The Rayleigh functions for , 2, ..., are defined as
where
are the zeros of the Bessel function
of the first kind
(Watson 1966, p. 502; Gupta and Muldoon 1999). They were used by Euler, Rayleigh,
and others to evaluate zeros of Bessel functions.
There is a convolution formula connecting Rayleigh functions of different orders,
Gupta, D. P. and Muldoon, M. E. "Riccati Equations and Convolution Formulas for Functions of Rayleigh Type." 24 Oct 1999.
http://arxiv.org/abs/math.CA/9910128.Ismail,
M. E. H. and Muldoon, M. E. "Bounds for the Small Real and Purely
Imaginary Zeros of Bessel and Related Functions." Meth. Appl. Anal.2,
1-21, 1995.Kishore, N. "The Rayleigh Function." Proc. Amer.
Math. Soc.14, 527-533, 1963.Obi, E. C. "The Complete
Monotonicity of the Rayleigh Function." J. Math. Anal. Appl.77,
465-468, 1980.Watson, G. N. A
Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge
University Press, 1966.
for 
, 2, ..., are defined as
are the zeros of the Bessel function
of the first kind 
(Watson 1966, p. 502; Gupta and Muldoon 1999). They were used by Euler, Rayleigh,
and others to evaluate zeros of Bessel functions.
