RECURRING SERIES. By a recurring series is meant one of the form ad infinitunt, in which the coefficients a„ ite. can each be expressed by means of certain preceding coefficients and constants in one uniform manner ; and it is usual to consider only such series as will admit of a linear relation (or one in which only first powers of coefficienth enter): thus the series 4- x + 4ritl + 13.0 +4a + 142.0 + • follows the linear law a,=.• + (4 = 3.1 + 1, 13 = 3.4 + 1, 43 = 3.13 + 4. &c.), and is what is commonly called a recurring series, though the following I 4- + + + 866.0 + ttc., in which a. = 4-1 + 4-elis equally recurring, according to the de finition. Tho recurrence alluded to is not that of terms, but of method of determining terms; and it would be desirable that the series which arc usually called recurring should be linearly recurring series, while all in which there is really recurrence (of law) should be called recurring.
Every linearly recurring series is the development of an algebraic function with a rational and integral numerator and denominator, and every such function can be developed into a linearly recurring series.
Thus, taking the first series mentioned, in which a„ = + we have a, = 3a, + a, a, = 3a, + a, .0 a, = 3a, x' + a, x', kc.
Let s lie the sum of a, + a, x , kc., ad in finilum. Then the pre ceding obviously gives + (a, - 3a,,) x or, 8 = 1 - 3x - IWo have here the value of any series in which this law of recur. rence prevails for all terms after the second ; and it cannot prevail before, since two terms must exist before a third can be expressed. In the case we chose, a, = 1, a = 1, whence the function of which the series was the development is (1 - 2x) (1 - 3x x'). Generally, a linear recurring series having the law of recurrence a. = p, a.-, + + + pe is the development of tho function A„ + A, x + A, + .. • . + Ar--1 1 - 2), x - p, - p. xt where A, = a„, A, = a, - p, A, = a, - p, a, - p„ . . . .
At-I = a1--1 - p, at-2 - Ps a t--3 - • • • • pe—, from which the inverse theorem may easily be derived, namely, that + A, X + A, x' . .