expansions in series of incomplete gamma functions

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11: 3.10 Continued Fractions

… ►Every convergent, asymptotic, or formal series … ►For several special functions the $S$-fractions are known explicitly, but in any case the coefficients $a_n$ can always be calculated from the power-series coefficients by means of the quotient-difference algorithm; see Table 3.10.1. … ►We say that it is associated with the formal power series $f(z)$ in (3.10.7) if the expansion of its $n$th convergent $C_n$ in ascending powers of $z$, agrees with (3.10.7) up to and including the term in $z^{2n-1}$, $n=1,2,3,\mathrm{\dots }$. … ►For special functions see §5.10 (gamma function), §7.9 (error function), §8.9 (incomplete gamma functions), §8.17(v) (incomplete beta function), §8.19(vii) (generalized exponential integral), §§10.10 and 10.33 (quotients of Bessel functions), §13.6 (quotients of confluent hypergeometric functions), §13.19 (quotients of Whittaker functions), and §15.7 (quotients of hypergeometric functions). … ►In Gautschi (1979c) the forward series algorithm is used for the evaluation of a continued fraction of an incomplete gamma function (see §8.9). …

12: Bibliography O

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Asymptotic expansions for $q$-gamma, $q$-exponential, and $q$-Bessel functions. J. Math. Anal. Appl. 186 (3), pp. 896–913.

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ISSN 0022-247X, Document, MathReview (P. K. Banerji), ZentralBlatt

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§5.18(ii).

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On the resurgence properties of the uniform asymptotic expansion of the incomplete gamma function. Methods Appl. Anal. 5 (4), pp. 425–438.

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ISSN 1073-2772, Pdf, MathReview (Éric Delabaere), ZentralBlatt

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§8.12.

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On the asymptotics for late coefficients in uniform asymptotic expansions of integrals with coalescing saddles. Methods Appl. Anal. 7 (4), pp. 727–745.

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ISSN 1073-2772, Pdf, MathReview (Raimundas Vidunas), ZentralBlatt

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§36.12(iii).

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Asymptotic expansions of the coefficients in asymptotic series solutions of linear differential equations. Methods Appl. Anal. 1 (1), pp. 1–13.

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ISSN 1073-2772, Pdf, MathReview (Archibald D. MacGillivray), ZentralBlatt

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§2.7(ii), §2.7(ii).

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On an asymptotic expansion of a ratio of gamma functions. Proc. Roy. Irish Acad. Sect. A 95 (1), pp. 5–9.

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ISSN 0035-8975, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§5.11(iii).

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13: 8.25 Methods of Computation

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§8.25(i) Series Expansions

►Although the series expansions in §§8.7, 8.19(iv), and 8.21(vi) converge for all finite values of $z$, they are cumbersome to use when $|z|$ is large owing to slowness of convergence and cancellation. … ►DiDonato and Morris (1986) describes an algorithm for computing $P(a,x)$ and $Q(a,x)$ for $a\ge 0$, $x\ge 0$, and $a+x\ne 0$ from the uniform expansions in §8.12. … ►The computation of $\gamma (a,z)$ and $\mathrm{\Gamma }(a,z)$ by means of continued fractions is described in Jones and Thron (1985) and Gautschi (1979b, §§4.3, 5). … ►Expansions involving incomplete gamma functions often require the generation of sequences $P(a+n,x)$, $Q(a+n,x)$, or ${\gamma }^{\ast }(a+n,x)$ for fixed $a$ and $n=0,1,2,\mathrm{\dots }$. …

14: 19.5 Maclaurin and Related Expansions

§19.5 Maclaurin and Related Expansions

… ► … ►Series expansions of $F(\varphi ,k)$ and $E(\varphi ,k)$ are surveyed and improved in Van de Vel (1969), and the case of $F(\varphi ,k)$ is summarized in Gautschi (1975, §1.3.2). For series expansions of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ when see Erdélyi et al. (1953b, §13.6(9)). …

15: Bibliography J

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Further results on the computation of incomplete gamma functions. In Analytic Theory of Continued Fractions, II (Pitlochry/Aviemore, 1985), W. J. Thron (Ed.), Lecture Notes in Math. 1199, pp. 67–89.

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MathReview (Marietta J. Tretter), ZentralBlatt

Referenced by:

§8.25(iv).

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Bounds on Dawson’s integral occurring in the analysis of a line distribution network for electric vehicles. Eurandom Preprint Series Technical Report 14, Eurandom, Eindhoven, The Netherlands.

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ISSN 1389-2355

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FullText

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§7.8, §7.8.

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Stirling Numbers, Lambert W and the Gamma Function. In Mathematical Aspects of Computer and Information Sciences, J. Blömer, I. S. Kotsireas, T. Kutsia, and D. E. Simos (Eds.), Cham, pp. 275–279.

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ISBN 978-3-319-72453-9, ZentralBlatt

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§4.13.

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Differential equations and mathematical biology. Chapman & Hall/CRC Mathematical Biology and Medicine Series, Chapman & Hall/CRC, Boca Raton, FL.

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Notes:

A second edition was published in 2010.

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ISBN 1-58488-296-4, MathReview (Jian Hong Wu), ZentralBlatt

Referenced by:

D. S. Jones, M. J. Plank, and B. D. Sleeman (2010).

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On the computation of incomplete gamma functions in the complex domain. J. Comput. Appl. Math. 12/13, pp. 401–417.

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ISSN 0377-0427, Document, MathReview (M. Marsden), ZentralBlatt

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§8.25(iv), §8.9.

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16: Bibliography N

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Uniform asymptotic expansion for the incomplete beta function. SIGMA Symmetry Integrability Geom. Methods Appl. 12, pp. 101, 5 pages.

