linguistic:voice_change
Examples

The next table provides examples of closed-form formulas for the component sequences found computationally (and subsequently proved correct in the cited references)in several special cases of the prescribed sequences, j, generated by the general expansions of the J -fractions defined in the first subsection. Here we define 0 < | a |, | b |, | q | < 1 and the parameters R, α ∈ Z + {\displaystyle R,\alpha \in \mathbb {Z} ^{+}} and x to be indeterminates with respect to these expansions, where the prescribed sequences enumerated by the expansions of these J -fractions are defined in terms of the q -Pochhammer symbol, Pochhammer symbol, and the binomial coefficients.

The radii of convergence of these series corresponding to the definition of the Jacobi-type J -fractions given above are in general different from that of the corresponding power series expansions defining the ordinary generating functions of these sequences.