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Far East Journal of Applied Mathematics

ds(t)/(t + z), the range from zero to ¥. Other continued fractions are related to this form and appear in the form

Second order continued fractions relate to a definite integral with integrand ds(t)/(t + z₁) (t + z₂). The Fibonacci c.f. is

or

with z = 1. The 2nd order c.f., relates to

Based on these 2nd order c.f.s, results are derived for properties of Fibonacci numbers and the squares of Fibonacci numbers.

Far East Journal of Applied Mathematics
Volume 20, Issue 1, Pages 17 - 31 (July 2005)

SECOND ORDER CONTINUED FRACTIONS AND FIBONACCI NUMBERS
L. R. Shenton (USA) and K. O. Bowman (USA)

Abstract:
Stieltjes transforms refer to integrals with integrand ds(t)/(t + z), the range from zero to ¥. Other continued fractions are related to this form and appear in the form

Second order continued fractions relate to a definite integral with integrand ds(t)/(t + z₁) (t + z₂). The Fibonacci c.f. is

or

with z = 1. The 2nd order c.f., relates to

Based on these 2nd order c.f.s, results are derived for properties of Fibonacci numbers and the squares of Fibonacci numbers.

Keywords and phrases: recurrence relations, S-fractions, Stieltjes integrals, Stieltjes moment problem.

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