On Generating Bessel Functions by Use of the Backward Recurrence Formula

摘要

In a previous paper by the author on the special functions, a class of rational approximations for the generalized hypergeometric function (sub p) F (sub q) was developed. These approximations depend on a number of free parameters. Since (I sub nu)(z) can be expressed in terms of a (sup O)F(sub l) or a (sub l)F(sub l), there is a particular rational approximation corresponding to each of these hypergeometric forms and a choice of the aforementioned free parameters. The idea of using the recursion formula for (I sub nu)(z) in the backward direction to generate values of (I sub nu)(z) is due to J. C. P. Miller. In a conversation Jerry L. Fields conjectured that the specific rational approximations noted above are identical to the certain rational approximations which emerge by use of the backward recurrence scheme noted above. In the present paper, the author verifies this conjecture. In addition, the author develops a new interpretation of the Miller method. The author also studies a third normalization technique which is sometimes used with the backward recursion scheme. (Author)