Simple Continued Fractions, Hill's Infinite Determinants, and Bessel Functions' Ratio

Convergence, Simple Continued Fractions, Hill's Infinite Determinants and Bessel Functions' Ratio

Update in June 2005 :

A new function : Simple_Cont_Frctn_b_1.m has been added.
This Programme computes a Simple Continued Fraction by making calls to Continuant_Poly_Kn.m ; in a way, this Programme is a Test Programme for Continuant_Poly_Kn.m

BslRat_ContFr_HillInfDet.zip is a suite of four *.m files which deal with Convergence, George William Hill's Infinite Determinants, Continued Fractions and Bessel Functions' Ratio.

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1) Hill_c_Theta_Inf_Det.m "analyses" the Hill Matrix (wrt Infinite Determinants) as given in :
Prob 21, Chapter II, P40 and Chapter 19.42, P415 in :
"A Course in Modern Analysis" by Whittaker and Watson.

2) Continuant_Poly_Kn.m calculates the Continuant Polynomial Kn (x1, x2, ... xn).
It is based on Eqn 5 / 4.5.3, P257 and Prob 3, 4.5.3, P373 in Vol 2 of :
"The Art of Computer Programming" by Donald Knuth.

3) Simple_Cont_Frctn_b_1.m is described above.

4) Thn_Ellip_BslRat_ContFr_InfDet.m was primarily written to understand the relationship between Bessel Functions, Continued Fractions, Infinite Determinant etc by verifying certain results of the foll article :
The lateral skin effect in a flat conductor by V. BELEVITCH, Philips tech. Rev. 32, 221-231, 1971, No. 6/7,18 at :
http://www.cvni.net/abc/rip2/r9/index.html
This function makes a call to Continuant_Poly_Kn.m

5) This ReadMe file

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PS : Limit of Convergence and Infinite Determinant :
It is one thing to prove convergence or divergence of a series, but finding the convergence value itself is probably tougher. Are there any clues / rules to actually get that final convergent value itself ? ? as for eg, in Prob 11, Chapter II, P39 in W&W, or the limit of convergence of an Infinite Determinant ? W&W discusses about the existence of D, the limit of convergence of an Infinite Determinant in P36-37, but how do we find that exact convergent value D itself, given an arbitrary Infinite Determinant with say, just the prior knowledge that yes, it converges ?
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