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ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§8.18(ii), §8.18(ii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v).

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Error bounds and exponential improvement for Hermite’s asymptotic expansion for the gamma function. Appl. Anal. Discrete Math. 7 (1), pp. 161–179.

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ISSN 1452-8630, Document, FullText, MathReview (Junesang Choi), ZentralBlatt

Referenced by:

§5.11(ii), §5.11(ii).

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The resurgence properties of the incomplete gamma function II. Stud. Appl. Math. 135 (1), pp. 86–116.

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ISSN 0022-2526, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§8.11(iii).

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The resurgence properties of the incomplete gamma function, I. Anal. Appl. (Singap.) 14 (5), pp. 631–677.

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ISSN 0219-5305, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§8.11(iii), §8.11(v), §8.11(v), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v).

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Inequalities and bounds for the incomplete gamma function. Results Math. 63 (3-4), pp. 1209–1214.

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ISSN 1422-6383, Document, FullText, MathReview (Pierpaolo Natalini), ZentralBlatt

Referenced by:

§8.10, §8.10.

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17: Bibliography S

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FGH, a code for the calculation of Coulomb radial wave functions from series expansions. Comput. Phys. Comm. 146 (2), pp. 250–253.

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Notes:

Single-precision Fortran code.

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Document, Code, ZentralBlatt

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10th item.

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On the expansion of the parabolic cylinder function in a series of the product of two parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 3, pp. 226–230.

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ZentralBlatt

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§12.13(i).

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On integral representation of Weber’s parabolic cylinder function and its expansion into an infinite series. J. Indian Math. Soc. (N. S.) 4, pp. 34–38.

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MathReview (M. C. Gray), ZentralBlatt

Referenced by:

§12.13(ii).

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The expansion of Lamé functions into series of associated Legendre functions of the second kind. Proc. Cambridge Philos. Soc. 62, pp. 441–452.

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Document, MathReview (F. M. Arscott), ZentralBlatt

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§29.17(i).

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Effective calculation of the incomplete gamma function for parameter values $\alpha =(2n+1)/2$, $n=0,\mathrm{\dots },5$ . Angew. Informatik 17, pp. 30–32.

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Notes:

Includes Fortran code.

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ZentralBlatt

Referenced by:

§8.28(ii).

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18: 8.12 Uniform Asymptotic Expansions for Large Parameter

§8.12 Uniform Asymptotic Expansions for Large Parameter

… ►The last reference also includes an exponentially-improved version (§2.11(iii)) of the expansions (8.12.4) and (8.12.7) for $Q(a,z)$. … ►Lastly, a uniform approximation for $\mathrm{\Gamma }(a,ax)$ for large $a$, with error bounds, can be found in Dunster (1996a). ►For other uniform asymptotic approximations of the incomplete gamma functions in terms of the function $\mathrm{erfc}$ see Paris (2002b) and Dunster (1996a). ►

Inverse Function

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19: Bibliography M

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Spherical Harmonics. An Elementary Treatise on Harmonic Functions with Applications. 3rd edition, International Series of Monographs in Pure and Applied Mathematics, Vol. 98, Pergamon Press, Oxford.

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MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

Ch.14.

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On the convergence of the Chebyshev series for functions possessing a singularity in the range of representation. SIAM J. Numer. Anal. 3 (3), pp. 390–409.

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ISSN 1095-7170, Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§3.11(ii).

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A $q$-analog of hypergeometric series well-poised in $\mathrm{𝑆𝑈}(n)$ and invariant $G$-functions. Adv. in Math. 58 (1), pp. 1–60.

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ISSN 0001-8708, Document, MathReview (Robert A. Gustafson), ZentralBlatt

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§17.15.

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Algorithm AS 187. Derivatives of the incomplete gamma integral. Appl. Statist. 31 (3), pp. 330–335.

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Notes:

Single-precision Fortran.

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FullText, Code, Document, ZentralBlatt

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4th item.

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Asymptotic expansions of ellipsoidal wave functions in terms of Hermite functions. Math. Nachr. 32, pp. 49–62.

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Document, MathReview (M. J. O. Strutt), ZentralBlatt

Referenced by:

§29.11, §29.7(ii).

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20: Bibliography P

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Tablitsy nepolnoi gamma-funktsii. Vyčisl. Centr Akad. Nauk SSSR, Moscow (Russian).

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Notes:

In Russian. Translated title: Tables of the Incomplete Gamma Function.

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MathReview (John Todd)

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2nd item.

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An asymptotic formula for the incomplete gamma function. Ž. Vyčisl. Mat. i Mat. Fiz. 5, pp. 118–121 (Russian).

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English translation in U.S.S.R. Computational Math. and Math. Phys. 5(1), pp. 162–166

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Document, MathReview (E. Lukacs), ZentralBlatt

Referenced by:

§8.11(iv).

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Error bounds for the uniform asymptotic expansion of the incomplete gamma function. J. Comput. Appl. Math. 147 (1), pp. 215–231.

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ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§8.12.

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A uniform asymptotic expansion for the incomplete gamma function. J. Comput. Appl. Math. 148 (2), pp. 323–339.

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ISSN 0377-0427, Document, MathReview (Dumitru Acu), ZentralBlatt

Referenced by:

(8.11.6), §8.11(iii), (8.12.18), §8.12, §8.12, §8.12, Erratum (V1.0.16) for Equation (8.12.18).

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The asymptotic expansion of a generalised incomplete gamma function. J. Comput. Appl. Math. 151 (2), pp. 297–306.

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ISSN 0377-0427, Document, MathReview (José Luis López), ZentralBlatt

Referenced by:

§8.16.

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