Formal theory of irregular linear difference equations

1. Cf. my paper,The Generalized Riemann Problem for linear Differential Equations and the Allied Problems for Linear Difference and q-Difference Equations, Proc. Am. Acad. Arts and Sciences, vol. 49 (1913), pp. 521–568.

2. Cf. N. E. Nörlund,Differenzenrechnung, Berlin, 1924, chap. 10.

3. C. R. AdamsOn the Irregular Cases of Linear Ordinary Difference Equations, Trans. Am. Math. Soc., vol. 30 (1928), pp. 507–541. In this paper references to the work of Barnes, Horn, Batchelder, Perron, and Galbrun may be found.

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4. General Theory of Linear Difference Equations, Trans. Am. Math. Soc., vol. 12 (1911), pp. 243–284.

5. Not even in the casen=2, in which many but not all cases have been treated by Batchelder. Batchelder has not published these results.

6. Generalized here to the extent that we allowp to exceed I.

7. Cf., for instance, N. E. Nörlund, Differenzenrechnung, pp. 312–313.

8. Sec, however, N. E. Nörlund, Differenzerechnung, chap. II, § I, where a specialized case (6″) of this logarithmic type is considered for those linear difference equations of ‘Fuchsian type’, in which the seriesa _i(x)/a_o(x) begin with a term of not higher than degree-i inx.

9. Note that this change of variables leaves the equation of the same general form (4), although the basic integerp may be altered.

10. As a matter of fact the symbolic factorization accomplished only involves powers ofx ^1/p in the coefficients, so that the stated reducibility is effective for the original basic integer. We omit, however, the proof of this fact, which is easily made directly.

11. Note the formal analogy here and later with the method used in the preceding paragraphs.

12. See N. E. Nörlund Differenzenrechnung, Chap. II.

13. All of these terms must appear when all of the roots are equal.

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