[math-fun] Ratio of two Bessel function values
I recently asked, on one of these networks, about the number whose continued fraction is {1,2,3,4,...} and someone was kind enough to provide an explicit answer, which I've stupidly deleted. Can it be repeated? Incidentally, is there a more general result, that a continued fraction whose partial quotients form an arithmetic progression (or a set of APs) can be expressed in terms of Bessel functions? There are a few examples which are rational functions of e. R.
I think a standard reference for this is: D.H. Lehmer. Continued Fractions containing arithmetic progressions. Scripta Math. 29, 17--24, (1973). Doug Bowman
I recently asked, on one of these networks, about the number whose continued fraction is {1,2,3,4,...} and someone was kind enough to provide an explicit answer, which I've stupidly deleted. Can it be repeated?
Incidentally, is there a more general result, that a continued fraction whose partial quotients form an arithmetic progression (or a set of APs) can be expressed in terms of Bessel functions?
There are a few examples which are rational functions of e. R.
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rkg>I recently asked, on one of these networks, about the number whose continued fraction is {1,2,3,4,...} and someone was kind enough to provide an explicit answer, which I've stupidly deleted. Can it be repeated? Incidentally, is there a more general result, that a continued fraction whose partial quotients form an arithmetic progression (or a set of APs) can be expressed in terms of Bessel functions? See Knuth, below. bessel_i (x) 2 4 6 8 1 (d133) [0, -, -, -, -,...] = ------------ x x x x bessel_i (x) 0 (c143) cf(subst(0.5d0,x,rhs(d133))) (d143) [0, 4, 8, 12, 16]
There are a few examples which are rational functions of e. R.
(c140) [0,1,3,5,7,9]/x = tanh(x) 1 3 5 7 9 (d140) [0, -, -, -, -, -] = tanh(x) x x x x x (c141) cf(subst(0.5,x,rhs(%))) (d141) [0, 2, 6, 10, 14] The former is the n=1/2 solution of d 2 2 n f(x) -- (f(x)) = - f (x) - -------- + 1, dx x the latter is from n=1. Knuth, Vol 2, ex 4.5.3.16 mentions Euler getting d 2 m - 1 m -- (f(x)) = c f (x) + b x f(x) + a x dx for a more general cf, but Macsyma seems unable to solve this for non0 m. --rwg
participants (3)
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Douglas Bowman -
R. William Gosper -
Richard Guy