The Bessel difference equation

Bohner, Martin; Cuchta, Tom

doi:10.1090/proc/13416

The Bessel difference equation
HTML articles powered by AMS MathViewer

by Martin Bohner and Tom Cuchta

Proc. Amer. Math. Soc. 145 (2017), 1567-1580

DOI: https://doi.org/10.1090/proc/13416

Published electronically: December 30, 2016

PDF | Request permission

Abstract:

We define a new difference equation analogue of the Bessel differential equation and investigate the properties of its solution, which we express using a ${}_2F_1$ hypergeometric function. We find analogous formulas for Bessel function recurrence relations, a summation transformation which is identical to the Laplace transform of classical Bessel functions, and oscillation.

References

Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964. For sale by the Superintendent of Documents. MR 0167642
George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
W. W. Bell, Special functions for scientists and engineers, Dover Publications, Inc., Mineola, NY, 2004. Reprint of the 1968 original. MR 2081241
Martin Bohner, Gusein Sh. Guseinov, and Başak Karpuz, Properties of the Laplace transform on time scales with arbitrary graininess, Integral Transforms Spec. Funct. 22 (2011), no. 11, 785–800. MR 2842561, DOI 10.1080/10652469.2010.548335
Martin Bohner and Allan Peterson, Dynamic equations on time scales, Birkhäuser Boston, Inc., Boston, MA, 2001. An introduction with applications. MR 1843232, DOI 10.1007/978-1-4612-0201-1
R. H. Boyer, Discrete Bessel functions, J. Math. Anal. Appl. 2 (1961), 509–524. MR 145218, DOI 10.1016/0022-247X(61)90026-9
Saber Elaydi, An introduction to difference equations, 3rd ed., Undergraduate Texts in Mathematics, Springer, New York, 2005. MR 2128146
Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vol. I, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981. Based on notes left by Harry Bateman; With a preface by Mina Rees; With a foreword by E. C. Watson; Reprint of the 1953 original. MR 698779
Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete mathematics, 2nd ed., Addison-Wesley Publishing Company, Reading, MA, 1994. A foundation for computer science. MR 1397498
Mourad E. H. Ismail, David R. Masson, and Sergei K. Suslov, The $q$-Bessel function on a $q$-quadratic grid, Algebraic methods and $q$-special functions (Montréal, QC, 1996) CRM Proc. Lecture Notes, vol. 22, Amer. Math. Soc., Providence, RI, 1999, pp. 183–200. MR 1726835, DOI 10.1090/crmp/022/10
N. N. Lebedev, Special functions and their applications, Revised English edition, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. Translated and edited by Richard A. Silverman. MR 0174795, DOI 10.1063/1.3047047
H. Levy and F. Lessman, Finite difference equations, Dover Publications, Inc., New York, 1992. Reprint of the 1961 edition. MR 1217083
Mansour Mahmoud, Generalized $q$-Bessel function and its properties, Adv. Difference Equ. , posted on (2013), 2013:121, 11. MR 3057261, DOI 10.1186/1687-1847-2013-121
Carl E. Pearson, Integral representation for solutions of Bessel’s difference equation, J. Math. and Phys. 39 (1960), 287–292. MR 123028, DOI 10.1002/sapm1960391287
Earl D. Rainville, Special functions, 1st ed., Chelsea Publishing Co., Bronx, N.Y., 1971. MR 0393590
Eric John Watson, Laplace transforms and applications, VNR New Mathematics Library, vol. 10, Van Nostrand Reinhold Co., New York-London, 1981. MR 622023
G. N. Watson, A treatise on the theory of Bessel functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR 1349110