On the Irregular Cases of the Linear Ordinary Difference Equation

r is a constant, in this and subsequent types.P(*_1) stands for a power series in x~l, namely l+s'x~1+s"x~t+ ••• ; the constant term may be taken as 1, since we are dealing with the homogeneous difference equation.Similarly, P(x_1'm) is used to represent a power series in x~llm, the first term being 1.The 7's, wherever they occur, are constants.In each type of series the constants are calculated 0>y formal substitution of the series in (1)) in the order in which they appear in the series as written; for example, in (6) the constants after pi are calculated in the order 7('"-1>, ■yO»-»,.. ., 7 ', r, s', s", • • • .The constants thus calculated are uniquely determined for each series of the type.* Cf. §5, in which this fact is pointed out more clearly in a particular case.* If n S4, the factor sin (ir/2-2r/n) occurring here is replaced by 1, since each term in the sum, for any p, has an argument between 0 and r/2 (inclusive of the latter).