ABSTRACT
This chapter deals with Stieltjes’ formula for any continued fraction coefficients. It is possible to derive an alternative form of the general coefficient an of continued fractions without using the method of the series inversion. This can be achieved simply if the first few explicit results for an (1 ≤ n ≤ 6) are rewritten in a way which would permit an inductive derivation of a general expression for any an (n ≥ 1). The recursion obtained is the analytical formula of Stieltjes derived in 1858 for any continued fraction coefficient am, in terms of the accompanying sequel which depends only upon the signal points {cn}.
