# asymptotic forms of higher coefficients

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##### 2: 2.9 Difference Equations
Formal solutions are … $c_{0}=1$, and higher coefficients are determined by formal substitution. … with $a_{0,j}=1$ and higher coefficients given by (2.9.7) (in the present case the coefficients of $a_{s,j}$ and $a_{s-1,j}$ are zero). … For analogous results for difference equations of the form
##### 3: 28.15 Expansions for Small $q$
###### §28.15 Expansions for Small $q$
28.15.1 $\lambda_{\nu}\left(q\right)=\nu^{2}+\frac{1}{2(\nu^{2}-1)}q^{2}+\frac{5\nu^{2}% +7}{32(\nu^{2}-1)^{3}(\nu^{2}-4)}q^{4}+\frac{9\nu^{4}+58\nu^{2}+29}{64(\nu^{2}% -1)^{5}(\nu^{2}-4)(\nu^{2}-9)}q^{6}+\cdots.$
Higher coefficients can be found by equating powers of $q$ in the following continued-fraction equation, with $a=\lambda_{\nu}\left(q\right)$: …
##### 4: 8.11 Asymptotic Approximations and Expansions
###### §8.11 Asymptotic Approximations and Expansions
For an exponentially-improved asymptotic expansion (§2.11(iii)) see Olver (1991a). … where … See Tricomi (1950b) for these approximations, together with higher terms and extensions to complex variables. …
##### 5: 2.11 Remainder Terms; Stokes Phenomenon
with … For higher-order Stokes phenomena see Olde Daalhuis (2004b) and Howls et al. (2004). … For higher-order differential equations, see Olde Daalhuis (1998a, b). … Furthermore, on proceeding to higher values of $n$ with higher precision, much more accuracy is achievable. … Their extrapolation is based on assumed forms of remainder terms that may not always be appropriate for asymptotic expansions. …
##### 6: 2.4 Contour Integrals
###### §2.4(i) Watson’s Lemma
For examples see Olver (1997b, pp. 315–320). … The final expansion then has the formHigher coefficients $b_{2s}$ in (2.4.15) can be found from (2.3.18) with $\lambda=1$, $\mu=2$, and $s$ replaced by $2s$. … For a symbolic method for evaluating the coefficients in the asymptotic expansions see Vidūnas and Temme (2002). …
##### 7: Bibliography R
• W. H. Reid (1974a) Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. I. Plane Couette flow. Studies in Appl. Math. 53, pp. 91–110.
• W. H. Reid (1974b) Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. II. The general theory. Studies in Appl. Math. 53, pp. 217–224.
• È. Ya. Riekstynš (1991) Asymptotics and Bounds of the Roots of Equations (Russian). Zinatne, Riga.
• W. Rudin (1973) Functional Analysis. McGraw-Hill Book Co., New York.
• J. Rushchitsky and S. Rushchitska (2000) On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.
• ##### 8: 9.7 Asymptotic Expansions
###### §9.7 Asymptotic Expansions
Also $u_{0}=v_{0}=1$ and for $k=1,2,\ldots$, … Corresponding bounds for the errors in (9.7.7) to (9.7.14) may be obtained by use of these results and those of §9.2(v) and their differentiated forms. … For higher re-expansions of the remainder terms see Olde Daalhuis (1995, 1996), and Olde Daalhuis and Olver (1995a).
##### 9: 3.6 Linear Difference Equations
Many special functions satisfy second-order recurrence relations, or difference equations, of the formThe values of $w_{N}$ and $w_{N+1}$ needed to begin the backward recursion may be available, for example, from asymptotic expansions (§2.9). … The process is then repeated with a higher value of $N$, and the normalized solutions compared. … The normalizing factor $\Lambda$ can be the true value of $w_{0}$ divided by its trial value, or $\Lambda$ can be chosen to satisfy a known property of the wanted solution of the formFor further information, including a more general form of normalizing condition, other examples, convergence proofs, and error analyses, see Olver (1967a), Olver and Sookne (1972), and Wimp (1984, Chapter 6). …
##### 10: Bibliography C
• B. C. Carlson (1972b) Intégrandes à deux formes quadratiques. C. R. Acad. Sci. Paris Sér. A–B 274 (15 May, 1972, Sér. A), pp. 1458–1461 (French).
• M. A. Chaudhry, N. M. Temme, and E. J. M. Veling (1996) Asymptotics and closed form of a generalized incomplete gamma function. J. Comput. Appl. Math. 67 (2), pp. 371–379.
• L. Chen, M. E. H. Ismail, and P. Simeonov (1999) Asymptotics of Racah coefficients and polynomials. J. Phys. A 32 (3), pp. 537–553.
• G. Chrystal (1959a) Algebra: An Elementary Textbook for the Higher Classes of Secondary Schools and for Colleges. 6th edition, Vol. 1, Chelsea Publishing Co., New York.
• G. Chrystal (1959b) Algebra: An Elementary Textbook for the Higher Classes of Secondary Schools and for Colleges. 6th edition, Vol. 2, Chelsea Publishing Co., New York.