Exact Transient Solution of a State-dependent Birth-death Process

Continued fractions (CFs) play a fundamental role in many investigations due to its applications to diverse fields like number theory, special functions, approximations, moment problems, digital networks, statistics, and signal processing. Their importance has grown further with the advent of fast computing facilities. The problem of converting a continued fraction into a power series is important in applications and has been studied by several authors for more than a century. The coefficients of a CF can be determined from the coefficients of a given power series through quotient of Hankel determinants (Vein and Dale [17]). Jones and Thron [7, page 227] describe a quotient-difference algorithm (qd algorithm) for computing the coefficients of continued fractions corresponding to a given power series. The J-fraction corresponding to a power series has been obtained from an addition formula by means of a decomposition (Goulden and Jackson [5, page 295]). Euler’s connection describes an equivalence between a T-fraction and a power series (Gill [4]). The converse relation to evaluate the power series coefficients from a known continued fraction expansion is pertinent and complicated. Rogers [15] obtained the first few coefficients and Ramanujan (Berndt [2, Entry 17]) has given a recursion. Wall [18, page 203] has presented an infinite Stieltjes matrix equation to obtain the coefficients of the power series expansion of a J-fraction. Zajta and Pandikow [19], Flajolet [3], and Goulden and Jackson [5] have employed a combinatorial approach. CF applications to the study of birth and death processes (BDPs) were initiated by Murphy and O’Donohoe [9] and later this concept has been discussed by several authors