) (1995), 904–907. https://doi.org/10.1109/8.402217
[8] L. Li, F. Li, F. B. Gross, A new polynomial approximation for Jn Bessel functions,Appl. Math. Comput., 183 (2006), 1220–1225. https://doi.org/10.1016/j.amc.2006.06.047
[9] V. Barsan, S. Cojocaru, Physical applications of a simple approximation of Bessel functions of integer order, Eur. J. Phys., 28(5) (2007), 983. https://doi.org/10.1088/0143-0807/28/5/021
[10] H. E. Fettis, Numerical calculation of certain definite integrals by Poisson’s summation formula, Math. Tables Aids. Comp., 9 (1955), 85–92. https://doi.org/10.2307/2002063
[11] W. Rehwald, Representation of Bessel functions through trigonometric functions or power series and asymtotic formulas with complex arguments (in German), Frequenz, 21 (1967), 321–328. https://doi.org/10.1515/FREQ.1967.21.10.320
[12] R. A. Waldron, Formulas for computation of approximate values of some Bessel functions, Proc. of the IEEE 69(12) (1981) 1586–1588. https://doi.org/10.1109/PROC.1981.12211
[13] N. M. Blachman, S. H. Mousavinezhad, Trigonometric Approximations for Bessel Functions, IEEE Transactions on Aerospace and Electronic Systems AES-22(1) (1986). https://doi.org/10.1109/TAES.1986.310686
[14] M. T. Abuelma’atti, Trigonometric approximations for some Bessel functions, Active and Passive Elec. Comp., 22 (1999), 75–85. https://downloads.hindawi.com/journals/apec/1999/015810.pdf
[15] A. Al-Jarrah, K. M. Dempsey, M. L. Glasser, Generalized series of Bessel functions, J. Comp. Appl. Math., 143 (2002), 1–8. https://doi.org/10.1016/S0377-0427(01)00505-2
[16] A. H. Stroud, J. P. Kohli, Computation of Jn by numerical integration, Commun. of the ACM, 12(4) (1969), 236–238. https://doi.org/10.1145/362912.362942
[17] P. Baratella, M. Garetto, G. Vinardi, Approximation of the Bessel function Jn (x) by numerical integration, J. Comp. Appl. Math., 7(2) (1981), 87–91. https://www.sciencedirect.com/science/article/pii/0771050X81900401
[18] L. N. Trefethen, J. A. C. Weideman, The exponentially convergent trapezoidal rule, SIAM Review, 56(3) (2014), 385–458. https://doi.org/10.1137/130932132
[19] L. G. Molinari, Some Neumann-Bessel series and the Laplacian on polygons, J. Math. Phys., 62(5) (2021), 053502, 7 pages. https://doi.org/10.1063/5.0037872
[20] H. Jelitto, Generalized trigonometric power sums covering the full circle, J. Appl. Math. Phys., 10 (2022), 405–414. https://doi.org/10.4236/jamp.2022.102031
[21] K. S. Williams and Z. Nan-Yue, Evaluation of two trigonometric sums, Math. Slovaca, 44(5) (1994) 575–583. http://dml.cz/dmlcz/136630.
[22] B. C. Berndt and B. P. Yeap, Explicit evaluations and reciprocity theorems for finite trigonometric sums, Adv. Appl. Math. 29 (2002) 358–385. https://www.sciencedirect.com/science/article/pii/S0196885802000209
[23] A. Gervois and M. L. Mehta, Some trigonometric identities encountered by McCoy and Orrick, J. Math. Phys. 36 (1995) 5098–5109. https://doi.org/10.1063/1.531345
[24] B. C. Berndt, S. Kim and A. Zaharescu, Exact evaluations and reciprocity theorems for finite trigonometric sums, arXiv:2210.00180[math.NT] (2022). https://doi.org/10.48550/arXiv.2210.00180
[25] A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series, Vol. 1: Elementary functions, Gordon and Breach, 1986.
[26] C. M. da Fonseca, M. L. Glasser, V. Kowalenko, Basic trigonometric power sums with applications, The Ramanujan Journal, 42(2) (2017), 401–428. https://link.springer.com/article/10.1007/s11139-016-9778-0
[27] M. Merca, A note on cosine power sums, Journal of Integer Sequences, 15 (2012), Article ID 12.5.3. https://cs.uwaterloo.ca/journals/JIS/VOL15/Merca1/merca6.pdf